\(\int x^2 (\pi +c^2 \pi x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [64]

Optimal result

Integrand size = 26, antiderivative size = 165 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \pi ^{3/2} x^2}{32 c}-\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 165, 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 = {5808, 5806, 5812, 5783, 30, 14} \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3}+\frac {\pi ^{3/2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} \pi x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{36} \pi ^{3/2} b c^3 x^6-\frac {7}{96} \pi ^{3/2} b c x^4-\frac {\pi ^{3/2} b x^2}{32 c} \]

[In]

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

[Out]

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

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*} \text {integral}& = \frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} \pi \int x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx-\frac {1}{6} \left (b c \pi ^{3/2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx \\ & = \frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} \pi ^{3/2} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{8} \left (b c \pi ^{3/2}\right ) \int x^3 \, dx-\frac {1}{6} \left (b c \pi ^{3/2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx \\ & = -\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2}-\frac {\left (b \pi ^{3/2}\right ) \int x \, dx}{16 c} \\ & = -\frac {b \pi ^{3/2} x^2}{32 c}-\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{3/2} \left (144 a c x \sqrt {1+c^2 x^2}+672 a c^3 x^3 \sqrt {1+c^2 x^2}+384 a c^5 x^5 \sqrt {1+c^2 x^2}-72 b \text {arcsinh}(c x)^2+18 b \cosh (2 \text {arcsinh}(c x))-9 b \cosh (4 \text {arcsinh}(c x))-2 b \cosh (6 \text {arcsinh}(c x))-12 \text {arcsinh}(c x) (12 a+3 b \sinh (2 \text {arcsinh}(c x))-3 b \sinh (4 \text {arcsinh}(c x))-b \sinh (6 \text {arcsinh}(c x)))\right )}{2304 c^3} \]

[In]

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

[Out]

(Pi^(3/2)*(144*a*c*x*Sqrt[1 + c^2*x^2] + 672*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 384*a*c^5*x^5*Sqrt[1 + c^2*x^2] - 7
2*b*ArcSinh[c*x]^2 + 18*b*Cosh[2*ArcSinh[c*x]] - 9*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] - 12*ArcS
inh[c*x]*(12*a + 3*b*Sinh[2*ArcSinh[c*x]] - 3*b*Sinh[4*ArcSinh[c*x]] - b*Sinh[6*ArcSinh[c*x]])))/(2304*c^3)

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.30

method	result	size
default	\(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {3}{2}} \left (-48 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}-84 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+21 c^{4} x^{4}-18 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+9 c^{2} x^{2}+9 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{288 c^{3}}\)	\(214\)
parts	\(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {3}{2}} \left (-48 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}-84 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+21 c^{4} x^{4}-18 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+9 c^{2} x^{2}+9 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{288 c^{3}}\)	\(214\)

[In]

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

[Out]

1/6*a*x*(Pi*c^2*x^2+Pi)^(5/2)/Pi/c^2-1/24*a/c^2*x*(Pi*c^2*x^2+Pi)^(3/2)-1/16*a/c^2*Pi*x*(Pi*c^2*x^2+Pi)^(1/2)-
1/16*a/c^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)-1/288*b*Pi^(3/2)*(-48*arcsinh
(c*x)*(c^2*x^2+1)^(1/2)*x^5*c^5+8*c^6*x^6-84*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3+21*c^4*x^4-18*arcsinh(c*x)
*c*x*(c^2*x^2+1)^(1/2)+9*c^2*x^2+9*arcsinh(c*x)^2-4)/c^3

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 5.49 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.59 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{6} + \frac {7 \pi ^{\frac {3}{2}} a x^{3} \sqrt {c^{2} x^{2} + 1}}{24} + \frac {\pi ^{\frac {3}{2}} a x \sqrt {c^{2} x^{2} + 1}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} a \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{6}}{36} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {7 \pi ^{\frac {3}{2}} b c x^{4}}{96} + \frac {7 \pi ^{\frac {3}{2}} b x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{24} - \frac {\pi ^{\frac {3}{2}} b x^{2}}{32 c} + \frac {\pi ^{\frac {3}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{3}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{3}}{3} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((pi**(3/2)*a*c**2*x**5*sqrt(c**2*x**2 + 1)/6 + 7*pi**(3/2)*a*x**3*sqrt(c**2*x**2 + 1)/24 + pi**(3/2)
*a*x*sqrt(c**2*x**2 + 1)/(16*c**2) - pi**(3/2)*a*asinh(c*x)/(16*c**3) - pi**(3/2)*b*c**3*x**6/36 + pi**(3/2)*b
*c**2*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/6 - 7*pi**(3/2)*b*c*x**4/96 + 7*pi**(3/2)*b*x**3*sqrt(c**2*x**2 + 1)
*asinh(c*x)/24 - pi**(3/2)*b*x**2/(32*c) + pi**(3/2)*b*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(16*c**2) - pi**(3/2)*
b*asinh(c*x)**2/(32*c**3), Ne(c, 0)), (pi**(3/2)*a*x**3/3, True))

Maxima [F(-2)]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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