Integrand size = 19, antiderivative size = 393 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b \left (1+c^2 x^2\right )}{3 c \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}-\frac {4 b \sqrt {-c^2} \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 c e \left (c^2 d^2+e^2\right ) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}+\frac {8 b \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2/3*d*(a+b*arccsch(c*x))/e^2/(e*x+d)^(3/2)-2*(a+b*arccsch(c*x))/e^2/(e*x+d )^(1/2)+4/3*b*(c^2*x^2+1)/c/(c^2*d^2+e^2)/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1 /2)+8/3*b*EllipticPi(1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2),2,2^(1/2)*(e/(d* (-c^2)^(1/2)+e))^(1/2))*(c^2*x^2+1)^(1/2)*((e*x+d)*(-c^2)^(1/2)/(d*(-c^2)^ (1/2)+e))^(1/2)/c/e^2/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)-4/3*b*EllipticE( 1/2*(1-x*(-c^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-c^2)^(1/2)/(c^2*d-e*(-c^2)^(1 /2)))^(1/2))*(-c^2)^(1/2)*(e*x+d)^(1/2)*(c^2*x^2+1)^(1/2)/c/e/(c^2*d^2+e^2 )/x/(1+1/c^2/x^2)^(1/2)/(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2)
Result contains complex when optimal does not.
Time = 13.93 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.99 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {2}{3} \left (\frac {2 b c \sqrt {1+\frac {1}{c^2 x^2}} x}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}-\frac {a (2 d+3 e x)}{e^2 (d+e x)^{3/2}}-\frac {b (2 d+3 e x) \text {csch}^{-1}(c x)}{e^2 (d+e x)^{3/2}}+\frac {2 i b \sqrt {-\frac {c}{c d-i e}} \sqrt {-\frac {e (-i+c x)}{c d+i e}} \sqrt {-\frac {e (i+c x)}{c d-i e}} \left (c d E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )-c d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )+2 (c d-i e) \operatorname {EllipticPi}\left (1-\frac {i e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )\right )}{c^2 d e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\right ) \]
(2*((2*b*c*Sqrt[1 + 1/(c^2*x^2)]*x)/((c^2*d^2 + e^2)*Sqrt[d + e*x]) - (a*( 2*d + 3*e*x))/(e^2*(d + e*x)^(3/2)) - (b*(2*d + 3*e*x)*ArcCsch[c*x])/(e^2* (d + e*x)^(3/2)) + ((2*I)*b*Sqrt[-(c/(c*d - I*e))]*Sqrt[-((e*(-I + c*x))/( c*d + I*e))]*Sqrt[-((e*(I + c*x))/(c*d - I*e))]*(c*d*EllipticE[I*ArcSinh[S qrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] - c*d*Ellip ticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I *e)] + 2*(c*d - I*e)*EllipticPi[1 - (I*e)/(c*d), I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c^2*d*e^2*Sqrt[1 + 1/( c^2*x^2)]*x)))/3
Leaf count is larger than twice the leaf count of optimal. \(2084\) vs. \(2(393)=786\).
Time = 3.83 (sec) , antiderivative size = 2084, normalized size of antiderivative = 5.30, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.105, Rules used = {6864, 27, 7272, 2351, 27, 498, 27, 507, 635, 25, 27, 498, 27, 507, 631, 1459, 1416, 1509, 1540, 1416, 2222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 6864 |
\(\displaystyle \frac {b \int -\frac {2 (2 d+3 e x)}{3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{c}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \int \frac {2 d+3 e x}{\sqrt {1+\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{3 c e^2}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 7272 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \int \frac {2 d+3 e x}{x (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 2351 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\int \frac {3 e}{(d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \int \frac {1}{(d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 498 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {c^2 x^2+1}}dx}{c^2 d^2+e^2}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {c^2 x^2+1}}dx}{c^2 d^2+e^2}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 507 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 635 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{d}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {c^2 x^2+1}}dx}{d}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 498 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{d}-\frac {e \left (-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {c^2 x^2+1}}dx}{c^2 d^2+e^2}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )+3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{d}-\frac {e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {c^2 x^2+1}}dx}{c^2 d^2+e^2}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )+3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 507 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{d}-\frac {e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 631 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {e \left (\frac {2 c^2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{d}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 1459 |
\(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \left (\frac {\sqrt {c^2 d^2+e^2} \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\sqrt {c^2 d^2+e^2} \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{d}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}-\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{d}-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}\right )\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
\(\Big \downarrow \) 1509 |
\(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{d}\right )\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
\(\Big \downarrow \) 1540 |
\(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \left (\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {c \left (c d-\sqrt {c^2 d^2+e^2}\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e^2}\right )}{d}\right )\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
\(\Big \downarrow \) 1416 |
\(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \left (\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \int -\frac {\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )}{d}\right )\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
\(\Big \downarrow \) 2222 |
\(\displaystyle -\frac {2 \left (a+b \text {csch}^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 b \sqrt {c^2 x^2+1} \left (3 e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )+2 d \left (-\frac {e \left (\frac {2 c^2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )}{e \left (c^2 d^2+e^2\right )}-\frac {2 e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) \sqrt {d+e x}}\right )}{d}-\frac {2 \left (\left (\frac {c^2 d^2}{e^2}+1\right ) \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {\left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right ) \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d} \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )}{2 \sqrt {d}}+\frac {\sqrt [4]{c^2 d^2+e^2} \left (1-\frac {c d}{\sqrt {c^2 d^2+e^2}}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c d+\sqrt {c^2 d^2+e^2}\right )^2}{4 c d \sqrt {c^2 d^2+e^2}},2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{4 \sqrt {c} d \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )-\frac {\sqrt {c} \sqrt [4]{c^2 d^2+e^2} \left (c d-\sqrt {c^2 d^2+e^2}\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 e^2 \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}\right )}{d}\right )\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(2*d*(a + b*ArcCsch[c*x]))/(3*e^2*(d + e*x)^(3/2)) - (2*(a + b*ArcCsch[c*x ]))/(e^2*Sqrt[d + e*x]) - (2*b*Sqrt[1 + c^2*x^2]*(3*e*((-2*e*Sqrt[1 + c^2* x^2])/((c^2*d^2 + e^2)*Sqrt[d + e*x]) + (2*c^2*(-((Sqrt[c^2*d^2 + e^2]*(-( (Sqrt[d + e*x]*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])/((1 + (c^2*d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2 ]))) + ((c^2*d^2 + e^2)^(1/4)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])*Sqrt [(1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2)/((1 + (c^2*d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2])^2)]*EllipticE[2* ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/Sqrt[c^2 *d^2 + e^2])/2])/(Sqrt[c]*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])))/c) + ((c^2*d^2 + e^2)^(3/4)*(1 + (c*(d + e*x) )/Sqrt[c^2*d^2 + e^2])*Sqrt[(1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2)/((1 + (c^2*d^2)/e^2)*(1 + (c*(d + e*x))/Sqrt[c^2*d ^2 + e^2])^2)]*EllipticF[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^ (1/4)], (1 + (c*d)/Sqrt[c^2*d^2 + e^2])/2])/(2*c^(3/2)*Sqrt[1 + (c^2*d^2)/ e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])))/(e*(c^2*d^2 + e^ 2))) + 2*d*(-((e*((-2*e*Sqrt[1 + c^2*x^2])/((c^2*d^2 + e^2)*Sqrt[d + e*x]) + (2*c^2*(-((Sqrt[c^2*d^2 + e^2]*(-((Sqrt[d + e*x]*Sqrt[1 + (c^2*d^2)/e^2 - (2*c^2*d*(d + e*x))/e^2 + (c^2*(d + e*x)^2)/e^2])/((1 + (c^2*d^2)/e^2)* (1 + (c*(d + e*x))/Sqrt[c^2*d^2 + e^2]))) + ((c^2*d^2 + e^2)^(1/4)*(1 +...
3.1.71.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[2/ d Subst[Int[x^2/Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)] , x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > Simp[-2 Subst[Int[1/((c - x^2)*Sqrt[(b*c^2 + a*d^2)/d^2 - 2*b*c*(x^2/d^ 2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && PosQ[b/a]
Int[((c_) + (d_.)*(x_))^(n_)/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[c^(n + 1/2) Int[1/(x*Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] + Int[( (c + d*x)^n/Sqrt[a + b*x^2])*ExpandToSum[(1 - c^(n + 1/2)*(c + d*x)^(-n - 1 /2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[n + 1/2, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c /a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ (2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) ], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[1/q Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[1/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q ^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 /(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2) Int[1 /Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2) I nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] /; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
Int[((Px_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.))/(x_), x_S ymbol] :> Int[PolynomialQuotient[Px, x, x]*(c + d*x)^n*(a + b*x^2)^p, x] + Simp[PolynomialRemainder[Px, x, x] Int[(c + d*x)^n*((a + b*x^2)^p/x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && PolynomialQ[Px, x]
Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHid e[u, x]}, Simp[(a + b*ArcCsch[c*x]) v, x] + Simp[b/c Int[SimplifyIntegr and[v/(x^2*Sqrt[1 + 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x] ] /; FreeQ[{a, b, c}, x]
Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] && ! IntegerQ[p] && ILtQ[n, 0] && !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
Result contains complex when optimal does not.
Time = 9.47 (sec) , antiderivative size = 2106, normalized size of antiderivative = 5.36
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(2106\) |
default | \(\text {Expression too large to display}\) | \(2106\) |
parts | \(\text {Expression too large to display}\) | \(2110\) |
2/e^2*(-a*(1/(e*x+d)^(1/2)-1/3*d/(e*x+d)^(3/2))-b*(1/(e*x+d)^(1/2)*arccsch (c*x)-1/3*arccsch(c*x)*d/(e*x+d)^(3/2)-2/3/c*(I*((c*d+I*e)*c/(c^2*d^2+e^2) )^(1/2)*(e*x+d)^2*c^2*d*e+2*I*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/ (c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e ^2))^(1/2)*EllipticPi((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),1/(c *d+I*e)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2)/((c*d+I*e)*c/ (c^2*d^2+e^2))^(1/2))*c^2*d^2*e*(e*x+d)^(1/2)+(-(I*c*e*(e*x+d)+c^2*d*(e*x+ d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2 +e^2)/(c^2*d^2+e^2))^(1/2)*EllipticF((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e ^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2))*c^3*d^3*(e*x+d) ^(1/2)-(-(I*c*e*(e*x+d)+c^2*d*(e*x+d)-c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*(( I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e^2)/(c^2*d^2+e^2))^(1/2)*EllipticE((e *x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2),(-(2*I*c*d*e-c^2*d^2+e^2)/(c ^2*d^2+e^2))^(1/2))*c^3*d^3*(e*x+d)^(1/2)-2*(-(I*c*e*(e*x+d)+c^2*d*(e*x+d) -c^2*d^2-e^2)/(c^2*d^2+e^2))^(1/2)*((I*c*e*(e*x+d)-c^2*d*(e*x+d)+c^2*d^2+e ^2)/(c^2*d^2+e^2))^(1/2)*EllipticPi((e*x+d)^(1/2)*((c*d+I*e)*c/(c^2*d^2+e^ 2))^(1/2),1/(c*d+I*e)/c*(c^2*d^2+e^2)/d,(-(I*e-c*d)*c/(c^2*d^2+e^2))^(1/2) /((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2))*c^3*d^3*(e*x+d)^(1/2)-((c*d+I*e)*c/(c^ 2*d^2+e^2))^(1/2)*c^3*d^2*(e*x+d)^2+I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^ 2*d^3*e+2*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^3*d^3*(e*x+d)+I*((c*d+I*e...
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
integral((b*x*arccsch(c*x) + a*x)*sqrt(e*x + d)/(e^3*x^3 + 3*d*e^2*x^2 + 3 *d^2*e*x + d^3), x)
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\left (d + e x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
-1/3*b*(2*(3*e*x + 2*d)*log(sqrt(c^2*x^2 + 1) + 1)/((e^3*x + d*e^2)*sqrt(e *x + d)) + 3*integrate(2/3*(3*c^2*e*x^2 + 2*c^2*d*x)/((c^2*e^3*x^3 + c^2*d *e^2*x^2 + e^3*x + d*e^2)*sqrt(c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^3*x^3 + c^2*d*e^2*x^2 + e^3*x + d*e^2)*sqrt(e*x + d)), x) + 3*integrate(-1/3*(10* c^2*d*e*x^2 - 3*(e^2*log(c) - 2*e^2)*c^2*x^3 + (4*c^2*d^2 - 3*e^2*log(c))* x - 3*(c^2*e^2*x^3 + e^2*x)*log(x))/((c^2*e^4*x^4 + 2*c^2*d*e^3*x^3 + 2*d* e^3*x + d^2*e^2 + (c^2*d^2*e^2 + e^4)*x^2)*sqrt(e*x + d)), x)) - 2/3*a*(3/ (sqrt(e*x + d)*e^2) - d/((e*x + d)^(3/2)*e^2))
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \]