Integrand size = 19, antiderivative size = 300 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b c \sqrt {1-\frac {1}{c^2 x^2}} x}{3 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 e \left (c^2 d^2-e^2\right ) \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {8 b \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \] Output:
4/3*b*c*(1-1/c^2/x^2)^(1/2)*x/(c^2*d^2-e^2)/(e*x+d)^(1/2)+2/3*d*(a+b*arccs c(c*x))/e^2/(e*x+d)^(3/2)-2*(a+b*arccsc(c*x))/e^2/(e*x+d)^(1/2)+4/3*b*(e*x +d)^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticE(1/2*(-c*x+1)^(1/2)*2^(1/2),2^(1/2)* (e/(c*d+e))^(1/2))/e/(c^2*d^2-e^2)/(1-1/c^2/x^2)^(1/2)/x/(c*(e*x+d)/(c*d+e ))^(1/2)+8/3*b*(c*(e*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticPi(1/2 *(-c*x+1)^(1/2)*2^(1/2),2,2^(1/2)*(e/(c*d+e))^(1/2))/c/e^2/(1-1/c^2/x^2)^( 1/2)/x/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 12.29 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.15 \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {4 b c \sqrt {1-\frac {1}{c^2 x^2}} x}{3 \left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 a (2 d+3 e x)}{3 e^2 (d+e x)^{3/2}}-\frac {2 b (2 d+3 e x) \csc ^{-1}(c x)}{3 e^2 (d+e x)^{3/2}}+\frac {4 i b \sqrt {-\frac {c}{c d+e}} \sqrt {\frac {e (1+c x)}{-c d+e}} \sqrt {\frac {e-c e x}{c d+e}} \left (c d E\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right )|\frac {c d+e}{c d-e}\right )-c d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )+2 (c d+e) \operatorname {EllipticPi}\left (1+\frac {e}{c d},i \text {arcsinh}\left (\sqrt {-\frac {c}{c d+e}} \sqrt {d+e x}\right ),\frac {c d+e}{c d-e}\right )\right )}{3 c^2 d e^2 \sqrt {1-\frac {1}{c^2 x^2}} x} \] Input:
Integrate[(x*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]
Output:
(4*b*c*Sqrt[1 - 1/(c^2*x^2)]*x)/(3*(c^2*d^2 - e^2)*Sqrt[d + e*x]) - (2*a*( 2*d + 3*e*x))/(3*e^2*(d + e*x)^(3/2)) - (2*b*(2*d + 3*e*x)*ArcCsc[c*x])/(3 *e^2*(d + e*x)^(3/2)) + (((4*I)/3)*b*Sqrt[-(c/(c*d + e))]*Sqrt[(e*(1 + c*x ))/(-(c*d) + e)]*Sqrt[(e - c*e*x)/(c*d + e)]*(c*d*EllipticE[I*ArcSinh[Sqrt [-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] - c*d*EllipticF[I*Ar cSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)] + 2*(c*d + e)*EllipticPi[1 + e/(c*d), I*ArcSinh[Sqrt[-(c/(c*d + e))]*Sqrt[d + e*x]], (c*d + e)/(c*d - e)]))/(c^2*d*e^2*Sqrt[1 - 1/(c^2*x^2)]*x)
Time = 1.95 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.38, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.053, Rules used = {5770, 27, 7272, 2351, 27, 498, 27, 508, 327, 635, 25, 27, 498, 27, 508, 327, 632, 186, 413, 412}
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.
| \(\displaystyle \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5770 |
| \(\displaystyle \frac {b \int -\frac {2 (2 d+3 e x)}{3 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{c}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \int \frac {2 d+3 e x}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 (d+e x)^{3/2}}dx}{3 c e^2}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 7272 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {2 d+3 e x}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 2351 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (\int \frac {3 e}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (3 e \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (3 e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \int \frac {1}{x (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 635 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\int -\frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx+\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\int \frac {e}{d (d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \int \frac {1}{(d+e x)^{3/2} \sqrt {1-c^2 x^2}}dx}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c^2 \int -\frac {\sqrt {d+e x}}{2 \sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {c^2 \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx}{c^2 d^2-e^2}+\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 508 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 632 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (\frac {\int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 186 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (-\frac {2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}}{d}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (2 d \left (-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}\right )+3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle -\frac {2 \left (a+b \csc ^{-1}(c x)\right )}{e^2 \sqrt {d+e x}}+\frac {2 d \left (a+b \csc ^{-1}(c x)\right )}{3 e^2 (d+e x)^{3/2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (3 e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )+2 d \left (-\frac {e \left (\frac {2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) \sqrt {d+e x}}-\frac {2 c \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{d}-\frac {2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{d \sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}\right )\right )}{3 c e^2 x \sqrt {1-\frac {1}{c^2 x^2}}}\) |
Input:
Int[(x*(a + b*ArcCsc[c*x]))/(d + e*x)^(5/2),x]
Output:
(2*d*(a + b*ArcCsc[c*x]))/(3*e^2*(d + e*x)^(3/2)) - (2*(a + b*ArcCsc[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*Sqrt[d + e*x]*EllipticE[ArcSin[S qrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/((c^2*d^2 - e^2)*Sqrt[(c*(d + e*x ))/(c*d + e)])) + 2*d*(-((e*((2*e*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*Sqrt [d + e*x]) - (2*c*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], ( 2*e)/(c*d + e)])/((c^2*d^2 - e^2)*Sqrt[(c*(d + e*x))/(c*d + e)])))/d) - (2 *Sqrt[1 - (e*(1 - c*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt [2]], (2*e)/(c*d + e)])/(d*Sqrt[d + e/c - (e*(1 - c*x))/c]))))/(3*c*e^2*Sq rt[1 - 1/(c^2*x^2)]*x)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ )]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2 Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c*f)/d, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
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] :> With[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c *q))])) Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
Int[1/((x_)*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] : > With[{q = Rt[-b/a, 2]}, Simp[1/Sqrt[a] Int[1/(x*Sqrt[c + d*x]*Sqrt[1 - q*x]*Sqrt[1 + q*x]), x], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ a, 0]
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[((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_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide [u, x]}, Simp[(a + b*ArcCsc[c*x]) v, x] + Simp[b/c Int[SimplifyIntegran d[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]
Leaf count of result is larger than twice the leaf count of optimal. \(899\) vs. \(2(270)=540\).
Time = 11.61 (sec) , antiderivative size = 900, normalized size of antiderivative = 3.00
| method | result | size |
| derivativedivides | \(\frac {-2 a \left (\frac {1}{\sqrt {e x +d}}-\frac {d}{3 \left (e x +d \right )^{\frac {3}{2}}}\right )-2 b \left (\frac {\operatorname {arccsc}\left (c x \right )}{\sqrt {e x +d}}-\frac {\operatorname {arccsc}\left (c x \right ) d}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}+2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2} d^{2} \sqrt {e x +d}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}-2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) e^{2} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c d \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) x \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}}}}\right )}{e^{2}}\) | \(900\) |
| default | \(\frac {-2 a \left (\frac {1}{\sqrt {e x +d}}-\frac {d}{3 \left (e x +d \right )^{\frac {3}{2}}}\right )-2 b \left (\frac {\operatorname {arccsc}\left (c x \right )}{\sqrt {e x +d}}-\frac {\operatorname {arccsc}\left (c x \right ) d}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2} \sqrt {e x +d}+2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c^{2} d^{2} \sqrt {e x +d}-2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )-\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e \sqrt {e x +d}+\sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}-2 \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) e^{2} \sqrt {e x +d}-\sqrt {\frac {c}{c d -e}}\, d \,e^{2}\right )}{3 c d \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) x \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}}}}\right )}{e^{2}}\) | \(900\) |
| parts | \(\frac {2 a \left (-\frac {1}{\sqrt {e x +d}}+\frac {d}{3 \left (e x +d \right )^{\frac {3}{2}}}\right )}{e^{2}}+\frac {2 b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{\sqrt {e x +d}}+\frac {\operatorname {arccsc}\left (c x \right ) d}{3 \left (e x +d \right )^{\frac {3}{2}}}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{2}}{3}-\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}}{3}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c^{2} d^{2}}{3}+\frac {4 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) \sqrt {e x +d}\, c^{2} d^{2}}{3}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \left (e x +d \right )}{3}-\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c d e}{3}+\frac {2 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) \sqrt {e x +d}\, c d e}{3}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{3}}{3}-\frac {4 \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) \sqrt {e x +d}\, e^{2}}{3}-\frac {2 \sqrt {\frac {c}{c d -e}}\, d \,e^{2}}{3}}{c d \left (c d -e \right ) \sqrt {\frac {c}{c d -e}}\, \sqrt {e x +d}\, \left (c d +e \right ) x \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}}}}\right )}{e^{2}}\) | \(916\) |
Input:
int(x*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
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)*arccsc( c*x)-1/3*arccsc(c*x)*d/(e*x+d)^(3/2)-2/3/c*((c/(c*d-e))^(1/2)*c^2*d*(e*x+d )^2-((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)* EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2 *(e*x+d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d +e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/ 2))*c^2*d^2*(e*x+d)^(1/2)+2*((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d )+c*d+e)/(c*d+e))^(1/2)*EllipticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c* d-e)/d,(c/(c*d+e))^(1/2)/(c/(c*d-e))^(1/2))*c^2*d^2*(e*x+d)^(1/2)-2*(c/(c* d-e))^(1/2)*c^2*d^2*(e*x+d)-((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d )+c*d+e)/(c*d+e))^(1/2)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e) /(c*d+e))^(1/2))*c*d*e*(e*x+d)^(1/2)+((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*(( -c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2) ,((c*d-e)/(c*d+e))^(1/2))*c*d*e*(e*x+d)^(1/2)+(c/(c*d-e))^(1/2)*c^2*d^3-2* ((-c*(e*x+d)+c*d-e)/(c*d-e))^(1/2)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*Elli pticPi((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),1/c*(c*d-e)/d,(c/(c*d+e))^(1/2)/(c/ (c*d-e))^(1/2))*e^2*(e*x+d)^(1/2)-(c/(c*d-e))^(1/2)*d*e^2)/d/(c*d-e)/(c/(c *d-e))^(1/2)/(e*x+d)^(1/2)/(c*d+e)/x/((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)))
Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate(x*(a+b*acsc(c*x))/(e*x+d)**(5/2),x)
Output:
Timed out
Exception generated. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e+c*d>0)', see `assume?` for mor e details)
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x}{{\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x*(a+b*arccsc(c*x))/(e*x+d)^(5/2),x, algorithm="giac")
Output:
integrate((b*arccsc(c*x) + a)*x/(e*x + d)^(5/2), x)
Timed out. \[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((x*(a + b*asin(1/(c*x))))/(d + e*x)^(5/2),x)
Output:
int((x*(a + b*asin(1/(c*x))))/(d + e*x)^(5/2), x)
\[ \int \frac {x \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{5/2}} \, dx=\frac {3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b d \,e^{2}+3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right ) x}{\sqrt {e x +d}\, d^{2}+2 \sqrt {e x +d}\, d e x +\sqrt {e x +d}\, e^{2} x^{2}}d x \right ) b \,e^{3} x -4 a d -6 a e x}{3 \sqrt {e x +d}\, e^{2} \left (e x +d \right )} \] Input:
int(x*(a+b*acsc(c*x))/(e*x+d)^(5/2),x)
Output:
(3*sqrt(d + e*x)*int((acsc(c*x)*x)/(sqrt(d + e*x)*d**2 + 2*sqrt(d + e*x)*d *e*x + sqrt(d + e*x)*e**2*x**2),x)*b*d*e**2 + 3*sqrt(d + e*x)*int((acsc(c* x)*x)/(sqrt(d + e*x)*d**2 + 2*sqrt(d + e*x)*d*e*x + sqrt(d + e*x)*e**2*x** 2),x)*b*e**3*x - 4*a*d - 6*a*e*x)/(3*sqrt(d + e*x)*e**2*(d + e*x))