\(\int \frac {x^4 (a+b \arccos (c x))^2}{(d-c^2 d x^2)^{5/2}} \, dx\) [257]

Optimal result

Integrand size = 29, antiderivative size = 421 \[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2 x}{3 c^4 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 \sqrt {1-c^2 x^2} \arccos (c x)}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {b x^2 (a+b \arccos (c x))}{3 c^3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {x (a+b \arccos (c x))^2}{c^4 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^5 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+e^{2 i \arccos (c x)}\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}}+\frac {4 i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )}{3 c^5 d^2 \sqrt {d-c^2 d x^2}} \] Output:

1/3*b^2*x/c^4/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b^2*(-c^2*x^2+1)^(1/2)*arccos(c 
*x)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*x^2*(a+b*arccos(c*x))/c^3/d^2/(-c^2 
*x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/3*x^3*(a+b*arccos(c*x))^2/c^2/d/(-c^2 
*d*x^2+d)^(3/2)-x*(a+b*arccos(c*x))^2/c^4/d^2/(-c^2*d*x^2+d)^(1/2)+4/3*I*( 
-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2/c^5/d^2/(-c^2*d*x^2+d)^(1/2)+1/3*(-c 
^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^3/b/c^5/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*b*( 
-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)/c^5 
/d^2/(-c^2*d*x^2+d)^(1/2)+4/3*I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,-(c*x+I*( 
-c^2*x^2+1)^(1/2))^2)/c^5/d^2/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.32 (sec) , antiderivative size = 349, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {a^2 c \sqrt {d} x \left (-3+4 c^2 x^2\right )+3 a^2 \left (-1+c^2 x^2\right ) \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+a b \sqrt {d} \left (\sqrt {1-c^2 x^2}+2 c x \arccos (c x)+8 c x \left (-1+c^2 x^2\right ) \arccos (c x)+\left (1-c^2 x^2\right )^{3/2} \left (-3 \arccos (c x)^2+4 \log \left (1-c^2 x^2\right )\right )\right )+b^2 \sqrt {d} \left (c x \left (1-c^2 x^2+\left (-3+4 c^2 x^2\right ) \arccos (c x)^2\right )+\sqrt {1-c^2 x^2} \arccos (c x) \left (1+\left (-1+c^2 x^2\right ) \arccos (c x) (4 i+\arccos (c x))-8 \left (-1+c^2 x^2\right ) \log \left (1-e^{2 i \arccos (c x)}\right )\right )-4 i \left (1-c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )}{3 c^5 d^{5/2} \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
 

Output:

(a^2*c*Sqrt[d]*x*(-3 + 4*c^2*x^2) + 3*a^2*(-1 + c^2*x^2)*Sqrt[d - c^2*d*x^ 
2]*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + a*b*Sqrt[d 
]*(Sqrt[1 - c^2*x^2] + 2*c*x*ArcCos[c*x] + 8*c*x*(-1 + c^2*x^2)*ArcCos[c*x 
] + (1 - c^2*x^2)^(3/2)*(-3*ArcCos[c*x]^2 + 4*Log[1 - c^2*x^2])) + b^2*Sqr 
t[d]*(c*x*(1 - c^2*x^2 + (-3 + 4*c^2*x^2)*ArcCos[c*x]^2) + Sqrt[1 - c^2*x^ 
2]*ArcCos[c*x]*(1 + (-1 + c^2*x^2)*ArcCos[c*x]*(4*I + ArcCos[c*x]) - 8*(-1 
 + c^2*x^2)*Log[1 - E^((2*I)*ArcCos[c*x])]) - (4*I)*(1 - c^2*x^2)^(3/2)*Po 
lyLog[2, E^((2*I)*ArcCos[c*x])]))/(3*c^5*d^(5/2)*(1 - c^2*x^2)*Sqrt[d - c^ 
2*d*x^2])
 

Rubi [A] (verified)

Time = 1.87 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.01, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.448, Rules used = {5207, 5207, 252, 223, 5153, 5181, 3042, 25, 4200, 25, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 5207

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

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

\(\Big \downarrow \) 5181

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\int -\left ((a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )\right )d\arccos (c x)}{c^4}+\frac {x^2 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {-\frac {2 b \sqrt {1-c^2 x^2} \int -\left ((a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )\right )d\arccos (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (-\frac {\int (a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{c^4}+\frac {x^2 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {\frac {2 b \sqrt {1-c^2 x^2} \int (a+b \arccos (c x)) \tan \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 4200

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2620

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {i (a+b \arccos (c x))^2}{2 b}}{c^4}+\frac {x^2 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{2} i b \int \log \left (1-e^{2 i \arccos (c x)}\right )d\arccos (c x)\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {i (a+b \arccos (c x))^2}{2 b}}{c^4}+\frac {x^2 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}-\frac {-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1-e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {x (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}+\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2 b \sqrt {1-c^2 x^2} \left (\frac {-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}}{c^4}+\frac {x^2 (a+b \arccos (c x))}{2 c^2 \left (1-c^2 x^2\right )}+\frac {b \left (\frac {x}{c^2 \sqrt {1-c^2 x^2}}-\frac {\arcsin (c x)}{c^3}\right )}{2 c}\right )}{3 c d^2 \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arccos (c x))^2}{3 c^2 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {\frac {x (a+b \arccos (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {2 b \sqrt {1-c^2 x^2} \left (-2 i \left (\frac {1}{2} i \log \left (1-e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}\right )}{c^3 d \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))^3}{3 b c^3 d \sqrt {d-c^2 d x^2}}}{c^2 d}\)

Input:

Int[(x^4*(a + b*ArcCos[c*x])^2)/(d - c^2*d*x^2)^(5/2),x]
 

Output:

(x^3*(a + b*ArcCos[c*x])^2)/(3*c^2*d*(d - c^2*d*x^2)^(3/2)) + (2*b*Sqrt[1 
- c^2*x^2]*((x^2*(a + b*ArcCos[c*x]))/(2*c^2*(1 - c^2*x^2)) + (b*(x/(c^2*S 
qrt[1 - c^2*x^2]) - ArcSin[c*x]/c^3))/(2*c) + (((-1/2*I)*(a + b*ArcCos[c*x 
])^2)/b - (2*I)*((I/2)*(a + b*ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])] 
+ (b*PolyLog[2, E^((2*I)*ArcCos[c*x])])/4))/c^4))/(3*c*d^2*Sqrt[d - c^2*d* 
x^2]) - ((x*(a + b*ArcCos[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - 
 c^2*x^2]*(a + b*ArcCos[c*x])^3)/(3*b*c^3*d*Sqrt[d - c^2*d*x^2]) - (2*b*Sq 
rt[1 - c^2*x^2]*(((-1/2*I)*(a + b*ArcCos[c*x])^2)/b - (2*I)*((I/2)*(a + b* 
ArcCos[c*x])*Log[1 - E^((2*I)*ArcCos[c*x])] + (b*PolyLog[2, E^((2*I)*ArcCo 
s[c*x])])/4)))/(c^3*d*Sqrt[d - c^2*d*x^2]))/(c^2*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[a, b, c, 2, m, p, x]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4200

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^ 
m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))), x] 
, x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]
 

rule 5153

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(-(b*c*(n + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2] 
]*(a + b*ArcCos[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^ 
2*d + e, 0] && NeQ[n, -1]
 

rule 5181

Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), 
x_Symbol] :> Simp[1/e   Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcCos[c*x]], 
x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]
 

rule 5207

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + 
 b*ArcCos[c*x])^n/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
 Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcCos[c*x])^n, x], x] - Simp 
[b*f*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[(f*x)^(m - 
 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[c*x])^(n - 1), x], x]) /; FreeQ[{ 
a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 1]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 4046 vs. \(2 (395 ) = 790\).

Time = 0.80 (sec) , antiderivative size = 4047, normalized size of antiderivative = 9.61

method	result	size
default	\(\text {Expression too large to display}\)	\(4047\)
parts	\(\text {Expression too large to display}\)	\(4047\)

Input:

int(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*b^2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^5/(c^2*x^2-1)*arcc 
os(c*x)^3-8/3*b^2*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^ 
4*x^4-71*c^2*x^2+16)/c^2*(-c^2*x^2+1)*x^3+128/3*I*a*b*(-d*(c^2*x^2-1))^(1/ 
2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^5*arccos(c*x)*( 
-c^2*x^2+1)^(1/2)+28/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6 
*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*(-c^2*x^2+1)*x^3+16/3*I*a*b*(-c^2*x^2+ 
1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^3/c^5/(c^2*x^2-1)*arccos(c*x)-4*I*a*b*(- 
d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16) 
/c^4*(-c^2*x^2+1)*x+8*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^ 
6+118*c^4*x^4-71*c^2*x^2+16)/c*(-c^2*x^2+1)^(1/2)*x^4+64*a*b*(-d*(c^2*x^2- 
1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)*c^2*arccos 
(c*x)*x^7-8/3*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^5/(c^2*x 
^2-1)*ln((c*x+I*(-c^2*x^2+1)^(1/2))^2-1)-13*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3 
/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^3*x^2*(-c^2*x^2+1)^(1 
/2)+a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^3/c^5/(c^2*x^2-1)*arcc 
os(c*x)^2-32*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4 
*x^4-71*c^2*x^2+16)/c^4*arccos(c*x)*x-40/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^ 
3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)/c^2*x^3-16/3*I*a*b*(-d 
*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+118*c^4*x^4-71*c^2*x^2+16)* 
c^2*x^7-16/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/d^3/(24*c^8*x^8-87*c^6*x^6+11...
 

Fricas [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="frica 
s")
 

Output:

integral(-(b^2*x^4*arccos(c*x)^2 + 2*a*b*x^4*arccos(c*x) + a^2*x^4)*sqrt(- 
c^2*d*x^2 + d)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

Sympy [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**4*(a+b*acos(c*x))**2/(-c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral(x**4*(a + b*acos(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxim 
a")
 

Output:

1/3*(x*(3*x^2/((-c^2*d*x^2 + d)^(3/2)*c^2*d) - 2/((-c^2*d*x^2 + d)^(3/2)*c 
^4*d)) - x/(sqrt(-c^2*d*x^2 + d)*c^4*d^2) + 3*arcsin(c*x)/(c^5*d^(5/2)))*a 
^2 - sqrt(d)*integrate((b^2*x^4*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) 
^2 + 2*a*b*x^4*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*s 
qrt(-c*x + 1)/(c^6*d^3*x^6 - 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 - d^3), x)
 

Giac [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arccos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac" 
)
 

Output:

integrate((b*arccos(c*x) + a)^2*x^4/(-c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \] Input:

int((x^4*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(5/2),x)
 

Output:

int((x^4*(a + b*acos(c*x))^2)/(d - c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \arccos (c x))^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2} c^{2} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2}+6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{7} x^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right ) x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) a b \,c^{5}+3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{7} x^{2}-3 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {acos} \left (c x \right )^{2} x^{4}}{\sqrt {-c^{2} x^{2}+1}\, c^{4} x^{4}-2 \sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c^{5}-4 a^{2} c^{3} x^{3}+3 a^{2} c x}{3 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} \left (c^{2} x^{2}-1\right )} \] Input:

int(x^4*(a+b*acos(c*x))^2/(-c^2*d*x^2+d)^(5/2),x)
 

Output:

(3*sqrt( - c**2*x**2 + 1)*asin(c*x)*a**2*c**2*x**2 - 3*sqrt( - c**2*x**2 + 
 1)*asin(c*x)*a**2 + 6*sqrt( - c**2*x**2 + 1)*int((acos(c*x)*x**4)/(sqrt( 
- c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - 
c**2*x**2 + 1)),x)*a*b*c**7*x**2 - 6*sqrt( - c**2*x**2 + 1)*int((acos(c*x) 
*x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 + 1)*c**2*x 
**2 + sqrt( - c**2*x**2 + 1)),x)*a*b*c**5 + 3*sqrt( - c**2*x**2 + 1)*int(( 
acos(c*x)**2*x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2*sqrt( - c**2*x**2 
 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b**2*c**7*x**2 - 3*sqrt( - c* 
*2*x**2 + 1)*int((acos(c*x)**2*x**4)/(sqrt( - c**2*x**2 + 1)*c**4*x**4 - 2 
*sqrt( - c**2*x**2 + 1)*c**2*x**2 + sqrt( - c**2*x**2 + 1)),x)*b**2*c**5 - 
 4*a**2*c**3*x**3 + 3*a**2*c*x)/(3*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**5*d** 
2*(c**2*x**2 - 1))