\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^3} \, dx\) [244]

Optimal result

Integrand size = 26, antiderivative size = 167 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {b^2}{12 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{2 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{3 c^4 d^3} \]

[Out]

-1/12*b^2/c^4/d^3/(c^2*x^2+1)+1/6*b*x^3*(a+b*arcsinh(c*x))/c/d^3/(c^2*x^2+1)^(3/2)-1/4*(a+b*arcsinh(c*x))^2/c^
4/d^3+1/4*x^4*(a+b*arcsinh(c*x))^2/d^3/(c^2*x^2+1)^2-1/3*b^2*ln(c^2*x^2+1)/c^4/d^3+1/2*b*x*(a+b*arcsinh(c*x))/
c^3/d^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5800, 5810, 5783, 266, 272, 45} \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (c^2 x^2+1\right )^2}+\frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (c^2 x^2+1\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{2 c^3 d^3 \sqrt {c^2 x^2+1}}-\frac {b^2}{12 c^4 d^3 \left (c^2 x^2+1\right )}-\frac {b^2 \log \left (c^2 x^2+1\right )}{3 c^4 d^3} \]

[In]

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

[Out]

-1/12*b^2/(c^4*d^3*(1 + c^2*x^2)) + (b*x^3*(a + b*ArcSinh[c*x]))/(6*c*d^3*(1 + c^2*x^2)^(3/2)) + (b*x*(a + b*A
rcSinh[c*x]))/(2*c^3*d^3*Sqrt[1 + c^2*x^2]) - (a + b*ArcSinh[c*x])^2/(4*c^4*d^3) + (x^4*(a + b*ArcSinh[c*x])^2
)/(4*d^3*(1 + c^2*x^2)^2) - (b^2*Log[1 + c^2*x^2])/(3*c^4*d^3)

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 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 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 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 5800

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 + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), 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, m, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -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^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{2 d^3} \\ & = \frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \int \frac {x^3}{\left (1+c^2 x^2\right )^2} \, dx}{6 d^3}-\frac {b \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 c d^3} \\ & = \frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{2 c^3 d^3 \sqrt {1+c^2 x^2}}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \text {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{12 d^3}-\frac {b \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 c^3 d^3}-\frac {b^2 \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2 d^3} \\ & = \frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{2 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{4 c^4 d^3}-\frac {b^2 \text {Subst}\left (\int \left (-\frac {1}{c^2 \left (1+c^2 x\right )^2}+\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{12 d^3} \\ & = -\frac {b^2}{12 c^4 d^3 \left (1+c^2 x^2\right )}+\frac {b x^3 (a+b \text {arcsinh}(c x))}{6 c d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {b x (a+b \text {arcsinh}(c x))}{2 c^3 d^3 \sqrt {1+c^2 x^2}}-\frac {(a+b \text {arcsinh}(c x))^2}{4 c^4 d^3}+\frac {x^4 (a+b \text {arcsinh}(c x))^2}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {b^2 \log \left (1+c^2 x^2\right )}{3 c^4 d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=-\frac {3 a^2+b^2+6 a^2 c^2 x^2+b^2 c^2 x^2-6 a b c x \sqrt {1+c^2 x^2}-8 a b c^3 x^3 \sqrt {1+c^2 x^2}+2 b \left (-b c x \sqrt {1+c^2 x^2} \left (3+4 c^2 x^2\right )+a \left (3+6 c^2 x^2\right )\right ) \text {arcsinh}(c x)+3 b^2 \left (1+2 c^2 x^2\right ) \text {arcsinh}(c x)^2+4 \left (b+b c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )}{12 c^4 d^3 \left (1+c^2 x^2\right )^2} \]

[In]

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

[Out]

-1/12*(3*a^2 + b^2 + 6*a^2*c^2*x^2 + b^2*c^2*x^2 - 6*a*b*c*x*Sqrt[1 + c^2*x^2] - 8*a*b*c^3*x^3*Sqrt[1 + c^2*x^
2] + 2*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(3 + 4*c^2*x^2)) + a*(3 + 6*c^2*x^2))*ArcSinh[c*x] + 3*b^2*(1 + 2*c^2*x^2)
*ArcSinh[c*x]^2 + 4*(b + b*c^2*x^2)^2*Log[1 + c^2*x^2])/(c^4*d^3*(1 + c^2*x^2)^2)

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.63

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{2 \left (c^{2} x^{2}+1\right )}\right )}{d^{3}}+\frac {b^{2} \left (\frac {4 \,\operatorname {arcsinh}\left (c x \right )}{3}-\frac {-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+8 \,\operatorname {arcsinh}\left (c x \right )+1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\)	\(272\)
default	\(\frac {\frac {a^{2} \left (\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {1}{2 \left (c^{2} x^{2}+1\right )}\right )}{d^{3}}+\frac {b^{2} \left (\frac {4 \,\operatorname {arcsinh}\left (c x \right )}{3}-\frac {-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+8 \,\operatorname {arcsinh}\left (c x \right )+1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3}}}{c^{4}}\)	\(272\)
parts	\(\frac {a^{2} \left (-\frac {1}{2 c^{4} \left (c^{2} x^{2}+1\right )}+\frac {1}{4 c^{4} \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b^{2} \left (\frac {4 \,\operatorname {arcsinh}\left (c x \right )}{3}-\frac {-8 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+6 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+16 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+8 \,\operatorname {arcsinh}\left (c x \right )+1}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )}-\frac {2 \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3}\right )}{d^{3} c^{4}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right )}{4 \left (c^{2} x^{2}+1\right )^{2}}-\frac {\operatorname {arcsinh}\left (c x \right )}{2 \left (c^{2} x^{2}+1\right )}-\frac {c x}{12 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {c x}{3 \sqrt {c^{2} x^{2}+1}}\right )}{d^{3} c^{4}}\)	\(280\)

[In]

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

[Out]

1/c^4*(a^2/d^3*(1/4/(c^2*x^2+1)^2-1/2/(c^2*x^2+1))+b^2/d^3*(4/3*arcsinh(c*x)-1/12*(-8*arcsinh(c*x)*(c^2*x^2+1)
^(1/2)*x^3*c^3+8*arcsinh(c*x)*c^4*x^4+6*arcsinh(c*x)^2*x^2*c^2-6*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)+16*arcsinh
(c*x)*c^2*x^2+3*arcsinh(c*x)^2+c^2*x^2+8*arcsinh(c*x)+1)/(c^4*x^4+2*c^2*x^2+1)-2/3*ln(1+(c*x+(c^2*x^2+1)^(1/2)
)^2))+2*a*b/d^3*(1/4/(c^2*x^2+1)^2*arcsinh(c*x)-1/2/(c^2*x^2+1)*arcsinh(c*x)-1/12/(c^2*x^2+1)^(3/2)*c*x+1/3*c*
x/(c^2*x^2+1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.69 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {8 \, a b c^{4} x^{4} - {\left (6 \, a^{2} - 16 \, a b + b^{2}\right )} c^{2} x^{2} - 3 \, {\left (2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} - 3 \, a^{2} + 8 \, a b - b^{2} - 4 \, {\left (b^{2} c^{4} x^{4} + 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{4} + {\left (4 \, b^{2} c^{3} x^{3} + 3 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 6 \, {\left (a b c^{4} x^{4} + 2 \, a b c^{2} x^{2} + a b\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (4 \, a b c^{3} x^{3} + 3 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} + 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]

[In]

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

[Out]

1/12*(8*a*b*c^4*x^4 - (6*a^2 - 16*a*b + b^2)*c^2*x^2 - 3*(2*b^2*c^2*x^2 + b^2)*log(c*x + sqrt(c^2*x^2 + 1))^2
- 3*a^2 + 8*a*b - b^2 - 4*(b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*log(c^2*x^2 + 1) + 2*(3*a*b*c^4*x^4 + (4*b^2*c^3
*x^3 + 3*b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 6*(a*b*c^4*x^4 + 2*a*b*c^2*x^2 + a*b)*log(
-c*x + sqrt(c^2*x^2 + 1)) + 2*(4*a*b*c^3*x^3 + 3*a*b*c*x)*sqrt(c^2*x^2 + 1))/(c^8*d^3*x^4 + 2*c^6*d^3*x^2 + c^
4*d^3)

Sympy [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a^{2} x^{3}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{6} + 3 c^{4} x^{4} + 3 c^{2} x^{2} + 1}\, dx}{d^{3}} \]

[In]

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

[Out]

(Integral(a**2*x**3/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(b**2*x**3*asinh(c*x)**2/(c**6*x
**6 + 3*c**4*x**4 + 3*c**2*x**2 + 1), x) + Integral(2*a*b*x**3*asinh(c*x)/(c**6*x**6 + 3*c**4*x**4 + 3*c**2*x*
*2 + 1), x))/d**3

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

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