\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [312]

Optimal result

Integrand size = 28, antiderivative size = 307 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {10 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \]

[Out]

-1/3*x^2*(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(3/2)-1/3*b^2/c^4/d^2/(c^2*d*x^2+d)^(1/2)-2/3*(a+b*arcsinh(c
*x))^2/c^4/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*x*(a+b*arcsinh(c*x))/c^3/d^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)+10
/3*b*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*d*x^2+d)^(1/2)-5/3*I*b^2*
polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*d*x^2+d)^(1/2)+5/3*I*b^2*polylog(2,I*(c*x
+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/2)/c^4/d^2/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5810, 5798, 5789, 4265, 2317, 2438, 267} \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {10 b \sqrt {c^2 x^2+1} \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {5 i b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}+\frac {5 i b^2 \sqrt {c^2 x^2+1} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2}{3 c^4 d^2 \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

-1/3*b^2/(c^4*d^2*Sqrt[d + c^2*d*x^2]) - (b*x*(a + b*ArcSinh[c*x]))/(3*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*
d*x^2]) - (x^2*(a + b*ArcSinh[c*x])^2)/(3*c^2*d*(d + c^2*d*x^2)^(3/2)) - (2*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2
*Sqrt[d + c^2*d*x^2]) + (10*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(3*c^4*d^2*Sqrt[d
+ c^2*d*x^2]) - (((5*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^4*d^2*Sqrt[d + c^2*d*x^2
]) + (((5*I)/3)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^4*d^2*Sqrt[d + c^2*d*x^2])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 4265

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

Rule 5789

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

Rule 5798

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

Rule 5810

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[
f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] + (-Dist[f^2*((m - 1)/(2*e*(p +
1))), Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(2*c*(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}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IGtQ[m, 1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{1+c^2 x^2} \, dx}{3 c^3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 c^2 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arcsinh}(c x))}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {10 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {10 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {b^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {10 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.12 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-a^2 \left (2+3 c^2 x^2\right )+a b \left (-2 \left (2+3 c^2 x^2\right ) \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (-c x+10 \left (1+c^2 x^2\right ) \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )\right )-b^2 \left (1+c^2 x^2+c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+2 \text {arcsinh}(c x)^2+3 c^2 x^2 \text {arcsinh}(c x)^2+5 i \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-5 i \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+5 i \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-5 i \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )}{3 c^4 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

(-(a^2*(2 + 3*c^2*x^2)) + a*b*(-2*(2 + 3*c^2*x^2)*ArcSinh[c*x] + Sqrt[1 + c^2*x^2]*(-(c*x) + 10*(1 + c^2*x^2)*
ArcTan[Tanh[ArcSinh[c*x]/2]])) - b^2*(1 + c^2*x^2 + c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 2*ArcSinh[c*x]^2 + 3*
c^2*x^2*ArcSinh[c*x]^2 + (5*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] - (5*I)*(1 + c^2*x^2
)^(3/2)*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] + (5*I)*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (
5*I)*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/E^ArcSinh[c*x]]))/(3*c^4*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (294 ) = 588\).

Time = 0.20 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.29

method	result	size
default	\(a^{2} \left (-\frac {x^{2}}{c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {4 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}+\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}-\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}\)	\(704\)
parts	\(a^{2} \left (-\frac {x^{2}}{c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {4 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}+\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}-\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{3 \sqrt {c^{2} x^{2}+1}\, c^{4} d^{3}}\)	\(704\)

[In]

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

[Out]

a^2*(-x^2/c^2/d/(c^2*d*x^2+d)^(3/2)-2/3/d/c^4/(c^2*d*x^2+d)^(3/2))-b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3
/c^2*arcsinh(c*x)^2*x^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(3/2)/d^3/c^3*arcsinh(c*x)*x-1/3*b^2*(d*(c^2
*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^2*x^2-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4*arcsinh(c*x)^2-1/
3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4-5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^4/d^3*ar
csinh(c*x)*ln(1+I*(c*x+(c^2*x^2+1)^(1/2)))+5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^4/d^3*arcsinh(c
*x)*ln(1-I*(c*x+(c^2*x^2+1)^(1/2)))-5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^4/d^3*dilog(1+I*(c*x+(
c^2*x^2+1)^(1/2)))+5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^4/d^3*dilog(1-I*(c*x+(c^2*x^2+1)^(1/2))
)-2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^2*arcsinh(c*x)*x^2-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)
^(3/2)/d^3/c^3*x-4/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4*arcsinh(c*x)+5/3*I*a*b*(d*(c^2*x^2+1))^(1
/2)/(c^2*x^2+1)^(1/2)/c^4/d^3*ln(c*x+(c^2*x^2+1)^(1/2)+I)-5/3*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^
4/d^3*ln(c*x+(c^2*x^2+1)^(1/2)-I)

Fricas [F]

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

[In]

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

[Out]

integral((b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^3)*sqrt(c^2*d*x^2 + d)/(c^6*d^3*x^6 + 3*c^4*
d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-1/3*a*b*c*(x/(c^6*d^(5/2)*x^2 + c^4*d^(5/2)) - 5*arctan(c*x)/(c^5*d^(5/2))) - 2/3*a*b*(3*x^2/((c^2*d*x^2 + d)
^(3/2)*c^2*d) + 2/((c^2*d*x^2 + d)^(3/2)*c^4*d))*arcsinh(c*x) - 1/3*a^2*(3*x^2/((c^2*d*x^2 + d)^(3/2)*c^2*d) +
2/((c^2*d*x^2 + d)^(3/2)*c^4*d)) + b^2*integrate(x^3*log(c*x + sqrt(c^2*x^2 + 1))^2/(c^2*d*x^2 + d)^(5/2), x)

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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