\(\int \frac {x^2 (a+b \csc ^{-1}(c x))}{(d+e x)^{3/2}} \, dx\) [64]

Optimal result

Integrand size = 21, antiderivative size = 369 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {4 b \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{3 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}+\frac {20 b d \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}+\frac {32 b d^2 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{3 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \] Output:

-2*d^2*(a+b*arccsc(c*x))/e^3/(e*x+d)^(1/2)-4*d*(e*x+d)^(1/2)*(a+b*arccsc(c 
*x))/e^3+2/3*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^3-4/3*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))/c^2/e^2/(1-1/c^2/x^2)^(1/2)/x/(c*(e*x+d)/(c*d+e))^(1/2)+20/3*b*d*(c*(e 
*x+d)/(c*d+e))^(1/2)*(-c^2*x^2+1)^(1/2)*EllipticF(1/2*(-c*x+1)^(1/2)*2^(1/ 
2),2^(1/2)*(e/(c*d+e))^(1/2))/c^2/e^2/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2)+ 
32/3*b*d^2*(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^3/(1-1/c^2/x^2)^(1/2) 
/x/(e*x+d)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4 in optimal.

Time = 33.58 (sec) , antiderivative size = 750, normalized size of antiderivative = 2.03 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=-\frac {a d^3 \left (1+\frac {e x}{d}\right )^{3/2} B_{-\frac {e x}{d}}\left (3,-\frac {1}{2}\right )}{e^3 (d+e x)^{3/2}}+\frac {b \left (-\frac {c^2 \left (e+\frac {d}{x}\right )^2 x^2 \left (-\frac {4 \sqrt {1-\frac {1}{c^2 x^2}}}{3 e^2}+\frac {16 c d \csc ^{-1}(c x)}{3 e^3}-\frac {2 c d \csc ^{-1}(c x)}{e^2 \left (e+\frac {d}{x}\right )}-\frac {2 c x \csc ^{-1}(c x)}{3 e^2}\right )}{(d+e x)^{3/2}}+\frac {2 \left (e+\frac {d}{x}\right )^{3/2} (c x)^{3/2} \left (\frac {10 c d e \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}+\frac {2 \left (8 c^2 d^2+e^2\right ) \sqrt {\frac {c d+c 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 )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} (c x)^{3/2}}-\frac {2 e \cos \left (2 \csc ^{-1}(c x)\right ) \left ((c d+c e x) \left (-1+c^2 x^2\right )+c^2 d x \sqrt {\frac {c d+c e x}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )-\frac {c x (1+c x) \sqrt {\frac {e-c e x}{c d+e}} \sqrt {\frac {c d+c e x}{c d-e}} \left ((c d+e) E\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right )|\frac {c d-e}{c d+e}\right )-e \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {c d+c e x}{c d-e}}\right ),\frac {c d-e}{c d+e}\right )\right )}{\sqrt {\frac {e (1+c x)}{-c d+e}}}+c e x \sqrt {\frac {c d+c 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 )\right )}{\sqrt {1-\frac {1}{c^2 x^2}} \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (-2+c^2 x^2\right )}\right )}{3 e^3 (d+e x)^{3/2}}\right )}{c^3} \] Input:

Integrate[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x)^(3/2),x]
 

Output:

-((a*d^3*(1 + (e*x)/d)^(3/2)*Beta[-((e*x)/d), 3, -1/2])/(e^3*(d + e*x)^(3/ 
2))) + (b*(-((c^2*(e + d/x)^2*x^2*((-4*Sqrt[1 - 1/(c^2*x^2)])/(3*e^2) + (1 
6*c*d*ArcCsc[c*x])/(3*e^3) - (2*c*d*ArcCsc[c*x])/(e^2*(e + d/x)) - (2*c*x* 
ArcCsc[c*x])/(3*e^2)))/(d + e*x)^(3/2)) + (2*(e + d/x)^(3/2)*(c*x)^(3/2)*( 
(10*c*d*e*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin 
[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + 
 d/x]*(c*x)^(3/2)) + (2*(8*c^2*d^2 + e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sq 
rt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + 
e)])/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*(c*x)^(3/2)) - (2*e*Cos[2*ArcCsc 
[c*x]]*((c*d + c*e*x)*(-1 + c^2*x^2) + c^2*d*x*Sqrt[(c*d + c*e*x)/(c*d + e 
)]*Sqrt[1 - c^2*x^2]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + 
 e)] - (c*x*(1 + c*x)*Sqrt[(e - c*e*x)/(c*d + e)]*Sqrt[(c*d + c*e*x)/(c*d 
- e)]*((c*d + e)*EllipticE[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e 
)/(c*d + e)] - e*EllipticF[ArcSin[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e 
)/(c*d + e)]))/Sqrt[(e*(1 + c*x))/(-(c*d) + e)] + c*e*x*Sqrt[(c*d + c*e*x) 
/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sqrt[2]], 
 (2*e)/(c*d + e)]))/(Sqrt[1 - 1/(c^2*x^2)]*Sqrt[e + d/x]*Sqrt[c*x]*(-2 + c 
^2*x^2))))/(3*e^3*(d + e*x)^(3/2))))/c^3
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.86, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.619, Rules used = {5770, 27, 7272, 2351, 600, 508, 327, 511, 321, 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.

\(\displaystyle \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 5770

\(\displaystyle \frac {b \int -\frac {2 \left (8 d^2+4 e x d-e^2 x^2\right )}{3 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{c}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 b \int \frac {8 d^2+4 e x d-e^2 x^2}{\sqrt {1-\frac {1}{c^2 x^2}} x^2 \sqrt {d+e x}}dx}{3 c e^3}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 7272

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \int \frac {8 d^2+4 e x d-e^2 x^2}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 2351

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\int \frac {4 d e-e^2 x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 600

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+5 d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-e \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+5 d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\frac {2 e \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+5 d e \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {10 d e \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {10 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 632

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {1}{x \sqrt {1-c x} \sqrt {c x+1} \sqrt {d+e x}}dx-\frac {10 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 186

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-16 d^2 \int \frac {1}{c x \sqrt {c x+1} \sqrt {d+\frac {e}{c}-\frac {e (1-c x)}{c}}}d\sqrt {1-c x}-\frac {10 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 413

\(\displaystyle -\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {16 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \int \frac {1}{c x \sqrt {c x+1} \sqrt {1-\frac {e (1-c x)}{c d+e}}}d\sqrt {1-c x}}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {10 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}-\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 412

\(\displaystyle -\frac {2 d^2 \left (a+b \csc ^{-1}(c x)\right )}{e^3 \sqrt {d+e x}}-\frac {4 d \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}-\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {16 d^2 \sqrt {1-\frac {e (1-c x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {-\frac {e (1-c x)}{c}+\frac {e}{c}+d}}-\frac {10 d e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}+\frac {2 e \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{c \sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{3 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}\)

Input:

Int[(x^2*(a + b*ArcCsc[c*x]))/(d + e*x)^(3/2),x]
 

Output:

(-2*d^2*(a + b*ArcCsc[c*x]))/(e^3*Sqrt[d + e*x]) - (4*d*Sqrt[d + e*x]*(a + 
 b*ArcCsc[c*x]))/e^3 + (2*(d + e*x)^(3/2)*(a + b*ArcCsc[c*x]))/(3*e^3) - ( 
2*b*Sqrt[1 - c^2*x^2]*((2*e*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/S 
qrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[(c*(d + e*x))/(c*d + e)]) - (10*d*e*Sqr 
t[(c*(d + e*x))/(c*d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/ 
(c*d + e)])/(c*Sqrt[d + e*x]) - (16*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)])/Sqrt[d + e/ 
c - (e*(1 - c*x))/c]))/(3*c*e^3*Sqrt[1 - 1/(c^2*x^2)]*x)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 186

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]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

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]
 

rule 412

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])
 

rule 413

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]
 

rule 508

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]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 632

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]
 

rule 2351

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]
 

rule 5770

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]
 

rule 7272

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]
 

Maple [A] (verified)

Time = 11.80 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}-2 \,\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \left (4 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -8 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{3}}\)	\(439\)
default	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )+2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}-2 \,\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \left (4 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -8 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {\frac {-c \left (e x +d \right )+c d +e}{c d +e}}\, \sqrt {\frac {-c \left (e x +d \right )+c d -e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{3}}\)	\(439\)
parts	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {3}{2}}}{3}-2 d \sqrt {e x +d}-\frac {d^{2}}{\sqrt {e x +d}}\right )}{e^{3}}+\frac {2 b \left (\frac {\left (e x +d \right )^{\frac {3}{2}} \operatorname {arccsc}\left (c x \right )}{3}-2 \,\operatorname {arccsc}\left (c x \right ) d \sqrt {e x +d}-\frac {\operatorname {arccsc}\left (c x \right ) d^{2}}{\sqrt {e x +d}}-\frac {2 \left (4 d \operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d -8 d \operatorname {EllipticPi}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \frac {c d -e}{c d}, \frac {\sqrt {\frac {c}{c d +e}}}{\sqrt {\frac {c}{c d -e}}}\right ) c -\operatorname {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e +\operatorname {EllipticE}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e \right ) \sqrt {-\frac {c \left (e x +d \right )-c d -e}{c d +e}}\, \sqrt {-\frac {c \left (e x +d \right )-c d +e}{c d -e}}}{3 c^{2} \sqrt {\frac {c}{c d -e}}\, x \sqrt {\frac {c^{2} \left (e x +d \right )^{2}-2 c^{2} d \left (e x +d \right )+c^{2} d^{2}-e^{2}}{c^{2} e^{2} x^{2}}}}\right )}{e^{3}}\)	\(444\)

Input:

int(x^2*(a+b*arccsc(c*x))/(e*x+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

2/e^3*(a*(1/3*(e*x+d)^(3/2)-2*d*(e*x+d)^(1/2)-d^2/(e*x+d)^(1/2))+b*(1/3*(e 
*x+d)^(3/2)*arccsc(c*x)-2*arccsc(c*x)*d*(e*x+d)^(1/2)-arccsc(c*x)*d^2/(e*x 
+d)^(1/2)-2/3/c^2*(4*d*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/ 
(c*d+e))^(1/2))*c+EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+ 
e))^(1/2))*c*d-8*d*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-EllipticF((e*x+d)^(1/2)*(c/(c*d-e 
))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e+EllipticE((e*x+d)^(1/2)*(c/(c*d-e))^(1 
/2),((c*d-e)/(c*d+e))^(1/2))*e)*((-c*(e*x+d)+c*d+e)/(c*d+e))^(1/2)*((-c*(e 
*x+d)+c*d-e)/(c*d-e))^(1/2)/(c/(c*d-e))^(1/2)/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)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(x**2*(a+b*acsc(c*x))/(e*x+d)**(3/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(3/2),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)
 

Giac [F]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{2}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(3/2),x, algorithm="giac")
 

Output:

integrate((b*arccsc(c*x) + a)*x^2/(e*x + d)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:

int((x^2*(a + b*asin(1/(c*x))))/(d + e*x)^(3/2),x)
 

Output:

int((x^2*(a + b*asin(1/(c*x))))/(d + e*x)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{(d+e x)^{3/2}} \, dx=\frac {3 \sqrt {e x +d}\, \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{2}}{\sqrt {e x +d}\, d +\sqrt {e x +d}\, e x}d x \right ) b \,e^{3}-16 a \,d^{2}-8 a d e x +2 a \,e^{2} x^{2}}{3 \sqrt {e x +d}\, e^{3}} \] Input:

int(x^2*(a+b*acsc(c*x))/(e*x+d)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(3*sqrt(d + e*x)*int((acsc(c*x)*x**2)/(sqrt(d + e*x)*d + sqrt(d + e*x)*e*x 
),x)*b*e**3 - 16*a*d**2 - 8*a*d*e*x + 2*a*e**2*x**2)/(3*sqrt(d + e*x)*e**3 
)