3.3.100 \(\int \frac {x^5 (a+b \sinh ^{-1}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [300]

Optimal. Leaf size=515 \[ \frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {32 b^2 \left (1+c^2 x^2\right )}{9 c^6 d \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^6 d \sqrt {d+c^2 d x^2}}+\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}} \]

[Out]

-32/9*b^2*(c^2*x^2+1)/c^6/d/(c^2*d*x^2+d)^(1/2)+2/27*b^2*(c^2*x^2+1)^2/c^6/d/(c^2*d*x^2+d)^(1/2)-x^4*(a+b*arcs
inh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(1/2)+16/3*a*b*x*(c^2*x^2+1)^(1/2)/c^5/d/(c^2*d*x^2+d)^(1/2)+16/3*b^2*x*arcsin
h(c*x)*(c^2*x^2+1)^(1/2)/c^5/d/(c^2*d*x^2+d)^(1/2)-2*b*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^5/d/(c^2*d*x^2
+d)^(1/2)-2/9*b*x^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3/d/(c^2*d*x^2+d)^(1/2)+4*b*(a+b*arcsinh(c*x))*arct
an(c*x+(c^2*x^2+1)^(1/2))*(c^2*x^2+1)^(1/2)/c^6/d/(c^2*d*x^2+d)^(1/2)-2*I*b^2*polylog(2,-I*(c*x+(c^2*x^2+1)^(1
/2)))*(c^2*x^2+1)^(1/2)/c^6/d/(c^2*d*x^2+d)^(1/2)+2*I*b^2*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))*(c^2*x^2+1)^(1/
2)/c^6/d/(c^2*d*x^2+d)^(1/2)-8/3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^6/d^2+4/3*x^2*(a+b*arcsinh(c*x))^2
*(c^2*d*x^2+d)^(1/2)/c^4/d^2

________________________________________________________________________________________

Rubi [A]
time = 0.51, antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5810, 5812, 5798, 5772, 267, 5776, 272, 45, 5789, 4265, 2317, 2438} \begin {gather*} \frac {4 b \sqrt {c^2 x^2+1} \text {ArcTan}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^6 d \sqrt {c^2 d x^2+d}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {c^2 d x^2+d}}-\frac {8 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {16 a b x \sqrt {c^2 x^2+1}}{3 c^5 d \sqrt {c^2 d x^2+d}}-\frac {2 b x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {c^2 d x^2+d}}+\frac {4 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac {2 b x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {c^2 d x^2+d}}-\frac {2 i b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {c^2 d x^2+d}}+\frac {2 i b^2 \sqrt {c^2 x^2+1} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )^2}{27 c^6 d \sqrt {c^2 d x^2+d}}-\frac {32 b^2 \left (c^2 x^2+1\right )}{9 c^6 d \sqrt {c^2 d x^2+d}}+\frac {16 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^5 d \sqrt {c^2 d x^2+d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^5*d*Sqrt[d + c^2*d*x^2]) - (32*b^2*(1 + c^2*x^2))/(9*c^6*d*Sqrt[d + c^2*d*x^
2]) + (2*b^2*(1 + c^2*x^2)^2)/(27*c^6*d*Sqrt[d + c^2*d*x^2]) + (16*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^
5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(c^5*d*Sqrt[d + c^2*d*x^2]) - (2*b*x
^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c^3*d*Sqrt[d + c^2*d*x^2]) - (x^4*(a + b*ArcSinh[c*x])^2)/(c^2*d
*Sqrt[d + c^2*d*x^2]) - (8*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^6*d^2) + (4*x^2*Sqrt[d + c^2*d*x^2
]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d^2) + (4*b*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]])/(c
^6*d*Sqrt[d + c^2*d*x^2]) - ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*
d*x^2]) + ((2*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I*E^ArcSinh[c*x]])/(c^6*d*Sqrt[d + c^2*d*x^2])

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 272

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

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 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[b*c*(n/(d*(m + 1))), Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[
1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

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]

Rule 5812

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/(e*(m + 2*p + 1))), x] + (-Dist[f^2*((m - 1)/(c^2*
(m + 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Dist[b*f*(n/(c*(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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0
]

Rubi steps

\begin {align*} \int \frac {x^5 \left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx &=-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 \int \frac {x^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}}\\ &=\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}-\frac {8 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^4 d}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (8 b \sqrt {1+c^2 x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{3 c^2 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {2 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (16 b \sqrt {1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^5 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{c^4 d \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{3 c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (8 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{9 c^2 d \sqrt {d+c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {2 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d \sqrt {d+c^2 d x^2}}+\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{3 c^2 d \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{9 c^2 d \sqrt {d+c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d \sqrt {d+c^2 d x^2}}+\frac {8 b^2 \left (1+c^2 x^2\right )}{3 c^6 d \sqrt {d+c^2 d x^2}}-\frac {2 b^2 \left (1+c^2 x^2\right )^2}{9 c^6 d \sqrt {d+c^2 d x^2}}+\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {\left (16 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^4 d \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{9 c^2 d \sqrt {d+c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {32 b^2 \left (1+c^2 x^2\right )}{9 c^6 d \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^6 d \sqrt {d+c^2 d x^2}}+\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}\\ &=\frac {16 a b x \sqrt {1+c^2 x^2}}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {32 b^2 \left (1+c^2 x^2\right )}{9 c^6 d \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^6 d \sqrt {d+c^2 d x^2}}+\frac {16 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{c^2 d \sqrt {d+c^2 d x^2}}-\frac {8 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^6 d^2}+\frac {4 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d^2}+\frac {4 b \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}-\frac {2 i b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}+\frac {2 i b^2 \sqrt {1+c^2 x^2} \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{c^6 d \sqrt {d+c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.38, size = 427, normalized size = 0.83 \begin {gather*} \frac {-72 a^2-94 b^2-36 a^2 c^2 x^2-92 b^2 c^2 x^2+9 a^2 c^4 x^4+2 b^2 c^4 x^4+90 a b c x \sqrt {1+c^2 x^2}-6 a b c^3 x^3 \sqrt {1+c^2 x^2}-144 a b \sinh ^{-1}(c x)-72 a b c^2 x^2 \sinh ^{-1}(c x)+18 a b c^4 x^4 \sinh ^{-1}(c x)+90 b^2 c x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-6 b^2 c^3 x^3 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-72 b^2 \sinh ^{-1}(c x)^2-36 b^2 c^2 x^2 \sinh ^{-1}(c x)^2+9 b^2 c^4 x^4 \sinh ^{-1}(c x)^2+108 a b \sqrt {1+c^2 x^2} \text {ArcTan}\left (\tanh \left (\frac {1}{2} \sinh ^{-1}(c x)\right )\right )-54 i b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \log \left (1-i e^{-\sinh ^{-1}(c x)}\right )+54 i b^2 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \log \left (1+i e^{-\sinh ^{-1}(c x)}\right )-54 i b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-i e^{-\sinh ^{-1}(c x)}\right )+54 i b^2 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,i e^{-\sinh ^{-1}(c x)}\right )}{27 c^6 d \sqrt {d+c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-72*a^2 - 94*b^2 - 36*a^2*c^2*x^2 - 92*b^2*c^2*x^2 + 9*a^2*c^4*x^4 + 2*b^2*c^4*x^4 + 90*a*b*c*x*Sqrt[1 + c^2*
x^2] - 6*a*b*c^3*x^3*Sqrt[1 + c^2*x^2] - 144*a*b*ArcSinh[c*x] - 72*a*b*c^2*x^2*ArcSinh[c*x] + 18*a*b*c^4*x^4*A
rcSinh[c*x] + 90*b^2*c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 6*b^2*c^3*x^3*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] - 72*b^
2*ArcSinh[c*x]^2 - 36*b^2*c^2*x^2*ArcSinh[c*x]^2 + 9*b^2*c^4*x^4*ArcSinh[c*x]^2 + 108*a*b*Sqrt[1 + c^2*x^2]*Ar
cTan[Tanh[ArcSinh[c*x]/2]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 - I/E^ArcSinh[c*x]] + (54*I)*b^2*
Sqrt[1 + c^2*x^2]*ArcSinh[c*x]*Log[1 + I/E^ArcSinh[c*x]] - (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, (-I)/E^ArcS
inh[c*x]] + (54*I)*b^2*Sqrt[1 + c^2*x^2]*PolyLog[2, I/E^ArcSinh[c*x]])/(27*c^6*d*Sqrt[d + c^2*d*x^2])

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Maple [A]
time = 3.96, size = 934, normalized size = 1.81

method	result	size
default	\(a^{2} \left (\frac {x^{4}}{3 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {4 \left (\frac {x^{2}}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {2}{d \,c^{4} \sqrt {c^{2} d \,x^{2}+d}}\right )}{3 c^{2}}\right )+\frac {2 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} d^{2}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{3}}{9 c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {10 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x}{3 c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}+\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{4}}{27 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {92 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{27 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} d^{2}}+\frac {2 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} d^{2}}-\frac {8 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{3 c^{6} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {94 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{27 c^{6} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{4}}{3 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {4 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} x^{2}}{3 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \dilog \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} d^{2}}+\frac {2 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1-i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} d^{2}}-\frac {2 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) \ln \left (1+i \left (c x +\sqrt {c^{2} x^{2}+1}\right )\right )}{\sqrt {c^{2} x^{2}+1}\, c^{6} d^{2}}-\frac {16 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )}{3 c^{6} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{4}}{3 c^{2} d^{2} \left (c^{2} x^{2}+1\right )}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{3}}{9 c^{3} d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {8 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right ) x^{2}}{3 c^{4} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {10 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{3 c^{5} d^{2} \sqrt {c^{2} x^{2}+1}}\)	\(934\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*(1/3*x^4/c^2/d/(c^2*d*x^2+d)^(1/2)-4/3/c^2*(x^2/c^2/d/(c^2*d*x^2+d)^(1/2)+2/d/c^4/(c^2*d*x^2+d)^(1/2)))-2*
I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*arcsinh(c*x)*ln(1+I*(c*x+(c^2*x^2+1)^(1/2)))-2/9*b^2*(d*
(c^2*x^2+1))^(1/2)/c^3/d^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*x^3+10/3*b^2*(d*(c^2*x^2+1))^(1/2)/c^5/d^2/(c^2*x^2+
1)^(1/2)*arcsinh(c*x)*x+2/27*b^2*(d*(c^2*x^2+1))^(1/2)/c^2/d^2/(c^2*x^2+1)*x^4-92/27*b^2*(d*(c^2*x^2+1))^(1/2)
/c^4/d^2/(c^2*x^2+1)*x^2+2*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*dilog(1-I*(c*x+(c^2*x^2+1)^(1
/2)))-2*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*ln(c*x+(c^2*x^2+1)^(1/2)-I)-8/3*b^2*(d*(c^2*x^2+
1))^(1/2)/c^6/d^2/(c^2*x^2+1)*arcsinh(c*x)^2-94/27*b^2*(d*(c^2*x^2+1))^(1/2)/c^6/d^2/(c^2*x^2+1)+1/3*b^2*(d*(c
^2*x^2+1))^(1/2)/c^2/d^2/(c^2*x^2+1)*arcsinh(c*x)^2*x^4-4/3*b^2*(d*(c^2*x^2+1))^(1/2)/c^4/d^2/(c^2*x^2+1)*arcs
inh(c*x)^2*x^2+2*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*ln(c*x+(c^2*x^2+1)^(1/2)+I)-2*I*b^2*(d*
(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^6/d^2*dilog(1+I*(c*x+(c^2*x^2+1)^(1/2)))+2*I*b^2*(d*(c^2*x^2+1))^(1/2)/
(c^2*x^2+1)^(1/2)/c^6/d^2*arcsinh(c*x)*ln(1-I*(c*x+(c^2*x^2+1)^(1/2)))-16/3*a*b*(d*(c^2*x^2+1))^(1/2)/c^6/d^2/
(c^2*x^2+1)*arcsinh(c*x)+2/3*a*b*(d*(c^2*x^2+1))^(1/2)/c^2/d^2/(c^2*x^2+1)*arcsinh(c*x)*x^4-2/9*a*b*(d*(c^2*x^
2+1))^(1/2)/c^3/d^2/(c^2*x^2+1)^(1/2)*x^3-8/3*a*b*(d*(c^2*x^2+1))^(1/2)/c^4/d^2/(c^2*x^2+1)*arcsinh(c*x)*x^2+1
0/3*a*b*(d*(c^2*x^2+1))^(1/2)/c^5/d^2/(c^2*x^2+1)^(1/2)*x

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*(x^4/(sqrt(c^2*d*x^2 + d)*c^2*d) - 4*x^2/(sqrt(c^2*d*x^2 + d)*c^4*d) - 8/(sqrt(c^2*d*x^2 + d)*c^6*d))
+ 1/3*(b^2*c^4*sqrt(d)*x^4 - 4*b^2*c^2*sqrt(d)*x^2 - 8*b^2*sqrt(d))*sqrt(c^2*x^2 + 1)*log(c*x + sqrt(c^2*x^2 +
1))^2/(c^8*d^2*x^2 + c^6*d^2) + integrate(2/3*((4*b^2*c^3*x^3 + (3*a*b*c^5 - b^2*c^5)*x^5 + 8*b^2*c*x)*(c^2*x
^2 + 1) + (3*b^2*c^4*x^4 + (3*a*b*c^6 - b^2*c^6)*x^6 + 12*b^2*c^2*x^2 + 8*b^2)*sqrt(c^2*x^2 + 1))*log(c*x + sq
rt(c^2*x^2 + 1))/(c^10*d^(3/2)*x^5 + 2*c^8*d^(3/2)*x^3 + c^6*d^(3/2)*x + (c^9*d^(3/2)*x^4 + 2*c^7*d^(3/2)*x^2
+ c^5*d^(3/2))*sqrt(c^2*x^2 + 1)), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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