\(\int (\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [253]

Optimal result

Integrand size = 25, antiderivative size = 210 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {15}{64} b^2 \pi ^{3/2} x \sqrt {1+c^2 x^2}+\frac {1}{32} b^2 \pi ^{3/2} x \left (1+c^2 x^2\right )^{3/2}-\frac {9 b^2 \pi ^{3/2} \text {arcsinh}(c x)}{64 c}-\frac {3}{8} b c \pi ^{3/2} x^2 (a+b \text {arcsinh}(c x))-\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c} \]

[Out]

1/32*b^2*Pi^(3/2)*x*(c^2*x^2+1)^(3/2)-9/64*b^2*Pi^(3/2)*arcsinh(c*x)/c-3/8*b*c*Pi^(3/2)*x^2*(a+b*arcsinh(c*x))
-1/8*b*Pi^(3/2)*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))/c+1/4*x*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))^2+1/8*Pi^(3/
2)*(a+b*arcsinh(c*x))^3/b/c+15/64*b^2*Pi^(3/2)*x*(c^2*x^2+1)^(1/2)+3/8*Pi*x*(a+b*arcsinh(c*x))^2*(Pi*c^2*x^2+P
i)^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5786, 5785, 5783, 5776, 327, 221, 5798, 201} \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} x \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {3}{8} \pi x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))^2-\frac {\pi ^{3/2} b \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{8 c}-\frac {3}{8} \pi ^{3/2} b c x^2 (a+b \text {arcsinh}(c x))+\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c}-\frac {9 \pi ^{3/2} b^2 \text {arcsinh}(c x)}{64 c}+\frac {1}{32} \pi ^{3/2} b^2 x \left (c^2 x^2+1\right )^{3/2}+\frac {15}{64} \pi ^{3/2} b^2 x \sqrt {c^2 x^2+1} \]

[In]

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

[Out]

(15*b^2*Pi^(3/2)*x*Sqrt[1 + c^2*x^2])/64 + (b^2*Pi^(3/2)*x*(1 + c^2*x^2)^(3/2))/32 - (9*b^2*Pi^(3/2)*ArcSinh[c
*x])/(64*c) - (3*b*c*Pi^(3/2)*x^2*(a + b*ArcSinh[c*x]))/8 - (b*Pi^(3/2)*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))/
(8*c) + (3*Pi*x*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x])^2)/8 + (x*(Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*
x])^2)/4 + (Pi^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c)

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 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 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 5786

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} (3 \pi ) \int \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{2} \left (b c \pi ^{3/2}\right ) \int x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{8} \left (3 \pi ^{3/2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{8} \left (b^2 \pi ^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx-\frac {1}{4} \left (3 b c \pi ^{3/2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx \\ & = \frac {1}{32} b^2 \pi ^{3/2} x \left (1+c^2 x^2\right )^{3/2}-\frac {3}{8} b c \pi ^{3/2} x^2 (a+b \text {arcsinh}(c x))-\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c}+\frac {1}{32} \left (3 b^2 \pi ^{3/2}\right ) \int \sqrt {1+c^2 x^2} \, dx+\frac {1}{8} \left (3 b^2 c^2 \pi ^{3/2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx \\ & = \frac {15}{64} b^2 \pi ^{3/2} x \sqrt {1+c^2 x^2}+\frac {1}{32} b^2 \pi ^{3/2} x \left (1+c^2 x^2\right )^{3/2}-\frac {3}{8} b c \pi ^{3/2} x^2 (a+b \text {arcsinh}(c x))-\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c}+\frac {1}{64} \left (3 b^2 \pi ^{3/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{16} \left (3 b^2 \pi ^{3/2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx \\ & = \frac {15}{64} b^2 \pi ^{3/2} x \sqrt {1+c^2 x^2}+\frac {1}{32} b^2 \pi ^{3/2} x \left (1+c^2 x^2\right )^{3/2}-\frac {9 b^2 \pi ^{3/2} \text {arcsinh}(c x)}{64 c}-\frac {3}{8} b c \pi ^{3/2} x^2 (a+b \text {arcsinh}(c x))-\frac {b \pi ^{3/2} \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{8 c}+\frac {3}{8} \pi x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{4} x \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^3}{8 b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.96 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\pi ^{3/2} \left (160 a^2 c x \sqrt {1+c^2 x^2}+64 a^2 c^3 x^3 \sqrt {1+c^2 x^2}+32 b^2 \text {arcsinh}(c x)^3-64 a b \cosh (2 \text {arcsinh}(c x))-4 a b \cosh (4 \text {arcsinh}(c x))+32 b^2 \sinh (2 \text {arcsinh}(c x))+b^2 \sinh (4 \text {arcsinh}(c x))+8 b \text {arcsinh}(c x)^2 (12 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))+4 \text {arcsinh}(c x) \left (-16 b^2 \cosh (2 \text {arcsinh}(c x))-b^2 \cosh (4 \text {arcsinh}(c x))+4 a (6 a+8 b \sinh (2 \text {arcsinh}(c x))+b \sinh (4 \text {arcsinh}(c x)))\right )\right )}{256 c} \]

[In]

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

[Out]

(Pi^(3/2)*(160*a^2*c*x*Sqrt[1 + c^2*x^2] + 64*a^2*c^3*x^3*Sqrt[1 + c^2*x^2] + 32*b^2*ArcSinh[c*x]^3 - 64*a*b*C
osh[2*ArcSinh[c*x]] - 4*a*b*Cosh[4*ArcSinh[c*x]] + 32*b^2*Sinh[2*ArcSinh[c*x]] + b^2*Sinh[4*ArcSinh[c*x]] + 8*
b*ArcSinh[c*x]^2*(12*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]) + 4*ArcSinh[c*x]*(-16*b^2*Cosh[2*A
rcSinh[c*x]] - b^2*Cosh[4*ArcSinh[c*x]] + 4*a*(6*a + 8*b*Sinh[2*ArcSinh[c*x]] + b*Sinh[4*ArcSinh[c*x]]))))/(25
6*c)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.38

method	result	size
default	\(\frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {3}{2}} \left (16 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+2 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+40 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x -40 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+17 c x \sqrt {c^{2} x^{2}+1}+8 \operatorname {arcsinh}\left (c x \right )^{3}-17 \,\operatorname {arcsinh}\left (c x \right )\right )}{64 c}+\frac {a b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}+10 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-5 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{8 c}\)	\(289\)
parts	\(\frac {a^{2} x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{4}+\frac {3 a^{2} \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{8}+\frac {3 a^{2} \pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{8 \sqrt {\pi \,c^{2}}}+\frac {b^{2} \pi ^{\frac {3}{2}} \left (16 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-8 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}+2 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+40 \operatorname {arcsinh}\left (c x \right )^{2} \sqrt {c^{2} x^{2}+1}\, c x -40 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+17 c x \sqrt {c^{2} x^{2}+1}+8 \operatorname {arcsinh}\left (c x \right )^{3}-17 \,\operatorname {arcsinh}\left (c x \right )\right )}{64 c}+\frac {a b \,\pi ^{\frac {3}{2}} \left (4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}-c^{4} x^{4}+10 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-5 c^{2} x^{2}+3 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{8 c}\)	\(289\)

[In]

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

[Out]

1/4*a^2*x*(Pi*c^2*x^2+Pi)^(3/2)+3/8*a^2*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)+3/8*a^2*Pi^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi
*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/64*b^2*Pi^(3/2)*(16*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*x^3*c^3-8*arcsinh(c*
x)*c^4*x^4+2*c^3*x^3*(c^2*x^2+1)^(1/2)+40*arcsinh(c*x)^2*(c^2*x^2+1)^(1/2)*c*x-40*arcsinh(c*x)*c^2*x^2+17*c*x*
(c^2*x^2+1)^(1/2)+8*arcsinh(c*x)^3-17*arcsinh(c*x))/c+1/8*a*b*Pi^(3/2)*(4*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c
^3-c^4*x^4+10*arcsinh(c*x)*c*x*(c^2*x^2+1)^(1/2)-5*c^2*x^2+3*arcsinh(c*x)^2-4)/c

Fricas [F]

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

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (197) = 394\).

Time = 2.91 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.93 \[ \int \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4} + \frac {5 \pi ^{\frac {3}{2}} a^{2} x \sqrt {c^{2} x^{2} + 1}}{8} + \frac {3 \pi ^{\frac {3}{2}} a^{2} \operatorname {asinh}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} a b c^{3} x^{4}}{8} + \frac {\pi ^{\frac {3}{2}} a b c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {5 \pi ^{\frac {3}{2}} a b c x^{2}}{8} + \frac {5 \pi ^{\frac {3}{2}} a b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4} + \frac {3 \pi ^{\frac {3}{2}} a b \operatorname {asinh}^{2}{\left (c x \right )}}{8 c} - \frac {\pi ^{\frac {3}{2}} b^{2} c^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {\pi ^{\frac {3}{2}} b^{2} c^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{32} - \frac {5 \pi ^{\frac {3}{2}} b^{2} c x^{2} \operatorname {asinh}{\left (c x \right )}}{8} + \frac {5 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {17 \pi ^{\frac {3}{2}} b^{2} x \sqrt {c^{2} x^{2} + 1}}{64} + \frac {\pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}^{3}{\left (c x \right )}}{8 c} - \frac {17 \pi ^{\frac {3}{2}} b^{2} \operatorname {asinh}{\left (c x \right )}}{64 c} & \text {for}\: c \neq 0 \\\pi ^{\frac {3}{2}} a^{2} x & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((pi**(3/2)*a**2*c**2*x**3*sqrt(c**2*x**2 + 1)/4 + 5*pi**(3/2)*a**2*x*sqrt(c**2*x**2 + 1)/8 + 3*pi**(
3/2)*a**2*asinh(c*x)/(8*c) - pi**(3/2)*a*b*c**3*x**4/8 + pi**(3/2)*a*b*c**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x
)/2 - 5*pi**(3/2)*a*b*c*x**2/8 + 5*pi**(3/2)*a*b*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/4 + 3*pi**(3/2)*a*b*asinh(c*
x)**2/(8*c) - pi**(3/2)*b**2*c**3*x**4*asinh(c*x)/8 + pi**(3/2)*b**2*c**2*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)*
*2/4 + pi**(3/2)*b**2*c**2*x**3*sqrt(c**2*x**2 + 1)/32 - 5*pi**(3/2)*b**2*c*x**2*asinh(c*x)/8 + 5*pi**(3/2)*b*
*2*x*sqrt(c**2*x**2 + 1)*asinh(c*x)**2/8 + 17*pi**(3/2)*b**2*x*sqrt(c**2*x**2 + 1)/64 + pi**(3/2)*b**2*asinh(c
*x)**3/(8*c) - 17*pi**(3/2)*b**2*asinh(c*x)/(64*c), Ne(c, 0)), (pi**(3/2)*a**2*x, True))

Maxima [F(-2)]

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

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

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

[In]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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