\(\int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx\) [587]

Optimal result

Integrand size = 35, antiderivative size = 505 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\frac {1}{4} b^2 c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {5 b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)}{4 \sqrt {1-c^2 x^2}}+\frac {3 b c^3 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))}{2 \sqrt {1-c^2 x^2}}+b c d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2} (a+b \arccos (c x))-\frac {3}{2} c^2 d e x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2-\frac {i c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^2}{\sqrt {1-c^2 x^2}}-\frac {d e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{x}-\frac {c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x))^3}{2 b \sqrt {1-c^2 x^2}}+\frac {2 b c d e \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arccos (c x)) \log \left (1-e^{2 i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b^2 c d e \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,e^{2 i \arccos (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 2.74 (sec) , antiderivative size = 538, normalized size of antiderivative = 1.07 \[ \int \frac {(d+c d x)^{3/2} (e-c e x)^{3/2} (a+b \arccos (c x))^2}{x^2} \, dx=\frac {-8 a^2 d e \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}-4 a^2 c^2 d e x^2 \sqrt {d+c d x} \sqrt {e-c e x} \sqrt {1-c^2 x^2}+4 b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^3+12 a^2 c d^{3/2} e^{3/2} x \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-2 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x} \cos (2 \arccos (c x))-16 a b c d e x \sqrt {d+c d x} \sqrt {e-c e x} \log (c x)+8 i b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )+b^2 c d e x \sqrt {d+c d x} \sqrt {e-c e x} \sin (2 \arccos (c x))-2 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x) \left (8 a \sqrt {1-c^2 x^2}+b c x \cos (2 \arccos (c x))+8 b c x \log \left (1+e^{2 i \arccos (c x)}\right )+2 a c x \sin (2 \arccos (c x))\right )-2 b d e \sqrt {d+c d x} \sqrt {e-c e x} \arccos (c x)^2 \left (-6 a c x-4 i b c x+4 b \sqrt {1-c^2 x^2}+b c x \sin (2 \arccos (c x))\right )}{8 x \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(-8*a^2*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] - 4*a^2*c^2* 
d*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sqrt[1 - c^2*x^2] + 4*b^2*c*d*e*x* 
Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcCos[c*x]^3 + 12*a^2*c*d^(3/2)*e^(3/2)*x 
*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*S 
qrt[e]*(-1 + c^2*x^2))] - 2*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Co 
s[2*ArcCos[c*x]] - 16*a*b*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Log[c*x] 
 + (8*I)*b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*PolyLog[2, -E^((2*I)* 
ArcCos[c*x])] + b^2*c*d*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Sin[2*ArcCos[c 
*x]] - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcCos[c*x]*(8*a*Sqrt[1 - c 
^2*x^2] + b*c*x*Cos[2*ArcCos[c*x]] + 8*b*c*x*Log[1 + E^((2*I)*ArcCos[c*x]) 
] + 2*a*c*x*Sin[2*ArcCos[c*x]]) - 2*b*d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]* 
ArcCos[c*x]^2*(-6*a*c*x - (4*I)*b*c*x + 4*b*Sqrt[1 - c^2*x^2] + b*c*x*Sin[ 
2*ArcCos[c*x]]))/(8*x*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.63, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {5239, 5201, 5157, 5139, 262, 223, 5153, 5189, 211, 223, 5137, 3042, 4202, 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.

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

\(\Big \downarrow \) 5239

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

\(\Big \downarrow \) 5201

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

\(\Big \downarrow \) 5157

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

\(\Big \downarrow \) 5139

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5153

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \int \frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))}{x}dx-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5189

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (\int \frac {a+b \arccos (c x)}{x}dx+\frac {1}{2} b c \int \sqrt {1-c^2 x^2}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (\int \frac {a+b \arccos (c x)}{x}dx+\frac {1}{2} b c \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (\int \frac {a+b \arccos (c x)}{x}dx+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5137

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (-\int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{c x}d\arccos (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (-\int (a+b \arccos (c x)) \tan (\arccos (c x))d\arccos (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 4202

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (2 i \int \frac {e^{2 i \arccos (c x)} (a+b \arccos (c x))}{1+e^{2 i \arccos (c x)}}d\arccos (c x)+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (2 i \left (\frac {1}{2} i b \int \log \left (1+e^{2 i \arccos (c x)}\right )d\arccos (c x)-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (2 i \left (\frac {1}{4} b \int e^{-2 i \arccos (c x)} \log \left (1+e^{2 i \arccos (c x)}\right )de^{2 i \arccos (c x)}-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))\right )+\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {d e \sqrt {c d x+d} \sqrt {e-c e x} \left (-2 b c \left (\frac {1}{2} \left (1-c^2 x^2\right ) (a+b \arccos (c x))+2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arccos (c x)}\right ) (a+b \arccos (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arccos (c x)}\right )\right )-\frac {i (a+b \arccos (c x))^2}{2 b}+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )\right )-3 c^2 \left (b c \left (\frac {1}{2} x^2 (a+b \arccos (c x))+\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2-\frac {(a+b \arccos (c x))^3}{6 b c}\right )-\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))^2}{x}\right )}{\sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(d*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-(((1 - c^2*x^2)^(3/2)*(a + b*ArcCos 
[c*x])^2)/x) - 3*c^2*((x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x])^2)/2 - (a + 
 b*ArcCos[c*x])^3/(6*b*c) + b*c*((x^2*(a + b*ArcCos[c*x]))/2 + (b*c*(-1/2* 
(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)) - 2*b*c*(((1 - c^2*x 
^2)*(a + b*ArcCos[c*x]))/2 - ((I/2)*(a + b*ArcCos[c*x])^2)/b + (b*c*((x*Sq 
rt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/2 + (2*I)*((-1/2*I)*(a + b*ArcCos 
[c*x])*Log[1 + E^((2*I)*ArcCos[c*x])] - (b*PolyLog[2, -E^((2*I)*ArcCos[c*x 
])])/4))))/Sqrt[1 - c^2*x^2]
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[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 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

rule 5137

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> -Subst[Int[ 
(a + b*x)^n*Tan[x], x], x, ArcCos[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0 
]
 

rule 5139

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

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 5157

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

rule 5189

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

rule 5201

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

rule 5239

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((h_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[((-d^2)*(g/e))^In 
tPart[q]*(d + e*x)^FracPart[q]*((f + g*x)^FracPart[q]/(1 - c^2*x^2)^FracPar 
t[q])   Int[(h*x)^m*(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcCos[c*x])^n 
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*f + d*g, 0] & 
& EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

Maple [A] (verified)

Time = 4.92 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.02

method	result	size
parts	\(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, d e \left (3 \arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}}\right ) x \,c^{2} d e +x^{2} c^{2} \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} d e}+2 \sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\right )}{2 \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} d e}\, x}-\frac {b^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-2 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-2 c^{3} x^{3} \arccos \left (c x \right )+2 \arccos \left (c x \right )^{3} c x +4 i \arccos \left (c x \right )^{2} x c +c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}-8 \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c +4 i \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c -4 \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}+c x \arccos \left (c x \right )\right ) d e}{4 \left (c^{2} x^{2}-1\right ) x}-\frac {a b \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (-4 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}-2 c^{3} x^{3}+6 \arccos \left (c x \right )^{2} c x +8 i \arccos \left (c x \right ) x c -8 \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c -8 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}+c x \right ) d e}{4 \left (c^{2} x^{2}-1\right ) x}\)	\(516\)
default	\(-\frac {a^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, d e \left (3 \arctan \left (\frac {\sqrt {c^{2} d e}\, x}{\sqrt {-d e \left (c^{2} x^{2}-1\right )}}\right ) x \,c^{2} d e +x^{2} c^{2} \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} d e}+2 \sqrt {c^{2} d e}\, \sqrt {-d e \left (c^{2} x^{2}-1\right )}\right )}{2 \sqrt {-d e \left (c^{2} x^{2}-1\right )}\, \sqrt {c^{2} d e}\, x}-\frac {i b^{2} \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (2 i \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}+2 i \arccos \left (c x \right ) x^{3} c^{3}-i \sqrt {-c^{2} x^{2}+1}\, x^{2} c^{2}-2 i \arccos \left (c x \right )^{3} x c +8 i \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) \arccos \left (c x \right ) x c +4 i \arccos \left (c x \right )^{2} \sqrt {-c^{2} x^{2}+1}-i \arccos \left (c x \right ) x c +4 \arccos \left (c x \right )^{2} c x +4 \operatorname {polylog}\left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c \right ) d e}{4 \left (c^{2} x^{2}-1\right ) x}+\frac {a b \sqrt {d \left (c x +1\right )}\, \sqrt {-e \left (c x -1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (4 \sqrt {-c^{2} x^{2}+1}\, \arccos \left (c x \right ) c^{2} x^{2}+2 c^{3} x^{3}-8 i \arccos \left (c x \right ) x c -6 \arccos \left (c x \right )^{2} c x +8 \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right ) x c +8 \arccos \left (c x \right ) \sqrt {-c^{2} x^{2}+1}-c x \right ) d e}{4 \left (c^{2} x^{2}-1\right ) x}\)	\(525\)

Input:

int((c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/x^2,x,method=_RET 
URNVERBOSE)
 

Output:

-1/2*a^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*d*e*(3*arctan((c^2*d*e)^(1/2 
)*x/(-d*e*(c^2*x^2-1))^(1/2))*x*c^2*d*e+x^2*c^2*(-d*e*(c^2*x^2-1))^(1/2)*( 
c^2*d*e)^(1/2)+2*(c^2*d*e)^(1/2)*(-d*e*(c^2*x^2-1))^(1/2))/(-d*e*(c^2*x^2- 
1))^(1/2)/(c^2*d*e)^(1/2)/x-1/4*b^2*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2)*( 
-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/x*(-2*arccos(c*x)^2*(-c^2*x^2+1)^(1/2)*x^2*c 
^2-2*c^3*x^3*arccos(c*x)+2*arccos(c*x)^3*c*x+4*I*arccos(c*x)^2*x*c+c^2*x^2 
*(-c^2*x^2+1)^(1/2)-8*arccos(c*x)*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))^2)*x*c+4 
*I*polylog(2,-(c*x+I*(-c^2*x^2+1)^(1/2))^2)*x*c-4*arccos(c*x)^2*(-c^2*x^2+ 
1)^(1/2)+c*x*arccos(c*x))*d*e-1/4*a*b*(d*(c*x+1))^(1/2)*(-e*(c*x-1))^(1/2) 
*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)/x*(-4*(-c^2*x^2+1)^(1/2)*arccos(c*x)*c^2*x 
^2-2*c^3*x^3+6*arccos(c*x)^2*c*x+8*I*arccos(c*x)*x*c-8*ln(1+(c*x+I*(-c^2*x 
^2+1)^(1/2))^2)*x*c-8*arccos(c*x)*(-c^2*x^2+1)^(1/2)+c*x)*d*e
 

Fricas [F]

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

integrate((c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/x^2,x, algo 
rithm="fricas")
                                                                                    
                                                                                    
 

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

integrate((c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/x^2,x, algo 
rithm="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>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

integrate((c*d*x+d)^(3/2)*(-c*e*x+e)^(3/2)*(a+b*arccos(c*x))^2/x^2,x, algo 
rithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*sqrt(d)*d*e*(6*asin(sqrt( - c*x + 1)/sqrt(2))*a**2*c*x - sqrt(c*x 
 + 1)*sqrt( - c*x + 1)*a**2*c**2*x**2 - 2*sqrt(c*x + 1)*sqrt( - c*x + 1)*a 
**2 + 4*int((sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x))/x**2,x)*a*b*x + 2*i 
nt((sqrt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x)**2)/x**2,x)*b**2*x - 4*int(sq 
rt(c*x + 1)*sqrt( - c*x + 1)*acos(c*x),x)*a*b*c**2*x - 2*int(sqrt(c*x + 1) 
*sqrt( - c*x + 1)*acos(c*x)**2,x)*b**2*c**2*x))/(2*x)