\(\int \frac {x^2 (a+b \csc ^{-1}(c x))}{\sqrt {d+e x}} \, dx\) [58]

Optimal result

Integrand size = 21, antiderivative size = 418 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {4 b \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}{15 c e}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}+\frac {4 b d \sqrt {d+e x} \sqrt {1-c^2 x^2} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{5 c^2 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {\frac {c (d+e x)}{c d+e}}}-\frac {4 b \left (7 c^2 d^2+e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c^4 e^2 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}}-\frac {32 b d^3 \sqrt {\frac {c (d+e x)}{c d+e}} \sqrt {1-c^2 x^2} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{15 c e^3 \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {d+e x}} \] Output:

4/15*b*(1-1/c^2/x^2)^(1/2)*x*(e*x+d)^(1/2)/c/e+2*d^2*(e*x+d)^(1/2)*(a+b*ar 
ccsc(c*x))/e^3-4/3*d*(e*x+d)^(3/2)*(a+b*arccsc(c*x))/e^3+2/5*(e*x+d)^(5/2) 
*(a+b*arccsc(c*x))/e^3+4/5*b*d*(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)-4/15*b*(7*c^2*d^2+e^2)*(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^4/e^2/(1-1/c^2/x^2)^(1/2)/x/(e*x+d)^(1/2)-32/15*b*d^ 
3*(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 = 33.41 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.88 \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=-\frac {a d^3 \sqrt {1+\frac {e x}{d}} B_{-\frac {e x}{d}}\left (3,\frac {1}{2}\right )}{e^3 \sqrt {d+e x}}+\frac {b \left (-\frac {c \left (e+\frac {d}{x}\right ) x \left (\frac {4 c d \sqrt {1-\frac {1}{c^2 x^2}}}{5 e^2}-\frac {16 c^2 d^2 \csc ^{-1}(c x)}{15 e^3}-\frac {2 c^2 x^2 \csc ^{-1}(c x)}{5 e}-\frac {4 c x \left (e \sqrt {1-\frac {1}{c^2 x^2}}-2 c d \csc ^{-1}(c x)\right )}{15 e^2}\right )}{\sqrt {d+e x}}-\frac {2 \sqrt {e+\frac {d}{x}} \sqrt {c x} \left (\frac {2 \left (7 c^2 d^2 e+e^3\right ) \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^3 d^3+3 c d 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 {6 c d 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 )}{15 e^3 \sqrt {d+e x}}\right )}{c^3} \] Input:

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

Output:

-((a*d^3*Sqrt[1 + (e*x)/d]*Beta[-((e*x)/d), 3, 1/2])/(e^3*Sqrt[d + e*x])) 
+ (b*(-((c*(e + d/x)*x*((4*c*d*Sqrt[1 - 1/(c^2*x^2)])/(5*e^2) - (16*c^2*d^ 
2*ArcCsc[c*x])/(15*e^3) - (2*c^2*x^2*ArcCsc[c*x])/(5*e) - (4*c*x*(e*Sqrt[1 
 - 1/(c^2*x^2)] - 2*c*d*ArcCsc[c*x]))/(15*e^2)))/Sqrt[d + e*x]) - (2*Sqrt[ 
e + d/x]*Sqrt[c*x]*((2*(7*c^2*d^2*e + e^3)*Sqrt[(c*d + c*e*x)/(c*d + e)]*S 
qrt[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^3*d^3 + 3*c* 
d*e^2)*Sqrt[(c*d + c*e*x)/(c*d + e)]*Sqrt[1 - c^2*x^2]*EllipticPi[2, ArcSi 
n[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)) - (6*c*d*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[ArcS 
in[Sqrt[(c*d + c*e*x)/(c*d - e)]], (c*d - e)/(c*d + e)] - e*EllipticF[ArcS 
in[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]*E 
llipticPi[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))))/(15*e^3*Sqrt[d + e*x 
])))/c^3
 

Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 414, normalized size of antiderivative = 0.99, 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, 637, 687, 27, 600, 508, 327, 511, 321, 2009}

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.

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

\(\Big \downarrow \) 5770

\(\displaystyle \frac {b \int \frac {2 \sqrt {d+e x} \left (8 d^2-4 e x d+3 e^2 x^2\right )}{15 e^3 \sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b \int \frac {\sqrt {d+e x} \left (8 d^2-4 e x d+3 e^2 x^2\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{15 c e^3}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 7272

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \int \frac {\sqrt {d+e x} \left (8 d^2-4 e x d+3 e^2 x^2\right )}{x \sqrt {1-c^2 x^2}}dx}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 2351

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \frac {\sqrt {d+e x}}{x \sqrt {1-c^2 x^2}}dx+\int \frac {\sqrt {d+e x} \left (3 e^2 x-4 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 637

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx+\int \frac {\sqrt {d+e x} \left (3 e^2 x-4 d e\right )}{\sqrt {1-c^2 x^2}}dx\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 687

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {2 \int \frac {3 e \left (4 d^2 c^2+3 d e x c^2-e^2\right )}{2 \sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{3 c^2}+8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \int \frac {4 d^2 c^2+3 d e x c^2-e^2}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx}{c^2}+8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 600

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx+3 c^2 d \int \frac {\sqrt {d+e x}}{\sqrt {1-c^2 x^2}}dx\right )}{c^2}+8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 508

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {6 c d \sqrt {d+e x} \int \frac {\sqrt {1-\frac {e (1-c x)}{c d+e}}}{\sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2}+8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (\left (c^2 d^2-e^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}dx-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2}+8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 511

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (-\frac {2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \int \frac {1}{\sqrt {1-\frac {e (1-c x)}{c d+e}} \sqrt {\frac {1}{2} (c x-1)+1}}d\frac {\sqrt {1-c x}}{\sqrt {2}}}{c \sqrt {d+e x}}-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2}+8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (8 d^2 \int \left (\frac {d}{x \sqrt {d+e x} \sqrt {1-c^2 x^2}}+\frac {e}{\sqrt {d+e x} \sqrt {1-c^2 x^2}}\right )dx-\frac {e \left (-\frac {2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2}-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 d^2 \sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{e^3}+\frac {2 (d+e x)^{5/2} \left (a+b \csc ^{-1}(c x)\right )}{5 e^3}-\frac {4 d (d+e x)^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{3 e^3}+\frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {e \left (-\frac {2 \left (c^2 d^2-e^2\right ) \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {6 c d \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right )|\frac {2 e}{c d+e}\right )}{\sqrt {\frac {c (d+e x)}{c d+e}}}\right )}{c^2}+8 d^2 \left (-\frac {2 e \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{c \sqrt {d+e x}}-\frac {2 d \sqrt {\frac {c (d+e x)}{c d+e}} \operatorname {EllipticPi}\left (2,\arcsin \left (\frac {\sqrt {1-c x}}{\sqrt {2}}\right ),\frac {2 e}{c d+e}\right )}{\sqrt {d+e x}}\right )-\frac {2 e^2 \sqrt {1-c^2 x^2} \sqrt {d+e x}}{c^2}\right )}{15 c e^3 x \sqrt {1-\frac {1}{c^2 x^2}}}\)

Input:

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

Output:

(2*d^2*Sqrt[d + e*x]*(a + b*ArcCsc[c*x]))/e^3 - (4*d*(d + e*x)^(3/2)*(a + 
b*ArcCsc[c*x]))/(3*e^3) + (2*(d + e*x)^(5/2)*(a + b*ArcCsc[c*x]))/(5*e^3) 
+ (2*b*Sqrt[1 - c^2*x^2]*((-2*e^2*Sqrt[d + e*x]*Sqrt[1 - c^2*x^2])/c^2 - ( 
e*((-6*c*d*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c 
*d + e)])/Sqrt[(c*(d + e*x))/(c*d + e)] - (2*(c^2*d^2 - e^2)*Sqrt[(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])))/c^2 + 8*d^2*((-2*e*Sqrt[(c*(d + e*x))/(c*d + e)]*Ell 
ipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*Sqrt[d + e*x]) 
- (2*d*Sqrt[(c*(d + e*x))/(c*d + e)]*EllipticPi[2, ArcSin[Sqrt[1 - c*x]/Sq 
rt[2]], (2*e)/(c*d + e)])/Sqrt[d + e*x])))/(15*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 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 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 637

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Int[ExpandIntegrand[(a + b*x^2)^p/Sqrt[c + d*x], x^m*(c + d*x)^(n + 1 
/2), x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[p + 1/2] && IntegerQ[n 
 + 1/2] && IntegerQ[m]
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(849\) vs. \(2(375)=750\).

Time = 13.47 (sec) , antiderivative size = 850, normalized size of antiderivative = 2.03

method	result	size
derivativedivides	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} d}{3}+d^{2} \sqrt {e x +d}\right )+2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {8 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 {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{15}+\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 ) c^{2} d^{2}}{5}-\frac {16 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}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\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 ) c d e}{5}+\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 ) c d e}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\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 ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \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}}\)	\(850\)
default	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} d}{3}+d^{2} \sqrt {e x +d}\right )+2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}+\frac {8 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 {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{15}+\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 ) c^{2} d^{2}}{5}-\frac {16 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}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}-\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 ) c d e}{5}+\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 ) c d e}{5}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}+\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 ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \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}}\)	\(850\)
parts	\(\frac {2 a \left (\frac {\left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \left (e x +d \right )^{\frac {3}{2}} d}{3}+d^{2} \sqrt {e x +d}\right )}{e^{3}}+\frac {2 b \left (\frac {\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {5}{2}}}{5}-\frac {2 \,\operatorname {arccsc}\left (c x \right ) \left (e x +d \right )^{\frac {3}{2}} d}{3}+\operatorname {arccsc}\left (c x \right ) d^{2} \sqrt {e x +d}+\frac {\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} \left (e x +d \right )^{\frac {5}{2}}}{15}-\frac {4 \sqrt {\frac {c}{c d -e}}\, c^{2} d \left (e x +d \right )^{\frac {3}{2}}}{15}+\frac {8 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 {EllipticF}\left (\sqrt {e x +d}\, \sqrt {\frac {c}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2}}{15}+\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 ) c^{2} d^{2}}{5}-\frac {16 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}}{15}+\frac {2 \sqrt {\frac {c}{c d -e}}\, c^{2} d^{2} \sqrt {e x +d}}{15}-\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 ) c d e}{5}+\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 ) c d e}{5}+\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 ) e^{2}}{15}-\frac {2 \sqrt {\frac {c}{c d -e}}\, e^{2} \sqrt {e x +d}}{15}}{c^{3} \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}}\)	\(865\)

Input:

int(x^2*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/e^3*(a*(1/5*(e*x+d)^(5/2)-2/3*(e*x+d)^(3/2)*d+d^2*(e*x+d)^(1/2))+b*(1/5* 
arccsc(c*x)*(e*x+d)^(5/2)-2/3*arccsc(c*x)*(e*x+d)^(3/2)*d+arccsc(c*x)*d^2* 
(e*x+d)^(1/2)+2/15/c^3*((c/(c*d-e))^(1/2)*c^2*(e*x+d)^(5/2)+4*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+3*((-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^2-8*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)*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^2-2*(c/(c*d-e))^(1/2)*c^2*d*(e*x+d)^(3/2)-3*((-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+3*((-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+(c/(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)*EllipticF((e*x+d)^(1/2)*(c/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e 
^2-(c/(c*d-e))^(1/2)*e^2*(e*x+d)^(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. \[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\text {Timed out} \] Input:

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

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

integrate(x**2*(a+b*acsc(c*x))/(e*x+d)**(1/2),x)
 

Output:

Integral(x**2*(a + b*acsc(c*x))/sqrt(d + e*x), x)
 

Maxima [F(-2)]

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

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(1/2),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]

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

integrate(x^2*(a+b*arccsc(c*x))/(e*x+d)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {d+e x}} \, dx=\frac {16 \sqrt {e x +d}\, a \,d^{2}-8 \sqrt {e x +d}\, a d e x +6 \sqrt {e x +d}\, a \,e^{2} x^{2}+15 \left (\int \frac {\mathit {acsc} \left (c x \right ) x^{2}}{\sqrt {e x +d}}d x \right ) b \,e^{3}}{15 e^{3}} \] Input:

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

Output:

(16*sqrt(d + e*x)*a*d**2 - 8*sqrt(d + e*x)*a*d*e*x + 6*sqrt(d + e*x)*a*e** 
2*x**2 + 15*int((acsc(c*x)*x**2)/sqrt(d + e*x),x)*b*e**3)/(15*e**3)