\(\int \frac {(d-c^2 d x^2)^{3/2} (a+b \arcsin (c x))^2}{x^3} \, dx\) [221]

Optimal result

Integrand size = 29, antiderivative size = 590 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^3} \, dx=2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \arcsin (c x)}{\sqrt {1-c^2 x^2}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x \sqrt {1-c^2 x^2}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2 \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {3 b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \] Output:

2*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)+3*a*b*c^3*d*x*(-c^2*d*x^2+d)^(1/2)/(-c^2* 
x^2+1)^(1/2)+3*b^2*c^3*d*x*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/(-c^2*x^2+1)^( 
1/2)-b*c*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x/(-c^2*x^2+1)^(1/2)-b*c 
^3*d*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2)-3/2*c^2*d 
*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2-1/2*(-c^2*d*x^2+d)^(3/2)*(a+b*ar 
csin(c*x))^2/x^2+3*c^2*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2*arctanh( 
I*c*x+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-b^2*c^2*d*(-c^2*d*x^2+d)^(1/2 
)*arctanh((-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-3*I*b*c^2*d*(-c^2*d*x^2+d 
)^(1/2)*(a+b*arcsin(c*x))*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1 
)^(1/2)+3*I*b*c^2*d*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))*polylog(2,I*c*x 
+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)+3*b^2*c^2*d*(-c^2*d*x^2+d)^(1/2)*p 
olylog(3,-I*c*x-(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-3*b^2*c^2*d*(-c^2*d 
*x^2+d)^(1/2)*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 7.00 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.45 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(-(a^2*c^2*d) - (a^2*d)/(2*x^2))*Sqrt[-(d*(-1 + c^2*x^2))] - (3*a^2*c^2*d^ 
(3/2)*Log[x])/2 + (3*a^2*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt[-(d*(-1 + c^2*x^ 
2))]])/2 - 2*a*b*c^2*d*Sqrt[d*(1 - c^2*x^2)]*(-((c*x)/Sqrt[1 - c^2*x^2]) + 
 ArcSin[c*x] + (ArcSin[c*x]*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*Arc 
Sin[c*x])]))/Sqrt[1 - c^2*x^2] + (I*(PolyLog[2, -E^(I*ArcSin[c*x])] - Poly 
Log[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) - b^2*c^2*d*Sqrt[d*(1 - c^2 
*x^2)]*(-2 - (2*c*x*ArcSin[c*x])/Sqrt[1 - c^2*x^2] + ArcSin[c*x]^2 + (ArcS 
in[c*x]^2*(Log[1 - E^(I*ArcSin[c*x])] - Log[1 + E^(I*ArcSin[c*x])]))/Sqrt[ 
1 - c^2*x^2] + ((2*I)*ArcSin[c*x]*(PolyLog[2, -E^(I*ArcSin[c*x])] - PolyLo 
g[2, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] + (2*(-PolyLog[3, -E^(I*ArcSin 
[c*x])] + PolyLog[3, E^(I*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2]) + (a*b*c^2*d^ 
2*Sqrt[1 - c^2*x^2]*(-2*Cot[ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2 
]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^( 
I*ArcSin[c*x])] - (4*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, 
E^(I*ArcSin[c*x])] + ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/ 
2]))/(4*Sqrt[d*(1 - c^2*x^2)]) + (b^2*c^2*d^2*Sqrt[1 - c^2*x^2]*(-4*ArcSin 
[c*x]*Cot[ArcSin[c*x]/2] - ArcSin[c*x]^2*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c 
*x]^2*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]^2*Log[1 + E^(I*ArcSin[c*x 
])] + 8*Log[Tan[ArcSin[c*x]/2]] - (8*I)*ArcSin[c*x]*PolyLog[2, -E^(I*ArcSi 
n[c*x])] + (8*I)*ArcSin[c*x]*PolyLog[2, E^(I*ArcSin[c*x])] + 8*PolyLog[...
 

Rubi [A] (verified)

Time = 2.03 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.64, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {5200, 5192, 25, 354, 90, 73, 221, 5198, 2009, 5218, 3042, 4671, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 5192

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 90

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5198

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

-1/2*((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/x^2 + (b*c*d*Sqrt[d - c 
^2*d*x^2]*(-((a + b*ArcSin[c*x])/x) - c^2*x*(a + b*ArcSin[c*x]) + (b*c*(-2 
*Sqrt[1 - c^2*x^2] - 2*ArcTanh[Sqrt[1 - c^2*x^2]]))/2))/Sqrt[1 - c^2*x^2] 
- (3*c^2*d*(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2 - (2*b*c*Sqrt[d - c^ 
2*d*x^2]*(a*x + (b*Sqrt[1 - c^2*x^2])/c + b*x*ArcSin[c*x]))/Sqrt[1 - c^2*x 
^2] + (Sqrt[d - c^2*d*x^2]*(-2*(a + b*ArcSin[c*x])^2*ArcTanh[E^(I*ArcSin[c 
*x])] + 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, -E^(I*ArcSin[c*x])] - b*Poly 
Log[3, -E^(I*ArcSin[c*x])]) - 2*b*(I*(a + b*ArcSin[c*x])*PolyLog[2, E^(I*A 
rcSin[c*x])] - b*PolyLog[3, E^(I*ArcSin[c*x])])))/Sqrt[1 - c^2*x^2]))/2
 

Defintions of rubi rules used

rule 25

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

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 90

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

rule 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 2009

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

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

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

rule 4671

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 
2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5192

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_) 
^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp[ 
(a + b*ArcSin[c*x])   u, x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 - c 
^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0 
] && IGtQ[p, 0]
 

rule 5198

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
(e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS 
in[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt[1 
 - c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x 
] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]]   Int[ 
(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, 
 f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 5200

Int[((a_.) + ArcSin[(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*ArcS 
in[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*ArcSin[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*ArcSin[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 5218

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

rule 7143

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

Maple [A] (verified)

Time = 0.83 (sec) , antiderivative size = 884, normalized size of antiderivative = 1.50

method	result	size
default	\(a^{2} \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}-\frac {3 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {d \left (c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {d \left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2} d}{2 c^{2} x^{2}-2}\right )\)	\(884\)
parts	\(a^{2} \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{2 d \,x^{2}}-\frac {3 c^{2} \left (\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )\right )}{2}\right )+b^{2} \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )^{2}-2+2 i \arcsin \left (c x \right )\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )^{2}-2-2 i \arcsin \left (c x \right )\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {d \left (c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \arcsin \left (c x \right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (3 \arcsin \left (c x \right )^{2} \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-3 \arcsin \left (c x \right )^{2} \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )+6 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )-4 \,\operatorname {arctanh}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+6 \operatorname {polylog}\left (3, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-6 \operatorname {polylog}\left (3, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}\right )+2 a b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) c^{2} d}{2 \left (c^{2} x^{2}-1\right )}-\frac {d \left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 \left (c^{2} x^{2}-1\right ) x^{2}}+\frac {3 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2} d}{2 c^{2} x^{2}-2}\right )\)	\(884\)

Input:

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

Output:

a^2*(-1/2/d/x^2*(-c^2*d*x^2+d)^(5/2)-3/2*c^2*(1/3*(-c^2*d*x^2+d)^(3/2)+d*( 
(-c^2*d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)))) 
+b^2*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*(ar 
csin(c*x)^2-2+2*I*arcsin(c*x))*c^2*d/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2 
)*(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(arcsin(c*x)^2-2-2*I*arcsin(c*x))*c 
^2*d/(c^2*x^2-1)-1/2*d*(c^2*x^2*arcsin(c*x)-2*c*x*(-c^2*x^2+1)^(1/2)-arcsi 
n(c*x))*arcsin(c*x)*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/x^2-1/2*(-d*(c^2*x^ 
2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(3*arcsin(c*x)^2*ln(1+I*c*x+(-c^2*x^2+1)^(1 
/2))-3*arcsin(c*x)^2*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-6*I*arcsin(c*x)*polylo 
g(2,-I*c*x-(-c^2*x^2+1)^(1/2))+6*I*arcsin(c*x)*polylog(2,I*c*x+(-c^2*x^2+1 
)^(1/2))-4*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))+6*polylog(3,-I*c*x-(-c^2*x^2+ 
1)^(1/2))-6*polylog(3,I*c*x+(-c^2*x^2+1)^(1/2)))*c^2*d/(c^2*x^2-1))+2*a*b* 
(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*c*x*(-c^2*x^2+1)^(1/2)-1)*(arcsin( 
c*x)+I)*c^2*d/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2) 
*c*x+c^2*x^2-1)*(arcsin(c*x)-I)*c^2*d/(c^2*x^2-1)-1/2*d*(c^2*x^2*arcsin(c* 
x)-c*x*(-c^2*x^2+1)^(1/2)-arcsin(c*x))*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^2-1)/ 
x^2+3*I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arcsin(c*x)*ln(1+I*c* 
x+(-c^2*x^2+1)^(1/2))-I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+polylog 
(2,-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))*c^2*d/( 
2*c^2*x^2-2))
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="frica 
s")
                                                                                    
                                                                                    
 

Output:

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

Sympy [F]

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

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

Output:

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*asin(c*x))**2/x**3, x)
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))^2/x^3,x, algorithm="maxim 
a")
 

Output:

1/2*(3*c^2*d^(3/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) 
 - (-c^2*d*x^2 + d)^(3/2)*c^2 - 3*sqrt(-c^2*d*x^2 + d)*c^2*d - (-c^2*d*x^2 
 + d)^(5/2)/(d*x^2))*a^2 - sqrt(d)*integrate(((b^2*c^2*d*x^2 - b^2*d)*arct 
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d*x^2 - a*b*d)*arcta 
n2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/x^3, x 
)
 

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))^2}{x^3} \, dx=\frac {\sqrt {d}\, d \left (-8 \sqrt {-c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-4 \sqrt {-c^{2} x^{2}+1}\, a^{2}+16 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{3}}d x \right ) a b \,x^{2}-16 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}-12 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2} c^{2} x^{2}+9 a^{2} c^{2} x^{2}\right )}{8 x^{2}} \] Input:

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

Output:

(sqrt(d)*d*( - 8*sqrt( - c**2*x**2 + 1)*a**2*c**2*x**2 - 4*sqrt( - c**2*x* 
*2 + 1)*a**2 + 16*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/x**3,x)*a*b*x**2 
- 16*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/x,x)*a*b*c**2*x**2 + 8*int((sq 
rt( - c**2*x**2 + 1)*asin(c*x)**2)/x**3,x)*b**2*x**2 - 8*int((sqrt( - c**2 
*x**2 + 1)*asin(c*x)**2)/x,x)*b**2*c**2*x**2 - 12*log(tan(asin(c*x)/2))*a* 
*2*c**2*x**2 + 9*a**2*c**2*x**2))/(8*x**2)