\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))}{x^5} \, dx\) [98]

Optimal result

Integrand size = 27, antiderivative size = 389 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^5} \, dx=-\frac {b c d^2 \sqrt {d-c^2 d x^2}}{12 x^3 \sqrt {1-c^2 x^2}}+\frac {9 b c^3 d^2 \sqrt {d-c^2 d x^2}}{8 x \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}+\frac {15}{8} c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {5 c^2 d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{8 x^2}-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{4 x^4}-\frac {15 c^4 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{4 \sqrt {1-c^2 x^2}}+\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{8 \sqrt {1-c^2 x^2}}-\frac {15 i b c^4 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{8 \sqrt {1-c^2 x^2}} \] Output:

-1/12*b*c*d^2*(-c^2*d*x^2+d)^(1/2)/x^3/(-c^2*x^2+1)^(1/2)+9/8*b*c^3*d^2*(- 
c^2*d*x^2+d)^(1/2)/x/(-c^2*x^2+1)^(1/2)-b*c^5*d^2*x*(-c^2*d*x^2+d)^(1/2)/( 
-c^2*x^2+1)^(1/2)+15/8*c^4*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))+5/8* 
c^2*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))/x^2-1/4*(-c^2*d*x^2+d)^(5/2)* 
(a+b*arcsin(c*x))/x^4-15/4*c^4*d^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))* 
arctanh(I*c*x+(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)+15/8*I*b*c^4*d^2*(-c^ 
2*d*x^2+d)^(1/2)*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))/(-c^2*x^2+1)^(1/2)-1 
5/8*I*b*c^4*d^2*(-c^2*d*x^2+d)^(1/2)*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))/( 
-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 4.43 (sec) , antiderivative size = 640, normalized size of antiderivative = 1.65 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^5} \, dx=\frac {a d^2 \sqrt {d-c^2 d x^2} \left (-2+9 c^2 x^2+8 c^4 x^4\right )}{8 x^4}+\frac {15}{8} a c^4 d^{5/2} \log (x)-\frac {15}{8} a c^4 d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b c^4 d^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \arcsin (c x)+\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b c^4 d^3 \sqrt {1-c^2 x^2} \left (-2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+4 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{4 \sqrt {d-c^2 d x^2}}+\frac {b c^4 d^2 \sqrt {d-c^2 d x^2} \left (8 \cot \left (\frac {1}{2} \arcsin (c x)\right )+6 \arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-c x \csc ^4\left (\frac {1}{2} \arcsin (c x)\right )-3 \arcsin (c x) \csc ^4\left (\frac {1}{2} \arcsin (c x)\right )-24 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+24 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-24 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+24 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-6 \arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )+3 \arcsin (c x) \sec ^4\left (\frac {1}{2} \arcsin (c x)\right )-\frac {16 \sin ^4\left (\frac {1}{2} \arcsin (c x)\right )}{c^3 x^3}+8 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{192 \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(a*d^2*Sqrt[d - c^2*d*x^2]*(-2 + 9*c^2*x^2 + 8*c^4*x^4))/(8*x^4) + (15*a*c 
^4*d^(5/2)*Log[x])/8 - (15*a*c^4*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^ 
2]])/8 + (b*c^4*d^2*Sqrt[d - c^2*d*x^2]*(-(c*x) + Sqrt[1 - c^2*x^2]*ArcSin 
[c*x] + ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - ArcSin[c*x]*Log[1 + E^(I* 
ArcSin[c*x])] + I*PolyLog[2, -E^(I*ArcSin[c*x])] - I*PolyLog[2, E^(I*ArcSi 
n[c*x])]))/Sqrt[1 - c^2*x^2] - (b*c^4*d^3*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*A 
rcSin[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]*S 
ec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/(4*Sqrt[d - c^2*d*x^2]) + (b* 
c^4*d^2*Sqrt[d - c^2*d*x^2]*(8*Cot[ArcSin[c*x]/2] + 6*ArcSin[c*x]*Csc[ArcS 
in[c*x]/2]^2 - c*x*Csc[ArcSin[c*x]/2]^4 - 3*ArcSin[c*x]*Csc[ArcSin[c*x]/2] 
^4 - 24*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 24*ArcSin[c*x]*Log[1 + E^ 
(I*ArcSin[c*x])] - (24*I)*PolyLog[2, -E^(I*ArcSin[c*x])] + (24*I)*PolyLog[ 
2, E^(I*ArcSin[c*x])] - 6*ArcSin[c*x]*Sec[ArcSin[c*x]/2]^2 + 3*ArcSin[c*x] 
*Sec[ArcSin[c*x]/2]^4 - (16*Sin[ArcSin[c*x]/2]^4)/(c^3*x^3) + 8*Tan[ArcSin 
[c*x]/2]))/(192*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.82 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.85, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {5200, 244, 2009, 5200, 244, 2009, 5198, 24, 5218, 3042, 4671, 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.

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5200

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

\(\Big \downarrow \) 244

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5198

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

\(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (\frac {\sqrt {d-c^2 d x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (\frac {\sqrt {d-c^2 d x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {5}{4} c^2 d \left (-\frac {3}{2} c^2 d \left (\frac {\sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}+\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x \sqrt {d-c^2 d x^2}}{\sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))}{2 x^2}+\frac {b c d \left (c^2 (-x)-\frac {1}{x}\right ) \sqrt {d-c^2 d x^2}}{2 \sqrt {1-c^2 x^2}}\right )-\frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{4 x^4}+\frac {b c d^2 \left (c^4 x+\frac {2 c^2}{x}-\frac {1}{3 x^3}\right ) \sqrt {d-c^2 d x^2}}{4 \sqrt {1-c^2 x^2}}\)

Input:

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

Output:

(b*c*d^2*(-1/3*1/x^3 + (2*c^2)/x + c^4*x)*Sqrt[d - c^2*d*x^2])/(4*Sqrt[1 - 
 c^2*x^2]) - ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/(4*x^4) - (5*c^2* 
d*((b*c*d*(-x^(-1) - c^2*x)*Sqrt[d - c^2*d*x^2])/(2*Sqrt[1 - c^2*x^2]) - ( 
(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(2*x^2) - (3*c^2*d*(-((b*c*x*Sq 
rt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2]) + Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[ 
c*x]) + (Sqrt[d - c^2*d*x^2]*(-2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c 
*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c* 
x])]))/Sqrt[1 - c^2*x^2]))/2))/4
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

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 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 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]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 529, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{4 d \,x^{4}}+\frac {3 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 d \,x^{2}}+\frac {3 a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8}+\frac {5 a \,c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {15 a \,c^{4} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}+\frac {15 a \,c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}{8}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right ) c^{4} d^{2}}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) c^{4} d^{2}}{2 c^{2} x^{2}-2}+\frac {d^{2} \left (27 c^{4} x^{4} \arcsin \left (c x \right )-27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c^{2} x^{2} \arcsin \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}+6 \arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{24 \left (c^{2} x^{2}-1\right ) x^{4}}-\frac {15 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{4} d^{2}}{8 c^{2} x^{2}-8}\right )\)	\(529\)
parts	\(-\frac {a \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{4 d \,x^{4}}+\frac {3 a \,c^{2} \left (-c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{8 d \,x^{2}}+\frac {3 a \,c^{4} \left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{8}+\frac {5 a \,c^{4} d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{8}-\frac {15 a \,c^{4} d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{8}+\frac {15 a \,c^{4} d^{2} \sqrt {-c^{2} d \,x^{2}+d}}{8}+b \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (c^{2} x^{2}-i \sqrt {-c^{2} x^{2}+1}\, c x -1\right ) \left (\arcsin \left (c x \right )+i\right ) c^{4} d^{2}}{2 c^{2} x^{2}-2}+\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \sqrt {-c^{2} x^{2}+1}\, c x +c^{2} x^{2}-1\right ) \left (\arcsin \left (c x \right )-i\right ) c^{4} d^{2}}{2 c^{2} x^{2}-2}+\frac {d^{2} \left (27 c^{4} x^{4} \arcsin \left (c x \right )-27 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-33 c^{2} x^{2} \arcsin \left (c x \right )+2 c x \sqrt {-c^{2} x^{2}+1}+6 \arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{24 \left (c^{2} x^{2}-1\right ) x^{4}}-\frac {15 i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{4} d^{2}}{8 c^{2} x^{2}-8}\right )\)	\(529\)

Input:

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

Output:

-1/4*a/d/x^4*(-c^2*d*x^2+d)^(7/2)+3/8*a*c^2/d/x^2*(-c^2*d*x^2+d)^(7/2)+3/8 
*a*c^4*(-c^2*d*x^2+d)^(5/2)+5/8*a*c^4*d*(-c^2*d*x^2+d)^(3/2)-15/8*a*c^4*d^ 
(5/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+15/8*a*c^4*d^2*(-c^2*d*x^ 
2+d)^(1/2)+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^4*d^2/(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^4*d^2/(c^2*x^2-1)+1/24*d^2 
*(27*c^4*x^4*arcsin(c*x)-27*c^3*x^3*(-c^2*x^2+1)^(1/2)-33*c^2*x^2*arcsin(c 
*x)+2*c*x*(-c^2*x^2+1)^(1/2)+6*arcsin(c*x))*(-d*(c^2*x^2-1))^(1/2)/(c^2*x^ 
2-1)/x^4-15*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))+p 
olylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))*c 
^4*d^2/(8*c^2*x^2-8))
 

Fricas [F]

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

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^5,x, algorithm="fricas" 
)
 

Output:

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c 
^2*d^2*x^2 + b*d^2)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d)/x^5, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^5,x, algorithm="maxima" 
)
 

Output:

b*sqrt(d)*integrate((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(c*x + 1)*sqrt 
(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x^5, x) - 1/8*(15*c^ 
4*d^(5/2)*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x)) - 3*(-c^ 
2*d*x^2 + d)^(5/2)*c^4 - 5*(-c^2*d*x^2 + d)^(3/2)*c^4*d - 15*sqrt(-c^2*d*x 
^2 + d)*c^4*d^2 - 3*(-c^2*d*x^2 + d)^(7/2)*c^2/(d*x^2) + 2*(-c^2*d*x^2 + d 
)^(7/2)/(d*x^4))*a
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^5} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x^5,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. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^5} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^5} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x^5} \, dx=\frac {\sqrt {d}\, d^{2} \left (8 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} x^{4}+9 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} x^{2}-2 \sqrt {-c^{2} x^{2}+1}\, a +8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{5}}d x \right ) b \,x^{4}-16 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x^{3}}d x \right ) b \,c^{2} x^{4}+8 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{x}d x \right ) b \,c^{4} x^{4}+15 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a \,c^{4} x^{4}-10 a \,c^{4} x^{4}\right )}{8 x^{4}} \] Input:

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

Output:

(sqrt(d)*d**2*(8*sqrt( - c**2*x**2 + 1)*a*c**4*x**4 + 9*sqrt( - c**2*x**2 
+ 1)*a*c**2*x**2 - 2*sqrt( - c**2*x**2 + 1)*a + 8*int((sqrt( - c**2*x**2 + 
 1)*asin(c*x))/x**5,x)*b*x**4 - 16*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/ 
x**3,x)*b*c**2*x**4 + 8*int((sqrt( - c**2*x**2 + 1)*asin(c*x))/x,x)*b*c**4 
*x**4 + 15*log(tan(asin(c*x)/2))*a*c**4*x**4 - 10*a*c**4*x**4))/(8*x**4)