3.2.29 \(\int x^2 (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x)) \, dx\) [129]

Optimal. Leaf size=254 \[ -\frac {b d x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

1/6*x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))+1/16*d*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+1/8*d*x^3*(
a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)-1/32*b*d*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-7/96*b*c*d*x^4*(c^2
*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/36*b*c^3*d*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/32*d*(a+b*arcsinh(c
*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)

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Rubi [A]
time = 0.22, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5808, 5806, 5812, 5783, 30, 14} \begin {gather*} \frac {d x \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} d x^3 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {c^2 x^2+1}}-\frac {b d x^2 \sqrt {c^2 d x^2+d}}{32 c \sqrt {c^2 x^2+1}}-\frac {7 b c d x^4 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {b c^3 d x^6 \sqrt {c^2 d x^2+d}}{36 \sqrt {c^2 x^2+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(b*d*x^2*Sqrt[d + c^2*d*x^2])/(c*Sqrt[1 + c^2*x^2]) - (7*b*c*d*x^4*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2
*x^2]) - (b*c^3*d*x^6*Sqrt[d + c^2*d*x^2])/(36*Sqrt[1 + c^2*x^2]) + (d*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/(16*c^2) + (d*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/8 + (x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh
[c*x]))/6 - (d*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c^3*Sqrt[1 + c^2*x^2])

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]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && 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 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 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 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 x^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{2} d \int x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{6 \sqrt {1+c^2 x^2}}\\ &=-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b d \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1+c^2 x^2}}\\ &=-\frac {b d x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {d \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt {1+c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.49, size = 251, normalized size = 0.99 \begin {gather*} \frac {48 a c d x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (3+14 c^2 x^2+8 c^4 x^4\right )-144 a d^{3/2} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-18 b d \sqrt {d+c^2 d x^2} \left (8 \sinh ^{-1}(c x)^2+\cosh \left (4 \sinh ^{-1}(c x)\right )-4 \sinh ^{-1}(c x) \sinh \left (4 \sinh ^{-1}(c x)\right )\right )+b d \sqrt {d+c^2 d x^2} \left (72 \sinh ^{-1}(c x)^2+18 \cosh \left (2 \sinh ^{-1}(c x)\right )+9 \cosh \left (4 \sinh ^{-1}(c x)\right )-2 \cosh \left (6 \sinh ^{-1}(c x)\right )+12 \sinh ^{-1}(c x) \left (-3 \sinh \left (2 \sinh ^{-1}(c x)\right )-3 \sinh \left (4 \sinh ^{-1}(c x)\right )+\sinh \left (6 \sinh ^{-1}(c x)\right )\right )\right )}{2304 c^3 \sqrt {1+c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(48*a*c*d*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(3 + 14*c^2*x^2 + 8*c^4*x^4) - 144*a*d^(3/2)*Sqrt[1 + c^2*x^
2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 18*b*d*Sqrt[d + c^2*d*x^2]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*
x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]) + b*d*Sqrt[d + c^2*d*x^2]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*ArcSinh[c*
x]] + 9*Cosh[4*ArcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] + 12*ArcSinh[c*x]*(-3*Sinh[2*ArcSinh[c*x]] - 3*Sinh[4*Ar
cSinh[c*x]] + Sinh[6*ArcSinh[c*x]])))/(2304*c^3*Sqrt[1 + c^2*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(798\) vs. \(2(218)=436\).
time = 2.43, size = 799, normalized size = 3.15

method	result	size
default	\(\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{6 c^{2} d}-\frac {a x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a d x \sqrt {c^{2} d \,x^{2}+d}}{16 c^{2}}-\frac {a \,d^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{16 c^{2} \sqrt {c^{2} d}}+b \left (-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2} d}{32 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 c^{7} x^{7}+32 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}+64 c^{5} x^{5}+48 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+38 c^{3} x^{3}+18 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+6 \arcsinh \left (c x \right )\right ) d}{2304 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \arcsinh \left (c x \right )\right ) d}{512 c^{3} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \arcsinh \left (c x \right )\right ) d}{256 c^{3} \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \arcsinh \left (c x \right )\right ) d}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \arcsinh \left (c x \right )\right ) d}{512 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (32 c^{7} x^{7}-32 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}+64 c^{5} x^{5}-48 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+38 c^{3} x^{3}-18 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+6 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+6 \arcsinh \left (c x \right )\right ) d}{2304 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(799\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*a*x*(c^2*d*x^2+d)^(5/2)/c^2/d-1/24*a/c^2*x*(c^2*d*x^2+d)^(3/2)-1/16*a/c^2*d*x*(c^2*d*x^2+d)^(1/2)-1/16*a/c
^2*d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b*(-1/32*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^
(1/2)/c^3*arcsinh(c*x)^2*d+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48
*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*(-1+6*arcsinh(c*x)
)*d/c^3/(c^2*x^2+1)+1/512*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c
^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(c*x))*d/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*
c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*(-1+2*arcsinh(c*x))*d/c^3/(c^2*x^2+1)-1/256*(d*(c
^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(c*x))*d/c^3/(c^2
*x^2+1)+1/512*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1
/2)+4*c*x-(c^2*x^2+1)^(1/2))*(1+4*arcsinh(c*x))*d/c^3/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*
c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*
x-(c^2*x^2+1)^(1/2))*(1+6*arcsinh(c*x))*d/c^3/(c^2*x^2+1))

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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