3.3.81 \(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x))^2}{x^4} \, dx\) [281]

Optimal. Leaf size=561 \[ \frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {7 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}} \]

[Out]

-5/3*c^2*d*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2/x-1/3*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3+7/12*b^
2*c^4*d^2*x*(c^2*d*x^2+d)^(1/2)-1/3*b^2*c^2*d^2*(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)/x-1/3*b*c*d^2*(c^2*x^2+1)^(3/2
)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/x^2+5/2*c^4*d^2*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)-23/12*b^2*
c^3*d^2*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-5/2*b*c^5*d^2*x^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^
(1/2)/(c^2*x^2+1)^(1/2)+7/3*c^3*d^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/6*c^3*d^2*(a+
b*arcsinh(c*x))^3*(c^2*d*x^2+d)^(1/2)/b/(c^2*x^2+1)^(1/2)+14/3*b*c^3*d^2*(a+b*arcsinh(c*x))*ln(1-1/(c*x+(c^2*x
^2+1)^(1/2))^2)*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-7/3*b^2*c^3*d^2*polylog(2,1/(c*x+(c^2*x^2+1)^(1/2))^2)*(
c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+7/3*b*c^3*d^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.63, antiderivative size = 561, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 15, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5807, 5785, 5783, 5776, 327, 221, 5801, 5775, 3797, 2221, 2317, 2438, 201, 5802, 283} \begin {gather*} -\frac {b c d^2 \left (c^2 x^2+1\right )^{3/2} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {5 c^2 d \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (c^2 d x^2+d\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}-\frac {5 b c^5 d^2 x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5}{2} c^4 d^2 x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {5 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {c^2 x^2+1}}+\frac {7 c^3 d^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {c^2 x^2+1}}+\frac {7}{3} b c^3 d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac {14 b c^3 d^2 \sqrt {c^2 d x^2+d} \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 \sqrt {c^2 x^2+1}}-\frac {b^2 c^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{3 x}+\frac {7}{12} b^2 c^4 d^2 x \sqrt {c^2 d x^2+d}-\frac {7 b^2 c^3 d^2 \sqrt {c^2 d x^2+d} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {c^2 x^2+1}}-\frac {23 b^2 c^3 d^2 \sqrt {c^2 d x^2+d} \sinh ^{-1}(c x)}{12 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

(7*b^2*c^4*d^2*x*Sqrt[d + c^2*d*x^2])/12 - (b^2*c^2*d^2*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(3*x) - (23*b^2*c^3
*d^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(12*Sqrt[1 + c^2*x^2]) - (5*b*c^5*d^2*x^2*Sqrt[d + c^2*d*x^2]*(a + b*Ar
cSinh[c*x]))/(2*Sqrt[1 + c^2*x^2]) + (7*b*c^3*d^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/
3 - (b*c*d^2*(1 + c^2*x^2)^(3/2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*x^2) + (5*c^4*d^2*x*Sqrt[d + c^2
*d*x^2]*(a + b*ArcSinh[c*x])^2)/2 + (7*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*Sqrt[1 + c^2*x^2
]) - (5*c^2*d*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/(3*x) - ((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]
)^2)/(3*x^3) + (5*c^3*d^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^3)/(6*b*Sqrt[1 + c^2*x^2]) + (14*b*c^3*d^2*
Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])*Log[1 - E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2]) - (7*b^2*c^3*d^2*
Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c*x])])/(3*Sqrt[1 + c^2*x^2])

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 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 3797

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((
c + d*x)^(m + 1)/(d*(m + 1))), x] + Dist[2*I, Int[((c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5775

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[1/b, Subst[Int[x^n*Coth[-a/b + x/b], x],
x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[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 5783

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

Rule 5785

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

Rule 5801

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

Rule 5802

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((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])/(f*(m + 1))), x] + (-Dist[b*c*(d^p/(f*(m + 1))), Int[(f*x)^(m + 1
)*(1 + c^2*x^2)^(p - 1/2), x], x] - Dist[2*e*(p/(f^2*(m + 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*A
rcSinh[c*x]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && ILtQ[(m + 1)/2, 0]

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]

Rubi steps

\begin {align*} \int \frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {1}{3} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{x^2} \, dx+\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\left (5 c^4 d^2\right ) \int \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^{3/2}}{x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (4 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (10 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{x} \, dx}{3 \sqrt {1+c^2 x^2}}\\ &=-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {\left (4 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (10 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (5 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} \, dx}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^5 d^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt {1+c^2 x^2}}\\ &=-\frac {2}{3} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {\left (4 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (10 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{6 \sqrt {1+c^2 x^2}}+\frac {\left (5 b^2 c^6 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {2 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{3 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}-\frac {\left (8 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (20 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^4 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{4 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (4 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (10 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}-\frac {\left (5 b^2 c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ &=\frac {7}{12} b^2 c^4 d^2 x \sqrt {d+c^2 d x^2}-\frac {b^2 c^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{3 x}-\frac {23 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)}{12 \sqrt {1+c^2 x^2}}-\frac {5 b c^5 d^2 x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {7}{3} b c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {b c d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 x^2}+\frac {5}{2} c^4 d^2 x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {7 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 \sqrt {1+c^2 x^2}}-\frac {5 c^2 d \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x}-\frac {\left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 x^3}+\frac {5 c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{6 b \sqrt {1+c^2 x^2}}+\frac {14 b c^3 d^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}+\frac {7 b^2 c^3 d^2 \sqrt {d+c^2 d x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 \sqrt {1+c^2 x^2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.61, size = 616, normalized size = 1.10 \begin {gather*} \frac {d^2 \left (-8 a b c x \sqrt {d+c^2 d x^2}-8 a^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-56 a^2 c^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-8 b^2 c^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+12 a^2 c^4 x^4 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+20 b^2 c^3 x^3 \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^3-6 a b c^3 x^3 \sqrt {d+c^2 d x^2} \cosh \left (2 \sinh ^{-1}(c x)\right )+112 a b c^3 x^3 \sqrt {d+c^2 d x^2} \log (c x)+60 a^2 c^3 \sqrt {d} x^3 \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-56 b^2 c^3 x^3 \sqrt {d+c^2 d x^2} \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )+3 b^2 c^3 x^3 \sqrt {d+c^2 d x^2} \sinh \left (2 \sinh ^{-1}(c x)\right )-2 b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x) \left (4 b c x+8 a \sqrt {1+c^2 x^2}+56 a c^2 x^2 \sqrt {1+c^2 x^2}+3 b c^3 x^3 \cosh \left (2 \sinh ^{-1}(c x)\right )-56 b c^3 x^3 \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-6 a c^3 x^3 \sinh \left (2 \sinh ^{-1}(c x)\right )\right )+2 b \sqrt {d+c^2 d x^2} \sinh ^{-1}(c x)^2 \left (30 a c^3 x^3-4 b \left (-7 c^3 x^3+\sqrt {1+c^2 x^2}+7 c^2 x^2 \sqrt {1+c^2 x^2}\right )+3 b c^3 x^3 \sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )}{24 x^3 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^4,x]

[Out]

(d^2*(-8*a*b*c*x*Sqrt[d + c^2*d*x^2] - 8*a^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] - 56*a^2*c^2*x^2*Sqrt[1 + c
^2*x^2]*Sqrt[d + c^2*d*x^2] - 8*b^2*c^2*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 12*a^2*c^4*x^4*Sqrt[1 + c^
2*x^2]*Sqrt[d + c^2*d*x^2] + 20*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 6*a*b*c^3*x^3*Sqrt[d + c^2*d*
x^2]*Cosh[2*ArcSinh[c*x]] + 112*a*b*c^3*x^3*Sqrt[d + c^2*d*x^2]*Log[c*x] + 60*a^2*c^3*Sqrt[d]*x^3*Sqrt[1 + c^2
*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 56*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*PolyLog[2, E^(-2*ArcSinh[c
*x])] + 3*b^2*c^3*x^3*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] - 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(4*b*c*x
+ 8*a*Sqrt[1 + c^2*x^2] + 56*a*c^2*x^2*Sqrt[1 + c^2*x^2] + 3*b*c^3*x^3*Cosh[2*ArcSinh[c*x]] - 56*b*c^3*x^3*Lo
g[1 - E^(-2*ArcSinh[c*x])] - 6*a*c^3*x^3*Sinh[2*ArcSinh[c*x]]) + 2*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(30*a*
c^3*x^3 - 4*b*(-7*c^3*x^3 + Sqrt[1 + c^2*x^2] + 7*c^2*x^2*Sqrt[1 + c^2*x^2]) + 3*b*c^3*x^3*Sinh[2*ArcSinh[c*x]
])))/(24*x^3*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3310\) vs. \(2(511)=1022\).
time = 4.11, size = 3311, normalized size = 5.90

method	result	size
default	\(\text {Expression too large to display}\)	\(3311\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

-1/3*a^2/d/x^3*(c^2*d*x^2+d)^(7/2)+4/3*a^2*c^4*x*(c^2*d*x^2+d)^(5/2)+70*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*
x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5-46/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2
*x^2+1)/x/(c^2*x^2+1)*arcsinh(c*x)*c^2+294*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^
2+1)^(1/2)*arcsinh(c*x)*c^7-294*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsin
h(c*x)*c^8-406*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6-380/3*
a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4-56/3*b^2*(d*(c^2*x^2+1)
)^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-71/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^
2*x^2+1)*x^3/(c^2*x^2+1)*c^6-16/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-1/
3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x/(c^2*x^2+1)*c^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/
(63*c^4*x^4+15*c^2*x^2+1)/x^3/(c^2*x^2+1)*arcsinh(c*x)^2+14/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcs
inh(c*x)*ln(1+c*x+(c^2*x^2+1)^(1/2))*d^2*c^3+1/2*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)^2*
x^3+1/2*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)^2*x+14/3*b^2*(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))*d^2*c^3-5*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*
x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^3-1/2*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^5/(c^2*x^2+1)^(1/2)*arcsinh(c*x)
*x^2+7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*c^3+5*b^2*(d
*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5+21*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(
63*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*c^7+49/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1
)*x^3*arcsinh(c*x)*c^6+7/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x*arcsinh(c*x)*c^4-7/3*b^2*
(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)*c^4-147*b^2*(d*(c^2*x^2+1))^(1/
2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*arcsinh(c*x)^2*c^8-203*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*
x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)^2*c^6-56/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2
+1)*x^3/(c^2*x^2+1)*arcsinh(c*x)*c^6-2/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x^3/(c^2*x^2+
1)*arcsinh(c*x)-1/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*c-21*a*b*(d*
(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*c^5-49/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/
(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*c^8-1/4*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^3/(c^2*x^2+1)^(1/2)+5/3*a^2*
c^4*d*x*(c^2*d*x^2+d)^(3/2)+5/2*a^2*c^4*d^2*x*(c^2*d*x^2+d)^(1/2)+5/2*a^2*c^4*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^
2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+14/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(c^2*x^2+1)^(1/2)
*arcsinh(c*x)*c^3+a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^3+a*b*(d*(c^2*x^2+1))^(1/2)*d^2
*c^4/(c^2*x^2+1)*arcsinh(c*x)*x-49/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^5/(c^2*x^2+1)*a
rcsinh(c*x)*c^8-1/2*a*b*(d*(c^2*x^2+1))^(1/2)*d^2*c^5/(c^2*x^2+1)^(1/2)*x^2+7/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/
(63*c^4*x^4+15*c^2*x^2+1)*x*c^4-28/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*d^2*c^3+49/3*a*b
*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3*c^6-5*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^
2*x^2+1)/(c^2*x^2+1)^(1/2)*c^3+5/2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^2*d^2*c^3+14/3*a*b
*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)*d^2*c^3-4/3*a^2*c^2/d/x*(c^2*d*x^2+d)
^(7/2)-56/3*a*b*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3/(c^2*x^2+1)*c^6-7/3*a*b*(d*(c^2*x^2+1)
)^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x/(c^2*x^2+1)*c^4-190/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2
*x^2+1)*x/(c^2*x^2+1)*arcsinh(c*x)^2*c^4-23/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x/(c^2*x
^2+1)*arcsinh(c*x)^2*c^2+35*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcs
inh(c*x)^2*c^5+147*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^4/(c^2*x^2+1)^(1/2)*arcsinh(c*x)^
2*c^7-21*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c^5-1/3*b^
2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/x^2/(c^2*x^2+1)^(1/2)*arcsinh(c*x)*c-7/3*b^2*(d*(c^2*x^2
+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)*x^3*c^6+1/3*b^2*(d*(c^2*x^2+1))^(1/2)*d^2/(63*c^4*x^4+15*c^2*x^2+1)/(
c^2*x^2+1)^(1/2)*c^3-1/4*b^2*(d*(c^2*x^2+1))^(1/2)*d^2*c^3/(c^2*x^2+1)^(1/2)*arcsinh(c*x)+1/4*b^2*(d*(c^2*x^2+
1))^(1/2)*d^2*c^6/(c^2*x^2+1)*x^3+14/3*b^2*(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))*d^2*c^3+14/3*b^2*(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))*d^2*c^3-14/3*
b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*arc...

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

________________________________________________________________________________________

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((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2/x**4,x)

[Out]

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

________________________________________________________________________________________

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((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^4,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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^4,x)

[Out]

int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^4, x)

________________________________________________________________________________________