3.1.96 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \arcsin (c x))}{x} \, dx\) [96]

3.1.96.1 Optimal result

Integrand size = 27, antiderivative size = 361 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x} \, dx=-\frac {23 b c d^2 x \sqrt {d-c^2 d x^2}}{15 \sqrt {1-c^2 x^2}}+\frac {11 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2}}{45 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d-c^2 d x^2}}{25 \sqrt {1-c^2 x^2}}+d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}+\frac {i b d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}} \]

output

1/3*d*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))+1/5*(-c^2*d*x^2+d)^(5/2)*(a+b 
*arcsin(c*x))+d^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)-23/15*b*c*d^2*x*( 
-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+11/45*b*c^3*d^2*x^3*(-c^2*d*x^2+d)^ 
(1/2)/(-c^2*x^2+1)^(1/2)-1/25*b*c^5*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2 
+1)^(1/2)-2*d^2*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2* 
d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+I*b*d^2*polylog(2,-I*c*x-(-c^2*x^2+1)^(1 
/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b*d^2*polylog(2,I*c*x+(-c^2 
*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
 

3.1.96.2 Mathematica [A] (verified)

Time = 1.36 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.09 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))}{x} \, dx=\frac {1}{15} a d^2 \sqrt {d-c^2 d x^2} \left (23-11 c^2 x^2+3 c^4 x^4\right )+a d^{5/2} \log (x)-a d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {1-c^2 x^2} \arcsin (c x)+\arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-\arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {1-c^2 x^2}}-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (9 c x-3 \arcsin (c x) \left (3 \sqrt {1-c^2 x^2}+\cos (3 \arcsin (c x))\right )+\sin (3 \arcsin (c x))\right )}{18 \sqrt {1-c^2 x^2}}+\frac {b d^2 \sqrt {d-c^2 d x^2} \left (450 c x-15 \arcsin (c x) \left (30 \sqrt {1-c^2 x^2}+5 \cos (3 \arcsin (c x))-3 \cos (5 \arcsin (c x))\right )+25 \sin (3 \arcsin (c x))-9 \sin (5 \arcsin (c x))\right )}{3600 \sqrt {1-c^2 x^2}} \]

input

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

output

(a*d^2*Sqrt[d - c^2*d*x^2]*(23 - 11*c^2*x^2 + 3*c^4*x^4))/15 + a*d^(5/2)*L 
og[x] - a*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*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*ArcSin[c*x])]))/Sqrt[1 - c^2*x^2] - 
 (b*d^2*Sqrt[d - c^2*d*x^2]*(9*c*x - 3*ArcSin[c*x]*(3*Sqrt[1 - c^2*x^2] + 
Cos[3*ArcSin[c*x]]) + Sin[3*ArcSin[c*x]]))/(18*Sqrt[1 - c^2*x^2]) + (b*d^2 
*Sqrt[d - c^2*d*x^2]*(450*c*x - 15*ArcSin[c*x]*(30*Sqrt[1 - c^2*x^2] + 5*C 
os[3*ArcSin[c*x]] - 3*Cos[5*ArcSin[c*x]]) + 25*Sin[3*ArcSin[c*x]] - 9*Sin[ 
5*ArcSin[c*x]]))/(3600*Sqrt[1 - c^2*x^2])
 

3.1.96.3 Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 315, normalized size of antiderivative = 0.87, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5202, 210, 2009, 5202, 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.

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

\(\Big \downarrow \) 5202

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

\(\Big \downarrow \) 210

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5202

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5198

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

\(\Big \downarrow \) 24

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

\(\Big \downarrow \) 5218

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

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

output

-1/5*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x - (2*c^2*x^3)/3 + (c^4*x^5)/5))/Sqrt[ 
1 - c^2*x^2] + ((d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/5 + d*(-1/3*(b* 
c*d*Sqrt[d - c^2*d*x^2]*(x - (c^2*x^3)/3))/Sqrt[1 - c^2*x^2] + ((d - c^2*d 
*x^2)^(3/2)*(a + b*ArcSin[c*x]))/3 + d*(-((b*c*x*Sqrt[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 
]))
 

3.1.96.3.1 Defintions of rubi rules used

rule 24

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

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, 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 5202

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 + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 
2*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*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] && 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]
 

3.1.96.4 Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 652, normalized size of antiderivative = 1.81

method	result	size
default	\(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a \,d^{2} \sqrt {-c^{2} d \,x^{2}+d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{6} c^{6}}{5 c^{2} x^{2}-5}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{4} c^{4}}{15 \left (c^{2} x^{2}-1\right )}+\frac {34 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{2} c^{2}}{15 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}}{25 c^{2} x^{2}-25}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{45 \left (c^{2} x^{2}-1\right )}+\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x c}{15 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right )}{15 \left (c^{2} x^{2}-1\right )}\)	\(652\)
parts	\(\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} a}{5}+\frac {a d \left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}-a \,d^{\frac {5}{2}} \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )+a \,d^{2} \sqrt {-c^{2} d \,x^{2}+d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{6} c^{6}}{5 c^{2} x^{2}-5}-\frac {14 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{4} c^{4}}{15 \left (c^{2} x^{2}-1\right )}+\frac {34 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right ) x^{2} c^{2}}{15 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{5} c^{5}}{25 c^{2} x^{2}-25}-\frac {11 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x^{3} c^{3}}{45 \left (c^{2} x^{2}-1\right )}+\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \sqrt {-c^{2} x^{2}+1}\, x c}{15 \left (c^{2} x^{2}-1\right )}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, d^{2} \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{c^{2} x^{2}-1}-\frac {23 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, d^{2} \arcsin \left (c x \right )}{15 \left (c^{2} x^{2}-1\right )}\)	\(652\)

input

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

output

1/5*(-c^2*d*x^2+d)^(5/2)*a+1/3*a*d*(-c^2*d*x^2+d)^(3/2)-a*d^(5/2)*ln((2*d+ 
2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)+a*d^2*(-c^2*d*x^2+d)^(1/2)-I*b*(-d*(c^2 
*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d^2*polylog(2,-I*c*x-(-c^2*x 
^2+1)^(1/2))+I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d^2 
*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))+1/5*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2 
*x^2-1)*arcsin(c*x)*x^6*c^6-14/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1) 
*arcsin(c*x)*x^4*c^4+34/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin 
(c*x)*x^2*c^2+1/25*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*(-c^2*x^2+1)^( 
1/2)*x^5*c^5-11/45*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*(-c^2*x^2+1)^( 
1/2)*x^3*c^3+23/15*b*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*(-c^2*x^2+1)^( 
1/2)*x*c+b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*d^2*arcsi 
n(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1 
)^(1/2)/(c^2*x^2-1)*d^2*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-23/15*b 
*(-d*(c^2*x^2-1))^(1/2)*d^2/(c^2*x^2-1)*arcsin(c*x)
 

3.1.96.5 Fricas [F]

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

input

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

3.1.96.6 Sympy [F(-1)]

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

input

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

output

Timed out
 

3.1.96.7 Maxima [F]

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

input

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x))/x,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, x) - 1/15*(15*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) - 5*(-c^2*d*x^2 + d)^(3/2)*d - 15*sqrt(-c^2*d*x^2 + d)*d^2)* 
a
 

3.1.96.8 Giac [F(-2)]

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

input

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

3.1.96.9 Mupad [F(-1)]

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

input

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

output

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