3.3.92 \(\int \frac {x^3 (a+b \sinh ^{-1}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [292]

Optimal. Leaf size=265 \[ \frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {14 b^2 \left (1+c^2 x^2\right )}{9 c^4 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^4 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d} \]

[Out]

-14/9*b^2*(c^2*x^2+1)/c^4/(c^2*d*x^2+d)^(1/2)+2/27*b^2*(c^2*x^2+1)^2/c^4/(c^2*d*x^2+d)^(1/2)+4/3*a*b*x*(c^2*x^
2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)+4/3*b^2*x*arcsinh(c*x)*(c^2*x^2+1)^(1/2)/c^3/(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/(c^2*d*x^2+d)^(1/2)-2/3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^4/d+
1/3*x^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^2/d

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Rubi [A]
time = 0.22, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5812, 5798, 5772, 267, 5776, 272, 45} \begin {gather*} \frac {x^2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {2 b x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{9 c \sqrt {c^2 d x^2+d}}-\frac {2 \sqrt {c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {4 a b x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}}+\frac {2 b^2 \left (c^2 x^2+1\right )^2}{27 c^4 \sqrt {c^2 d x^2+d}}-\frac {14 b^2 \left (c^2 x^2+1\right )}{9 c^4 \sqrt {c^2 d x^2+d}}+\frac {4 b^2 x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)}{3 c^3 \sqrt {c^2 d x^2+d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*b*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d*x^2]) - (14*b^2*(1 + c^2*x^2))/(9*c^4*Sqrt[d + c^2*d*x^2]) +
(2*b^2*(1 + c^2*x^2)^2)/(27*c^4*Sqrt[d + c^2*d*x^2]) + (4*b^2*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(3*c^3*Sqrt[d
+ c^2*d*x^2]) - (2*b*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c*Sqrt[d + c^2*d*x^2]) - (2*Sqrt[d + c^2*
d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^4*d) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^2*d)

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 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 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 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^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}-\frac {2 \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx}{3 c^2}-\frac {\left (2 b \sqrt {1+c^2 x^2}\right ) \int x^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c \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 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\frac {\left (2 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{9 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b \sqrt {1+c^2 x^2}\right ) \int \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 c^3 \sqrt {d+c^2 d x^2}}\\ &=\frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \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 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\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 )}{9 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \int \sinh ^{-1}(c x) \, dx}{3 c^3 \sqrt {d+c^2 d x^2}}\\ &=\frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^3 \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 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}+\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 )}{9 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{3 c^2 \sqrt {d+c^2 d x^2}}\\ &=\frac {4 a b x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {14 b^2 \left (1+c^2 x^2\right )}{9 c^4 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^2}{27 c^4 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 x \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)}{3 c^3 \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 \sqrt {d+c^2 d x^2}}-\frac {2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^4 d}+\frac {x^2 \sqrt {d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 c^2 d}\\ \end {align*}

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Mathematica [A]
time = 0.17, size = 176, normalized size = 0.66 \begin {gather*} \frac {-6 a b c x \left (-6+c^2 x^2\right ) \sqrt {1+c^2 x^2}+2 b^2 \left (-20-19 c^2 x^2+c^4 x^4\right )+9 a^2 \left (-2-c^2 x^2+c^4 x^4\right )-6 b \left (b c x \left (-6+c^2 x^2\right ) \sqrt {1+c^2 x^2}+a \left (6+3 c^2 x^2-3 c^4 x^4\right )\right ) \sinh ^{-1}(c x)+9 b^2 \left (-2-c^2 x^2+c^4 x^4\right ) \sinh ^{-1}(c x)^2}{27 c^4 \sqrt {d+c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*a*b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + 2*b^2*(-20 - 19*c^2*x^2 + c^4*x^4) + 9*a^2*(-2 - c^2*x^2 + c^4*
x^4) - 6*b*(b*c*x*(-6 + c^2*x^2)*Sqrt[1 + c^2*x^2] + a*(6 + 3*c^2*x^2 - 3*c^4*x^4))*ArcSinh[c*x] + 9*b^2*(-2 -
c^2*x^2 + c^4*x^4)*ArcSinh[c*x]^2)/(27*c^4*Sqrt[d + c^2*d*x^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(705\) vs. \(2(231)=462\).
time = 3.43, size = 706, normalized size = 2.66

method	result	size
default	\(a^{2} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (9 \arcsinh \left (c x \right )^{2}-6 \arcsinh \left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (\arcsinh \left (c x \right )^{2}-2 \arcsinh \left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (\arcsinh \left (c x \right )^{2}+2 \arcsinh \left (c x \right )+2\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (9 \arcsinh \left (c x \right )^{2}+6 \arcsinh \left (c x \right )+2\right )}{216 c^{4} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (-1+3 \arcsinh \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (\arcsinh \left (c x \right )-1\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}-\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+\arcsinh \left (c x \right )\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, c x +1\right ) \left (1+3 \arcsinh \left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

a^2*(1/3*x^2/c^2/d*(c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2))+b^2*(1/216*(d*(c^2*x^2+1))^(1/2)*(4*c^4*
x^4+4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*c*x+1)*(9*arcsinh(c*x)^2-6*arcsinh(c*x)+2)/c^4/d
/(c^2*x^2+1)-3/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2)/c^4
/d/(c^2*x^2+1)-3/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)^2+2*arcsinh(c*x)+2)/c
^4/d/(c^2*x^2+1)+1/216*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2-3*(c^2*x^2+1)^(1
/2)*c*x+1)*(9*arcsinh(c*x)^2+6*arcsinh(c*x)+2)/c^4/d/(c^2*x^2+1))+2*a*b*(1/72*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4
+4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2+3*(c^2*x^2+1)^(1/2)*c*x+1)*(-1+3*arcsinh(c*x))/c^4/d/(c^2*x^2+1)-3/8*(d
*(c^2*x^2+1))^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*c*x+1)*(arcsinh(c*x)-1)/c^4/d/(c^2*x^2+1)-3/8*(d*(c^2*x^2+1))^(
1/2)*(c^2*x^2-(c^2*x^2+1)^(1/2)*c*x+1)*(1+arcsinh(c*x))/c^4/d/(c^2*x^2+1)+1/72*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^
4-4*(c^2*x^2+1)^(1/2)*x^3*c^3+5*c^2*x^2-3*(c^2*x^2+1)^(1/2)*c*x+1)*(1+3*arcsinh(c*x))/c^4/d/(c^2*x^2+1))

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Maxima [A]
time = 0.29, size = 243, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, b^{2} {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{3} \, a b {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, a^{2} {\left (\frac {\sqrt {c^{2} d x^{2} + d} x^{2}}{c^{2} d} - \frac {2 \, \sqrt {c^{2} d x^{2} + d}}{c^{4} d}\right )} + \frac {2}{27} \, b^{2} {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2} - \frac {20 \, \sqrt {c^{2} x^{2} + 1}}{c^{2}}}{c^{2} \sqrt {d}} - \frac {3 \, {\left (c^{2} x^{3} - 6 \, x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3} \sqrt {d}}\right )} - \frac {2 \, {\left (c^{2} x^{3} - 6 \, x\right )} a b}{9 \, c^{3} \sqrt {d}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

1/3*b^2*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d) - 2*sqrt(c^2*d*x^2 + d)/(c^4*d))*arcsinh(c*x)^2 + 2/3*a*b*(sqrt(c^2*d
*x^2 + d)*x^2/(c^2*d) - 2*sqrt(c^2*d*x^2 + d)/(c^4*d))*arcsinh(c*x) + 1/3*a^2*(sqrt(c^2*d*x^2 + d)*x^2/(c^2*d)
- 2*sqrt(c^2*d*x^2 + d)/(c^4*d)) + 2/27*b^2*((sqrt(c^2*x^2 + 1)*x^2 - 20*sqrt(c^2*x^2 + 1)/c^2)/(c^2*sqrt(d))
- 3*(c^2*x^3 - 6*x)*arcsinh(c*x)/(c^3*sqrt(d))) - 2/9*(c^2*x^3 - 6*x)*a*b/(c^3*sqrt(d))

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Fricas [A]
time = 0.39, size = 254, normalized size = 0.96 \begin {gather*} \frac {9 \, {\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{4} x^{4} - 3 \, a b c^{2} x^{2} - 6 \, a b - {\left (b^{2} c^{3} x^{3} - 6 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} x^{4} - {\left (9 \, a^{2} + 38 \, b^{2}\right )} c^{2} x^{2} - 18 \, a^{2} - 40 \, b^{2} - 6 \, {\left (a b c^{3} x^{3} - 6 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{27 \, {\left (c^{6} d x^{2} + c^{4} d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

1/27*(9*(b^2*c^4*x^4 - b^2*c^2*x^2 - 2*b^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 6*(3*a*b*c^4*
x^4 - 3*a*b*c^2*x^2 - 6*a*b - (b^2*c^3*x^3 - 6*b^2*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(
c^2*x^2 + 1)) + ((9*a^2 + 2*b^2)*c^4*x^4 - (9*a^2 + 38*b^2)*c^2*x^2 - 18*a^2 - 40*b^2 - 6*(a*b*c^3*x^3 - 6*a*b
*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*d*x^2 + c^4*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), 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^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),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^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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