3.134 \(\int \frac{(d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=270 \[ -\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3}{2} c^2 d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac{3 c^2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{b c^3 d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{b c d \sqrt{c^2 d x^2+d}}{2 x \sqrt{c^2 x^2+1}} \]

[Out]

-(b*c*d*Sqrt[d + c^2*d*x^2])/(2*x*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (3*
c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*x^2) - (3*
c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (3*b*c^2*d*Sqrt[d
+ c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^
ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2])

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Rubi [A]  time = 0.31003, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {5739, 5742, 5760, 4182, 2279, 2391, 8, 14} \[ -\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3 b c^2 d \sqrt{c^2 d x^2+d} \text{PolyLog}\left (2,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{c^2 x^2+1}}+\frac{3}{2} c^2 d \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac{3 c^2 d \sqrt{c^2 d x^2+d} \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt{c^2 x^2+1}}-\frac{b c^3 d x \sqrt{c^2 d x^2+d}}{\sqrt{c^2 x^2+1}}-\frac{b c d \sqrt{c^2 d x^2+d}}{2 x \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*c*d*Sqrt[d + c^2*d*x^2])/(2*x*Sqrt[1 + c^2*x^2]) - (b*c^3*d*x*Sqrt[d + c^2*d*x^2])/Sqrt[1 + c^2*x^2] + (3*
c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/2 - ((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(2*x^2) - (3*
c^2*d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*ArcTanh[E^ArcSinh[c*x]])/Sqrt[1 + c^2*x^2] - (3*b*c^2*d*Sqrt[d
+ c^2*d*x^2]*PolyLog[2, -E^ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2]) + (3*b*c^2*d*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^
ArcSinh[c*x]])/(2*Sqrt[1 + c^2*x^2])

Rule 5739

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*d^IntPart[p]*(d + e*x^2)^FracPart[p
])/(f*(m + 1)*(1 + c^2*x^2)^FracPart[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 5742

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[Sqrt[d + e*x^2]/((m + 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*Sqrt[d + e*x^2])/(f*
(m + 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] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 5760

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

Rule 4182

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 2279

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 2391

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 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int \frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac{1}{2} \left (3 c^2 d\right ) \int \frac{\sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx+\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int \frac{1+c^2 x^2}{x^2} \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=\frac{3}{2} c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac{\left (b c d \sqrt{d+c^2 d x^2}\right ) \int \left (c^2+\frac{1}{x^2}\right ) \, dx}{2 \sqrt{1+c^2 x^2}}+\frac{\left (3 c^2 d \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{x \sqrt{1+c^2 x^2}} \, dx}{2 \sqrt{1+c^2 x^2}}-\frac{\left (3 b c^3 d \sqrt{d+c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d \sqrt{d+c^2 d x^2}}{2 x \sqrt{1+c^2 x^2}}-\frac{b c^3 d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{3}{2} c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}+\frac{\left (3 c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int (a+b x) \text{csch}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d \sqrt{d+c^2 d x^2}}{2 x \sqrt{1+c^2 x^2}}-\frac{b c^3 d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{3}{2} c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac{3 c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (3 b c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}+\frac{\left (3 b c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d \sqrt{d+c^2 d x^2}}{2 x \sqrt{1+c^2 x^2}}-\frac{b c^3 d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{3}{2} c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac{3 c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{\left (3 b c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{1+c^2 x^2}}+\frac{\left (3 b c^2 d \sqrt{d+c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{1+c^2 x^2}}\\ &=-\frac{b c d \sqrt{d+c^2 d x^2}}{2 x \sqrt{1+c^2 x^2}}-\frac{b c^3 d x \sqrt{d+c^2 d x^2}}{\sqrt{1+c^2 x^2}}+\frac{3}{2} c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac{\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{2 x^2}-\frac{3 c^2 d \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{\sqrt{1+c^2 x^2}}-\frac{3 b c^2 d \sqrt{d+c^2 d x^2} \text{Li}_2\left (-e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{1+c^2 x^2}}+\frac{3 b c^2 d \sqrt{d+c^2 d x^2} \text{Li}_2\left (e^{\sinh ^{-1}(c x)}\right )}{2 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 4.07929, size = 352, normalized size = 1.3 \[ \frac{b c^2 d \sqrt{c^2 d x^2+d} \left (\text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-\text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+\sqrt{c^2 x^2+1} \sinh ^{-1}(c x)-c x+\sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-\sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )\right )}{\sqrt{c^2 x^2+1}}+\frac{b c^2 d \sqrt{c^2 d x^2+d} \left (4 \text{PolyLog}\left (2,-e^{-\sinh ^{-1}(c x)}\right )-4 \text{PolyLog}\left (2,e^{-\sinh ^{-1}(c x)}\right )+4 \sinh ^{-1}(c x) \log \left (1-e^{-\sinh ^{-1}(c x)}\right )-4 \sinh ^{-1}(c x) \log \left (e^{-\sinh ^{-1}(c x)}+1\right )+2 \tanh \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-2 \coth \left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{csch}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )-\sinh ^{-1}(c x) \text{sech}^2\left (\frac{1}{2} \sinh ^{-1}(c x)\right )\right )}{8 \sqrt{c^2 x^2+1}}-\frac{3}{2} a c^2 d^{3/2} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+d\right )+\frac{3}{2} a c^2 d^{3/2} \log (x)+a \left (c^2 d-\frac{d}{2 x^2}\right ) \sqrt{c^2 d x^2+d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a*(c^2*d - d/(2*x^2))*Sqrt[d + c^2*d*x^2] + (3*a*c^2*d^(3/2)*Log[x])/2 - (3*a*c^2*d^(3/2)*Log[d + Sqrt[d]*Sqrt
[d + c^2*d*x^2]])/2 + (b*c^2*d*Sqrt[d + c^2*d*x^2]*(-(c*x) + Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log
[1 - E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - PolyLog[2
, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (b*c^2*d*Sqrt[d + c^2*d*x^2]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x
]*Csch[ArcSinh[c*x]/2]^2 + 4*ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[c*x]
)] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x])] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2
+ 2*Tanh[ArcSinh[c*x]/2]))/(8*Sqrt[1 + c^2*x^2])

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Maple [A]  time = 0.184, size = 472, normalized size = 1.8 \begin{align*} -{\frac{a}{2\,d{x}^{2}} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{5}{2}}}}+{\frac{a{c}^{2}}{2} \left ({c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,a{c}^{2}}{2}{d}^{{\frac{3}{2}}}\ln \left ({\frac{1}{x} \left ( 2\,d+2\,\sqrt{d}\sqrt{{c}^{2}d{x}^{2}+d} \right ) } \right ) }+{\frac{3\,a{c}^{2}d}{2}\sqrt{{c}^{2}d{x}^{2}+d}}+{\frac{b{c}^{4}d{\it Arcsinh} \left ( cx \right ){x}^{2}}{{c}^{2}{x}^{2}+1}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{b{c}^{3}dx\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{{c}^{2}db{\it Arcsinh} \left ( cx \right ) }{2\,{c}^{2}{x}^{2}+2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{bdc}{2\,x}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{bd{\it Arcsinh} \left ( cx \right ) }{2\,{x}^{2} \left ({c}^{2}{x}^{2}+1 \right ) }\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }}-{\frac{3\,{c}^{2}db{\it Arcsinh} \left ( cx \right ) }{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1+cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}-{\frac{3\,{c}^{2}db}{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{3\,{c}^{2}db{\it Arcsinh} \left ( cx \right ) }{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }\ln \left ( 1-cx-\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}+{\frac{3\,{c}^{2}db}{2}\sqrt{d \left ({c}^{2}{x}^{2}+1 \right ) }{\it polylog} \left ( 2,cx+\sqrt{{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a c^{2} d x^{2} + a d +{\left (b c^{2} d x^{2} + b d\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{asinh}{\left (c x \right )}\right )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arsinh}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c^2*d*x^2 + d)^(3/2)*(b*arcsinh(c*x) + a)/x^3, x)