Integrand size = 19, antiderivative size = 679 \[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {4 b \sqrt {d+e x} \left (1+c^2 x^2\right )}{15 c^3 \sqrt {1+\frac {1}{c^2 x^2}} x}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}+\frac {8 b c d \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {8 b c d^2 \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {4 b c \left (c^2 d^2+e^2\right ) \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{15 \left (-c^2\right )^{5/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^3 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{15 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
-2/3*d*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^2+2/5*(e*x+d)^(5/2)*(a+b*arccsch (c*x))/e^2+4/15*b*(c^2*x^2+1)*(e*x+d)^(1/2)/c^3/x/(1+1/c^2/x^2)^(1/2)+8/15 *b*d^3*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)+8/15*b*c*d*Elliptic E(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))*(e*x+d)^(1/2)*(c^2*x^2+1)^(1/2)/(-c^2)^(3/2)/e/x/(1+1/c^2/x ^2)^(1/2)/(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2)-8/15*b*c*d^2*Elliptic F(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*x^2+1)^(1/2)*(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1/2)))^(1/2 )/(-c^2)^(3/2)/e/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)+4/15*b*c*(c^2*d^2+e^2 )*EllipticF(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*x^2+1)^(1/2)*(c^2*(e*x+d)/(c^2*d-e*(-c^2)^(1 /2)))^(1/2)/(-c^2)^(5/2)/e/x/(1+1/c^2/x^2)^(1/2)/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 13.33 (sec) , antiderivative size = 418, normalized size of antiderivative = 0.62 \[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\frac {1}{15} \left (\frac {4 b \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{c}+\frac {2 a \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right )}{e^2}+\frac {2 b \sqrt {d+e x} \left (-2 d^2+d e x+3 e^2 x^2\right ) \text {csch}^{-1}(c x)}{e^2}+\frac {4 i b \sqrt {-\frac {e (-i+c x)}{c d+i e}} \sqrt {-\frac {e (i+c x)}{c d-i e}} \left (2 c d (c d+i e) E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right )|\frac {c d-i e}{c d+i e}\right )+\left (c^2 d^2-2 i c d e+e^2\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )-2 c^2 d^2 \operatorname {EllipticPi}\left (1-\frac {i e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d-i e}} \sqrt {d+e x}\right ),\frac {c d-i e}{c d+i e}\right )\right )}{c^3 \sqrt {-\frac {c}{c d-i e}} e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\right ) \]
((4*b*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[d + e*x])/c + (2*a*Sqrt[d + e*x]*(-2*d^ 2 + d*e*x + 3*e^2*x^2))/e^2 + (2*b*Sqrt[d + e*x]*(-2*d^2 + d*e*x + 3*e^2*x ^2)*ArcCsch[c*x])/e^2 + ((4*I)*b*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*Sqrt[ -((e*(I + c*x))/(c*d - I*e))]*(2*c*d*(c*d + I*e)*EllipticE[I*ArcSinh[Sqrt[ -(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)] + (c^2*d^2 - (2 *I)*c*d*e + e^2)*EllipticF[I*ArcSinh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]] , (c*d - I*e)/(c*d + I*e)] - 2*c^2*d^2*EllipticPi[1 - (I*e)/(c*d), I*ArcSi nh[Sqrt[-(c/(c*d - I*e))]*Sqrt[d + e*x]], (c*d - I*e)/(c*d + I*e)]))/(c^3* Sqrt[-(c/(c*d - I*e))]*e^2*Sqrt[1 + 1/(c^2*x^2)]*x))/15
Leaf count is larger than twice the leaf count of optimal. \(2043\) vs. \(2(679)=1358\).
Time = 3.82 (sec) , antiderivative size = 2043, normalized size of antiderivative = 3.01, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {6864, 27, 7272, 2351, 27, 497, 27, 599, 27, 634, 599, 27, 631, 1511, 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.
| \(\displaystyle \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx\) |
| \(\Big \downarrow \) 6864 |
| \(\displaystyle \frac {b \int -\frac {2 (2 d-3 e x) (d+e x)^{3/2}}{15 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {(2 d-3 e x) (d+e x)^{3/2}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{15 c e^2}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \int \frac {(2 d-3 e x) (d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\int -\frac {3 e (d+e x)^{3/2}}{\sqrt {c^2 x^2+1}}dx+2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx-3 e \int \frac {(d+e x)^{3/2}}{\sqrt {c^2 x^2+1}}dx\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 497 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx-3 e \left (\frac {2 \int \frac {3 d^2 c^2+4 d e x c^2-e^2}{2 \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx-3 e \left (\frac {\int \frac {3 d^2 c^2+4 d e x c^2-e^2}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \int \frac {e \left (d^2 c^2-4 d (d+e x) c^2+e^2\right )}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 c^2 e^2}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {(d+e x)^{3/2}}{x \sqrt {c^2 x^2+1}}dx-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \int \frac {d^2 c^2-4 d (d+e x) c^2+e^2}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 c^2 e}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 634 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx-\int \frac {-x e^2-2 d e}{\sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \int \frac {d^2 c^2-4 d (d+e x) c^2+e^2}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 c^2 e}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 599 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (\frac {2 \int \frac {e^2 (2 d+e x)}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{e^2}+d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \int \frac {d^2 c^2-4 d (d+e x) c^2+e^2}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 c^2 e}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \int \frac {2 d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}+d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {c^2 x^2+1}}dx\right )-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \int \frac {d^2 c^2-4 d (d+e x) c^2+e^2}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 c^2 e}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 631 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \int \frac {2 d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-2 d^2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \int \frac {d^2 c^2-4 d (d+e x) c^2+e^2}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{3 c^2 e}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 1511 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \left (\frac {\left (\sqrt {c^2 d^2+e^2}+c d\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )-2 d^2 \int -\frac {1}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )-3 e \left (\frac {2 e \sqrt {c^2 x^2+1} \sqrt {d+e x}}{3 c^2}-\frac {2 \left (4 c d \sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-\sqrt {c^2 d^2+e^2} \left (4 c d-\sqrt {c^2 d^2+e^2}\right ) \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c^2 e}\right )\right )}{15 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{5/2} \left (a+b \text {csch}^{-1}(c x)\right )}{5 e^2}-\frac {2 d (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{5/2}}{5 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \left (\frac {\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 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 )-2 d^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}\right )-3 e \left (\frac {2 e \sqrt {d+e x} \sqrt {c^2 x^2+1}}{3 c^2}-\frac {2 \left (4 c d \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}-\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (4 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 \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}}\right )}{3 c^2 e}\right )\right )}{15 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 ) (d+e x)^{5/2}}{5 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \left (\frac {\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 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 )-2 d^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}\right )-3 e \left (\frac {2 e \sqrt {d+e x} \sqrt {c^2 x^2+1}}{3 c^2}-\frac {2 \left (4 c d \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 )-\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (4 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 \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}}\right )}{3 c^2 e}\right )\right )}{15 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 ) (d+e x)^{5/2}}{5 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \left (\frac {\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 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 )-2 d^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 )\right )-3 e \left (\frac {2 e \sqrt {d+e x} \sqrt {c^2 x^2+1}}{3 c^2}-\frac {2 \left (4 c d \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 )-\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (4 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 \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}}\right )}{3 c^2 e}\right )\right )}{15 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 ) (d+e x)^{5/2}}{5 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \left (\frac {\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 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 )-2 d^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 )\right )-3 e \left (\frac {2 e \sqrt {d+e x} \sqrt {c^2 x^2+1}}{3 c^2}-\frac {2 \left (4 c d \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 )-\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (4 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 \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}}\right )}{3 c^2 e}\right )\right )}{15 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 ) (d+e x)^{5/2}}{5 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \left (2 \left (\frac {\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 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 )-2 d^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 )\right )-3 e \left (\frac {2 e \sqrt {d+e x} \sqrt {c^2 x^2+1}}{3 c^2}-\frac {2 \left (4 c d \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 )-\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (4 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 \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}}\right )}{3 c^2 e}\right )\right )}{15 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(-2*d*(d + e*x)^(3/2)*(a + b*ArcCsch[c*x]))/(3*e^2) + (2*(d + e*x)^(5/2)*( a + b*ArcCsch[c*x]))/(5*e^2) - (2*b*Sqrt[1 + c^2*x^2]*(-3*e*((2*e*Sqrt[d + e*x]*Sqrt[1 + c^2*x^2])/(3*c^2) - (2*(4*c*d*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^2*d^2 + e^2)^(3/4)*(4*c*d - Sqrt[c^2*d^2 + e^2 ])*(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*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] )))/(3*c^2*e)) + 2*d*(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 ...
3.1.52.3.1 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)/(b*(n + 2*p + 1))), x] + Simp[1/(b *(n + 2*p + 1)) Int[(c + d*x)^(n - 2)*(a + b*x^2)^p*Simp[b*c^2*(n + 2*p + 1) - a*d^2*(n - 1) + 2*b*c*d*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, n , p}, x] && If[RationalQ[n], GtQ[n, 1], SumSimplerQ[n, -2]] && NeQ[n + 2*p + 1, 0] && IntQuadraticQ[a, 0, b, c, d, n, p, x]
Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] ), x_Symbol] :> Simp[-2/d^2 Subst[Int[(B*c - A*d - B*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] /; Fr eeQ[{a, b, c, d, A, B}, 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[( 1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]))*ExpandToSum[(c^(n + 1/2) - (c + d*x)^(n + 1/2))/x, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[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[((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[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q Int[1/Sqrt[a + b*x^2 + c*x^ 4], x], x] - Simp[e/q Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos Q[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 = 9.06 (sec) , antiderivative size = 1963, normalized size of antiderivative = 2.89
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1963\) |
| default | \(\text {Expression too large to display}\) | \(1963\) |
| parts | \(\text {Expression too large to display}\) | \(1967\) |
2/e^2*(-a*(-1/5*(e*x+d)^(5/2)+1/3*(e*x+d)^(3/2)*d)-b*(-1/5*arccsch(c*x)*(e *x+d)^(5/2)+1/3*arccsch(c*x)*(e*x+d)^(3/2)*d+2/15/c^3*(-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+((c*d+I*e)*c /(c^2*d^2+e^2))^(1/2)*c^3*d*(e*x+d)^(5/2)-I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1 /2)*e^3*(e*x+d)^(1/2)-I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^2*e*(e*x+d)^(5 /2)+2*I*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)*c^2*d*e*(e*x+d)^(3/2)-(-(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^3-2*(-(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)*Ellipt icE((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+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^3-2*((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)...
Timed out. \[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\text {Timed out} \]
\[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x}\, dx \]
\[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x \,d x } \]
-1/225*(3375*c^2*e^2*integrate(1/15*sqrt(e*x + d)*x^3*log(x)/(c^2*e^2*x^2 + e^2), x) - 60*c^2*d^2*(e*integrate(((e*x + d)*c^2*d - c^2*d^2 - e^2)/((( e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2)*sqrt(e*x + d)), x)/c^2 + 2*sqrt(e*x + d)/c^2)/e^2 + 3375*e^2*integrate(1/15*sqrt(e*x + d)*x*log( x)/(c^2*e^2*x^2 + e^2), x) - 10*(3*e^3*integrate(sqrt(e*x + d)/((e*x + d)^ 2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2), x)/c^2 - 2*(e*x + d)^(3/2)/c^2 )*c^2*d/e^2 + 225*(e*integrate(((e*x + d)*c^2*d - c^2*d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2)*sqrt(e*x + d)), x)/c^2 + 2*s qrt(e*x + d)/c^2)*log(c) - 15*c^2*(15*e^3*integrate(((e*x + d)*c^2*d - c^2 *d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2)*sqrt(e* x + d)), x)/c^4 - 2*(3*(e*x + d)^(5/2)*c^2 - 5*(e*x + d)^(3/2)*c^2*d - 15* sqrt(e*x + d)*e^2)/c^4)*log(c)/e^2 - 6*c^2*(15*e^3*integrate(((e*x + d)*c^ 2*d - c^2*d^2 - e^2)/(((e*x + d)^2*c^2 - 2*(e*x + d)*c^2*d + c^2*d^2 + e^2 )*sqrt(e*x + d)), x)/c^4 - 2*(3*(e*x + d)^(5/2)*c^2 - 5*(e*x + d)^(3/2)*c^ 2*d - 15*sqrt(e*x + d)*e^2)/c^4)/e^2 - 30*(3*e^2*x^2 + d*e*x - 2*d^2)*sqrt (e*x + d)*log(sqrt(c^2*x^2 + 1) + 1)/e^2 - 225*integrate(2/15*(3*c^2*e^2*x ^3 + c^2*d*e*x^2 - 2*c^2*d^2*x)*sqrt(e*x + d)/(c^2*e^2*x^2 + e^2 + (c^2*e^ 2*x^2 + e^2)*sqrt(c^2*x^2 + 1)), x))*b + 2/15*a*(3*(e*x + d)^(5/2)/e^2 - 5 *(e*x + d)^(3/2)*d/e^2)
\[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int { \sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x \,d x } \]
Timed out. \[ \int x \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right ) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x} \,d x \]