Integrand size = 19, antiderivative size = 474 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}+\frac {4 b c \sqrt {d+e x} \sqrt {1+c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right )|-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}}}-\frac {8 b c d \sqrt {\frac {c^2 (d+e x)}{c^2 d-\sqrt {-c^2} e}} \sqrt {1+c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),-\frac {2 \sqrt {-c^2} e}{c^2 d-\sqrt {-c^2} e}\right )}{3 \left (-c^2\right )^{3/2} e \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {8 b d^2 \sqrt {\frac {\sqrt {-c^2} (d+e x)}{\sqrt {-c^2} d+e}} \sqrt {1+c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-\sqrt {-c^2} x}}{\sqrt {2}}\right ),\frac {2 e}{\sqrt {-c^2} d+e}\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \]
2/3*(e*x+d)^(3/2)*(a+b*arccsch(c*x))/e^2-2*d*(a+b*arccsch(c*x))*(e*x+d)^(1 /2)/e^2+8/3*b*d^2*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*c* 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))*(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)-8/3*b*c*d*Ell ipticF(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)
Result contains complex when optimal does not.
Time = 2.00 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.88 \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {\frac {2 a (-2 d+e x) (d+e x)}{e^2}+\frac {2 b (-2 d+e x) (d+e x) \text {csch}^{-1}(c x)}{e^2}+\frac {2 b (i c d+e) \sqrt {1+\frac {1}{c^2 x^2}} x \sqrt {\frac {c e (i+c x) (d+e x)}{(i c d+e)^2}} \left (2 i (c d+i e) e \sqrt {-\frac {e (-i+c x)}{c d+i e}} E\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-i e}}\right )|\frac {c d-i e}{c d+i e}\right )-i c d e \sqrt {2+2 i c x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )+2 e^2 \sqrt {-\frac {e (-i+c x)}{c d+i e}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c (d+e x)}{c d-i e}}\right ),\frac {c d-i e}{c d+i e}\right )+2 c^2 d^2 \sqrt {2+2 i c x} \operatorname {EllipticPi}\left (1+\frac {i c d}{e},\arcsin \left (\sqrt {-\frac {e (i+c x)}{c d-i e}}\right ),\frac {i c d+e}{2 e}\right )\right )}{e^3 \left (c+c^3 x^2\right )}}{3 \sqrt {d+e x}} \]
((2*a*(-2*d + e*x)*(d + e*x))/e^2 + (2*b*(-2*d + e*x)*(d + e*x)*ArcCsch[c* x])/e^2 + (2*b*(I*c*d + e)*Sqrt[1 + 1/(c^2*x^2)]*x*Sqrt[(c*e*(I + c*x)*(d + e*x))/(I*c*d + e)^2]*((2*I)*(c*d + I*e)*e*Sqrt[-((e*(-I + c*x))/(c*d + I *e))]*EllipticE[ArcSin[Sqrt[(c*(d + e*x))/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] - I*c*d*e*Sqrt[2 + (2*I)*c*x]*EllipticF[ArcSin[Sqrt[-((e*(I + c*x) )/(c*d - I*e))]], (I*c*d + e)/(2*e)] + 2*e^2*Sqrt[-((e*(-I + c*x))/(c*d + I*e))]*EllipticF[ArcSin[Sqrt[(c*(d + e*x))/(c*d - I*e)]], (c*d - I*e)/(c*d + I*e)] + 2*c^2*d^2*Sqrt[2 + (2*I)*c*x]*EllipticPi[1 + (I*c*d)/e, ArcSin[ Sqrt[-((e*(I + c*x))/(c*d - I*e))]], (I*c*d + e)/(2*e)]))/(e^3*(c + c^3*x^ 2)))/(3*Sqrt[d + e*x])
Leaf count is larger than twice the leaf count of optimal. \(1377\) vs. \(2(474)=948\).
Time = 2.93 (sec) , antiderivative size = 1377, normalized size of antiderivative = 2.91, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {6864, 27, 7272, 2351, 25, 27, 507, 630, 1459, 1416, 1509, 1656, 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.
| \(\displaystyle \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 6864 |
| \(\displaystyle \frac {b \int -\frac {2 (2 d-e x) \sqrt {d+e x}}{3 e^2 \sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {(2 d-e x) \sqrt {d+e x}}{\sqrt {1+\frac {1}{c^2 x^2}} x^2}dx}{3 c e^2}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \int \frac {(2 d-e x) \sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\int -\frac {e \sqrt {d+e x}}{\sqrt {c^2 x^2+1}}dx+2 d \int \frac {\sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx-\int \frac {e \sqrt {d+e x}}{\sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx-e \int \frac {\sqrt {d+e x}}{\sqrt {c^2 x^2+1}}dx\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 507 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (2 d \int \frac {\sqrt {d+e x}}{x \sqrt {c^2 x^2+1}}dx-2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 630 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-2 \int \frac {d+e x}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-4 d \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 1459 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-4 d \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-2 \left (\frac {\sqrt {c^2 d^2+e^2} \int \frac {1}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 1416 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \int \frac {1-\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}}{\sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}}{c}\right )-4 d \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 1509 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (-4 d \int -\frac {d+e x}{e x \sqrt {\frac {(d+e x)^2 c^2}{e^2}-\frac {2 d (d+e x) c^2}{e^2}+\frac {d^2 c^2}{e^2}+1}}d\sqrt {d+e x}-2 \left (\frac {\left (c^2 d^2+e^2\right )^{3/4} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right ),\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{2 c^{3/2} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-\frac {\sqrt {c^2 d^2+e^2} \left (\frac {\sqrt [4]{c^2 d^2+e^2} \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right ) \sqrt {\frac {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )^2}} E\left (2 \arctan \left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt [4]{c^2 d^2+e^2}}\right )|\frac {1}{2} \left (\frac {c d}{\sqrt {c^2 d^2+e^2}}+1\right )\right )}{\sqrt {c} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}-\frac {\sqrt {d+e x} \sqrt {\frac {c^2 d^2}{e^2}+\frac {c^2 (d+e x)^2}{e^2}-\frac {2 c^2 d (d+e x)}{e^2}+1}}{\left (\frac {c^2 d^2}{e^2}+1\right ) \left (\frac {c (d+e x)}{\sqrt {c^2 d^2+e^2}}+1\right )}\right )}{c}\right )\right )}{3 c e^2 x \sqrt {\frac {1}{c^2 x^2}+1}}+\frac {2 (d+e x)^{3/2} \left (a+b \text {csch}^{-1}(c x)\right )}{3 e^2}-\frac {2 d \sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{e^2}\) |
| \(\Big \downarrow \) 1656 |
| \(\displaystyle \frac {2 \left (a+b \text {csch}^{-1}(c x)\right ) (d+e x)^{3/2}}{3 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (-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 )-4 d \left (d \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}-\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 {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 )\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 ) (d+e x)^{3/2}}{3 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (-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 )-4 d \left (d \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 {\left (\frac {c^2 d^2}{e^2}+1\right ) \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 {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 )\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 ) (d+e x)^{3/2}}{3 e^2}-\frac {2 d \left (a+b \text {csch}^{-1}(c x)\right ) \sqrt {d+e x}}{e^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (-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 )-4 d \left (d \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 {\left (\frac {c^2 d^2}{e^2}+1\right ) \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 {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 )\right )}{3 c e^2 \sqrt {1+\frac {1}{c^2 x^2}} x}\) |
(-2*d*Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/e^2 + (2*(d + e*x)^(3/2)*(a + b* ArcCsch[c*x]))/(3*e^2) - (2*b*Sqrt[1 + c^2*x^2]*(-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)]*Ellipti cE[2*ArcTan[(Sqrt[c]*Sqrt[d + e*x])/(c^2*d^2 + e^2)^(1/4)], (1 + (c*d)/Sqr t[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])) - 4*d*(-1/2* ((1 + (c^2*d^2)/e^2)*(c^2*d^2 + e^2)^(1/4)*(1 - (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])/(Sqrt[c]*...
3.1.59.3.1 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[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[Sqrt[(c_) + (d_.)*(x_)]/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp[-2 Subst[Int[x^2/((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[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[(x_)^2/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) , x_Symbol] :> With[{q = Rt[c/a, 2]}, Simp[(-a)*((e + d*q)/(c*d^2 - a*e^2)) Int[1/Sqrt[a + b*x^2 + c*x^4], x], x] + Simp[a*d*((e + d*q)/(c*d^2 - a*e ^2)) Int[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; Fr eeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a] && NeQ[c*d^2 - a*e^2, 0]
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 = 8.00 (sec) , antiderivative size = 868, normalized size of antiderivative = 1.83
| method | result | size |
| derivativedivides | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+d \sqrt {e x +d}\right )-2 b \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}+\operatorname {arccsch}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (2 i \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -2 i \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{2}}\) | \(868\) |
| default | \(\frac {-2 a \left (-\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}+d \sqrt {e x +d}\right )-2 b \left (-\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}+\operatorname {arccsch}\left (c x \right ) d \sqrt {e x +d}+\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (2 i \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -2 i \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{2}}\) | \(868\) |
| parts | \(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-d \sqrt {e x +d}\right )}{e^{2}}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsch}\left (c x \right )}{3}-\operatorname {arccsch}\left (c x \right ) d \sqrt {e x +d}-\frac {2 \sqrt {-\frac {i c e \left (e x +d \right )+c^{2} d \left (e x +d \right )-c^{2} d^{2}-e^{2}}{c^{2} d^{2}+e^{2}}}\, \sqrt {\frac {i c e \left (e x +d \right )-c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} d^{2}+e^{2}}}\, \left (2 i \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c d e -2 i \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c d e -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) c^{2} d^{2}+2 \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \frac {c^{2} d^{2}+e^{2}}{\left (c d +i e \right ) c d}, \frac {\sqrt {-\frac {\left (-c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}{\sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}}\right ) c^{2} d^{2}+\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}-\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}, \sqrt {-\frac {-c^{2} d^{2}+2 i c d e +e^{2}}{c^{2} d^{2}+e^{2}}}\right ) e^{2}\right )}{3 c^{2} \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}+e^{2}}{c^{2} e^{2} x^{2}}}\, x \sqrt {\frac {\left (c d +i e \right ) c}{c^{2} d^{2}+e^{2}}}\, \left (-c d +i e \right )}\right )}{e^{2}}\) | \(871\) |
2/e^2*(-a*(-1/3*(e*x+d)^(3/2)+d*(e*x+d)^(1/2))-b*(-1/3*(e*x+d)^(3/2)*arccs ch(c*x)+arccsch(c*x)*d*(e*x+d)^(1/2)+2/3/c^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)*(2*I*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*d*e-2*I*E llipticPi((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*d*e-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^2*d^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^2*d^2+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+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)) *e^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))*e^2)/((c^2*(e*x+d)^2-2*c^2*d*(e*x+d)+ c^2*d^2+e^2)/c^2/e^2/x^2)^(1/2)/x/((c*d+I*e)*c/(c^2*d^2+e^2))^(1/2)/(I*e-c *d)))
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \]
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x \left (a + b \operatorname {acsch}{\left (c x \right )}\right )}{\sqrt {d + e x}}\, dx \]
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \]
2/3*a*((e*x + d)^(3/2)/e^2 - 3*sqrt(e*x + d)*d/e^2) + 1/3*b*(2*(e^2*x^2 - d*e*x - 2*d^2)*log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e^2) + 3*integrat e(2/3*(c^2*e^2*x^3 - c^2*d*e*x^2 - 2*c^2*d^2*x)/((c^2*e^2*x^2 + e^2)*sqrt( c^2*x^2 + 1)*sqrt(e*x + d) + (c^2*e^2*x^2 + e^2)*sqrt(e*x + d)), x) - 3*in tegrate(-1/3*(2*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^2*x^2 + e^2)*s qrt(e*x + d)), x))
\[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int { \frac {{\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x}{\sqrt {e x + d}} \,d x } \]
Timed out. \[ \int \frac {x \left (a+b \text {csch}^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {d+e\,x}} \,d x \]