\(\int \frac {x^2 (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\) [93]

Optimal result

Integrand size = 26, antiderivative size = 80 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]

[Out]

1/2*(a+b*arcsinh(c*x))^2/b/c^3/Pi^(3/2)+1/2*b*ln(c^2*x^2+1)/c^3/Pi^(3/2)-x*(a+b*arcsinh(c*x))/c^2/Pi/(Pi*c^2*x
^2+Pi)^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5810, 5783, 266} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{2 \pi ^{3/2} b c^3}-\frac {x (a+b \text {arcsinh}(c x))}{\pi c^2 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \log \left (c^2 x^2+1\right )}{2 \pi ^{3/2} c^3} \]

[In]

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

[Out]

-((x*(a + b*ArcSinh[c*x]))/(c^2*Pi*Sqrt[Pi + c^2*Pi*x^2])) + (a + b*ArcSinh[c*x])^2/(2*b*c^3*Pi^(3/2)) + (b*Lo
g[1 + c^2*x^2])/(2*c^3*Pi^(3/2))

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 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 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 (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {b \int \frac {x}{1+c^2 x^2} \, dx}{c \pi ^{3/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^2 \pi } \\ & = -\frac {x (a+b \text {arcsinh}(c x))}{c^2 \pi \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^3 \pi ^{3/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.98 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-\frac {2 a c x}{\sqrt {1+c^2 x^2}}+\left (2 a-\frac {2 b c x}{\sqrt {1+c^2 x^2}}\right ) \text {arcsinh}(c x)+b \text {arcsinh}(c x)^2+b \log \left (1+c^2 x^2\right )}{2 c^3 \pi ^{3/2}} \]

[In]

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

[Out]

((-2*a*c*x)/Sqrt[1 + c^2*x^2] + (2*a - (2*b*c*x)/Sqrt[1 + c^2*x^2])*ArcSinh[c*x] + b*ArcSinh[c*x]^2 + b*Log[1
+ c^2*x^2])/(2*c^3*Pi^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(70)=140\).

Time = 0.21 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.10

method	result	size
default	\(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\right )\)	\(168\)
parts	\(-\frac {a x}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{\pi \,c^{2} \sqrt {\pi \,c^{2}}}+b \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{3} \pi ^{\frac {3}{2}}}-\frac {2 \,\operatorname {arcsinh}\left (c x \right )}{c^{3} \pi ^{\frac {3}{2}}}+\frac {\left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) \operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {3}{2}} c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{c^{3} \pi ^{\frac {3}{2}}}\right )\)	\(168\)

[In]

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

[Out]

-a*x/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)+a/Pi/c^2*ln(Pi*c^2*x/(Pi*c^2)^(1/2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+b*
(1/2/c^3/Pi^(3/2)*arcsinh(c*x)^2-2/c^3/Pi^(3/2)*arcsinh(c*x)+1/Pi^(3/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*arcs
inh(c*x)/c^3/(c^2*x^2+1)+1/c^3/Pi^(3/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)