\(\int \frac {(\pi +c^2 \pi x^2)^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx\) [77]

Optimal result

Integrand size = 26, antiderivative size = 205 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {b c \pi ^{5/2}}{2 x}-\frac {7}{3} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-5 c^2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {5}{2} b c^2 \pi ^{5/2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{2} b c^2 \pi ^{5/2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]

[Out]

-1/2*b*c*Pi^(5/2)/x-7/3*b*c^3*Pi^(5/2)*x-1/9*b*c^5*Pi^(5/2)*x^3+5/6*c^2*Pi*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(
c*x))-1/2*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/x^2-5*c^2*Pi^(5/2)*(a+b*arcsinh(c*x))*arctanh(c*x+(c^2*x^2+
1)^(1/2))-5/2*b*c^2*Pi^(5/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))+5/2*b*c^2*Pi^(5/2)*polylog(2,c*x+(c^2*x^2+1)^(1
/2))+5/2*c^2*Pi^2*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5807, 5808, 5806, 5816, 4267, 2317, 2438, 8, 276} \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-5 \pi ^{5/2} c^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {5}{6} \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {5}{2} \pi ^2 c^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {\left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {5}{2} \pi ^{5/2} b c^2 \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{2} \pi ^{5/2} b c^2 \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )-\frac {1}{9} \pi ^{5/2} b c^5 x^3-\frac {7}{3} \pi ^{5/2} b c^3 x-\frac {\pi ^{5/2} b c}{2 x} \]

[In]

Int[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3,x]

[Out]

-1/2*(b*c*Pi^(5/2))/x - (7*b*c^3*Pi^(5/2)*x)/3 - (b*c^5*Pi^(5/2)*x^3)/9 + (5*c^2*Pi^2*Sqrt[Pi + c^2*Pi*x^2]*(a
+ b*ArcSinh[c*x]))/2 + (5*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/6 - ((Pi + c^2*Pi*x^2)^(5/2)*(
a + b*ArcSinh[c*x]))/(2*x^2) - 5*c^2*Pi^(5/2)*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]] - (5*b*c^2*Pi^(5/2)
*PolyLog[2, -E^ArcSinh[c*x]])/2 + (5*b*c^2*Pi^(5/2)*PolyLog[2, E^ArcSinh[c*x]])/2

Rule 8

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

Rule 276

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

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[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 2438

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 4267

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

Rule 5806

Int[((a_.) + ArcSinh[(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*ArcSinh[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]
/Sqrt[1 + c^2*x^2]], Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x] - Dist[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])

Rule 5807

Int[((a_.) + ArcSinh[(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*ArcSinh[c*x])^n/(f*(m + 1))), x] + (-Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x
)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b,
c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]

Rule 5808

Int[((a_.) + ArcSinh[(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*ArcSinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Dist[2*d*(p/(m + 2*p + 1)), Int
[(f*x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{
a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]

Rule 5816

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} \left (5 c^2 \pi \right ) \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{2} \left (b c \pi ^{5/2}\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^2} \, dx \\ & = \frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} \left (5 c^2 \pi ^2\right ) \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {1}{2} \left (b c \pi ^{5/2}\right ) \int \left (2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx-\frac {1}{6} \left (5 b c^3 \pi ^{5/2}\right ) \int \left (1+c^2 x^2\right ) \, dx \\ & = -\frac {b c \pi ^{5/2}}{2 x}+\frac {1}{6} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} \left (5 c^2 \pi ^{5/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\frac {1}{2} \left (5 b c^3 \pi ^{5/2}\right ) \int 1 \, dx \\ & = -\frac {b c \pi ^{5/2}}{2 x}-\frac {7}{3} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} \left (5 c^2 \pi ^{5/2}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x)) \\ & = -\frac {b c \pi ^{5/2}}{2 x}-\frac {7}{3} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-5 c^2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{2} \left (5 b c^2 \pi ^{5/2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\frac {1}{2} \left (5 b c^2 \pi ^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {b c \pi ^{5/2}}{2 x}-\frac {7}{3} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-5 c^2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {1}{2} \left (5 b c^2 \pi ^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\frac {1}{2} \left (5 b c^2 \pi ^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = -\frac {b c \pi ^{5/2}}{2 x}-\frac {7}{3} b c^3 \pi ^{5/2} x-\frac {1}{9} b c^5 \pi ^{5/2} x^3+\frac {5}{2} c^2 \pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-5 c^2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\frac {5}{2} b c^2 \pi ^{5/2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\frac {5}{2} b c^2 \pi ^{5/2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.49 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.70 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {\pi ^{5/2} \left (-168 b c^3 x^3-8 b c^5 x^5-36 a \sqrt {1+c^2 x^2}+168 a c^2 x^2 \sqrt {1+c^2 x^2}+24 a c^4 x^4 \sqrt {1+c^2 x^2}+168 b c^2 x^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+24 b c^4 x^4 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-9 b c^3 x^3 \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-9 b c^2 x^2 \text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+180 b c^2 x^2 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-180 b c^2 x^2 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+180 a c^2 x^2 \log (x)-180 a c^2 x^2 \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+180 b c^2 x^2 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-180 b c^2 x^2 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )+36 b c x \sinh ^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )-36 b \text {arcsinh}(c x) \sinh ^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{72 x^2} \]

[In]

Integrate[((Pi + c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/x^3,x]

[Out]

(Pi^(5/2)*(-168*b*c^3*x^3 - 8*b*c^5*x^5 - 36*a*Sqrt[1 + c^2*x^2] + 168*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 24*a*c^4*
x^4*Sqrt[1 + c^2*x^2] + 168*b*c^2*x^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 24*b*c^4*x^4*Sqrt[1 + c^2*x^2]*ArcSinh[
c*x] - 9*b*c^3*x^3*Csch[ArcSinh[c*x]/2]^2 - 9*b*c^2*x^2*ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 180*b*c^2*x^2*Ar
cSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 180*b*c^2*x^2*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + 180*a*c^2*x^2*
Log[x] - 180*a*c^2*x^2*Log[Pi*(1 + Sqrt[1 + c^2*x^2])] + 180*b*c^2*x^2*PolyLog[2, -E^(-ArcSinh[c*x])] - 180*b*
c^2*x^2*PolyLog[2, E^(-ArcSinh[c*x])] + 36*b*c*x*Sinh[ArcSinh[c*x]/2]^2 - 36*b*ArcSinh[c*x]*Sinh[ArcSinh[c*x]/
2]^2))/(72*x^2)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.70

method	result	size
default	\(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{2 \pi \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )}{2}\right )-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x^{2} c^{4}}{3}+\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{3}}{9}-\frac {7 b \,c^{3} \pi ^{\frac {5}{2}} x}{3}+\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b c \,\pi ^{\frac {5}{2}}}{2 x}-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right )}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {7 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} c^{2}}{3}\)	\(348\)
parts	\(a \left (-\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{2 \pi \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )}{2}\right )-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) c^{2}}{2 \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x^{2} c^{4}}{3}+\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{3}}{9}-\frac {7 b \,c^{3} \pi ^{\frac {5}{2}} x}{3}+\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {5 b \,c^{2} \pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )}{2}-\frac {b c \,\pi ^{\frac {5}{2}}}{2 x}-\frac {b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right )}{2 \sqrt {c^{2} x^{2}+1}\, x^{2}}+\frac {7 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} c^{2}}{3}\)	\(348\)

[In]

int((Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,x,method=_RETURNVERBOSE)

[Out]

a*(-1/2/Pi/x^2*(Pi*c^2*x^2+Pi)^(7/2)+5/2*c^2*(1/5*(Pi*c^2*x^2+Pi)^(5/2)+Pi*(1/3*(Pi*c^2*x^2+Pi)^(3/2)+Pi*((Pi*
c^2*x^2+Pi)^(1/2)-Pi^(1/2)*arctanh(Pi^(1/2)/(Pi*c^2*x^2+Pi)^(1/2))))))-1/2*b*Pi^(5/2)/(c^2*x^2+1)^(1/2)*arcsin
h(c*x)*c^2+1/3*b*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*Pi^(5/2)*x^2*c^4+5/2*b*c^2*Pi^(5/2)*arcsinh(c*x)*ln(1-c*x-(c^2
*x^2+1)^(1/2))-1/9*b*c^5*Pi^(5/2)*x^3-7/3*b*c^3*Pi^(5/2)*x+5/2*b*c^2*Pi^(5/2)*polylog(2,c*x+(c^2*x^2+1)^(1/2))
-5/2*b*c^2*Pi^(5/2)*polylog(2,-c*x-(c^2*x^2+1)^(1/2))-5/2*b*c^2*Pi^(5/2)*arcsinh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/
2))-1/2*b*c*Pi^(5/2)/x-1/2*b*Pi^(5/2)/(c^2*x^2+1)^(1/2)/x^2*arcsinh(c*x)+7/3*b*(c^2*x^2+1)^(1/2)*arcsinh(c*x)*
Pi^(5/2)*c^2

Fricas [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="fricas")

[Out]

integral(sqrt(pi + pi*c^2*x^2)*(pi^2*a*c^4*x^4 + 2*pi^2*a*c^2*x^2 + pi^2*a + (pi^2*b*c^4*x^4 + 2*pi^2*b*c^2*x^
2 + pi^2*b)*arcsinh(c*x))/x^3, x)

Sympy [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\pi ^{\frac {5}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x^{3}}\, dx + \int \frac {2 a c^{2} \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int a c^{4} x \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x^{3}}\, dx + \int \frac {2 b c^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int b c^{4} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]

[In]

integrate((pi*c**2*x**2+pi)**(5/2)*(a+b*asinh(c*x))/x**3,x)

[Out]

pi**(5/2)*(Integral(a*sqrt(c**2*x**2 + 1)/x**3, x) + Integral(2*a*c**2*sqrt(c**2*x**2 + 1)/x, x) + Integral(a*
c**4*x*sqrt(c**2*x**2 + 1), x) + Integral(b*sqrt(c**2*x**2 + 1)*asinh(c*x)/x**3, x) + Integral(2*b*c**2*sqrt(c
**2*x**2 + 1)*asinh(c*x)/x, x) + Integral(b*c**4*x*sqrt(c**2*x**2 + 1)*asinh(c*x), x))

Maxima [F]

\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]

[In]

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="maxima")

[Out]

-1/6*(15*pi^(5/2)*c^2*arcsinh(1/(c*abs(x))) - 15*pi^2*sqrt(pi + pi*c^2*x^2)*c^2 - 5*pi*(pi + pi*c^2*x^2)^(3/2)
*c^2 - 3*(pi + pi*c^2*x^2)^(5/2)*c^2 + 3*(pi + pi*c^2*x^2)^(7/2)/(pi*x^2))*a + b*integrate((pi + pi*c^2*x^2)^(
5/2)*log(c*x + sqrt(c^2*x^2 + 1))/x^3, x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((pi*c^2*x^2+pi)^(5/2)*(a+b*arcsinh(c*x))/x^3,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x^3} \,d x \]

[In]

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^3,x)

[Out]

int(((a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(5/2))/x^3, x)