\(\int \frac {x^4 (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{5/2}} \, dx\) [102]

Optimal result

Integrand size = 26, antiderivative size = 139 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^5 \pi ^{5/2}}+\frac {2 b \log \left (1+c^2 x^2\right )}{3 c^5 \pi ^{5/2}} \]

[Out]

1/6*b/c^5/Pi^(5/2)/(c^2*x^2+1)-1/3*x^3*(a+b*arcsinh(c*x))/c^2/Pi/(Pi*c^2*x^2+Pi)^(3/2)+1/2*(a+b*arcsinh(c*x))^
2/b/c^5/Pi^(5/2)+2/3*b*ln(c^2*x^2+1)/c^5/Pi^(5/2)-x*(a+b*arcsinh(c*x))/c^4/Pi^2/(Pi*c^2*x^2+Pi)^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {5810, 5783, 266, 272, 45} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {(a+b \text {arcsinh}(c x))^2}{2 \pi ^{5/2} b c^5}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 \pi c^2 \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{\pi ^2 c^4 \sqrt {\pi c^2 x^2+\pi }}+\frac {b}{6 \pi ^{5/2} c^5 \left (c^2 x^2+1\right )}+\frac {2 b \log \left (c^2 x^2+1\right )}{3 \pi ^{5/2} c^5} \]

[In]

Int[(x^4*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]

[Out]

b/(6*c^5*Pi^(5/2)*(1 + c^2*x^2)) - (x^3*(a + b*ArcSinh[c*x]))/(3*c^2*Pi*(Pi + c^2*Pi*x^2)^(3/2)) - (x*(a + b*A
rcSinh[c*x]))/(c^4*Pi^2*Sqrt[Pi + c^2*Pi*x^2]) + (a + b*ArcSinh[c*x])^2/(2*b*c^5*Pi^(5/2)) + (2*b*Log[1 + c^2*
x^2])/(3*c^5*Pi^(5/2))

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 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^3 (a+b \text {arcsinh}(c x))}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}+\frac {b \int \frac {x^3}{\left (1+c^2 x^2\right )^2} \, dx}{3 c \pi ^{5/2}}+\frac {\int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx}{c^2 \pi } \\ & = -\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b \int \frac {x}{1+c^2 x^2} \, dx}{c^3 \pi ^{5/2}}+\frac {b \text {Subst}\left (\int \frac {x}{\left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 c \pi ^{5/2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{c^4 \pi ^2} \\ & = -\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^5 \pi ^{5/2}}+\frac {b \log \left (1+c^2 x^2\right )}{2 c^5 \pi ^{5/2}}+\frac {b \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 )}{6 c \pi ^{5/2}} \\ & = \frac {b}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {x^3 (a+b \text {arcsinh}(c x))}{3 c^2 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))}{c^4 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {(a+b \text {arcsinh}(c x))^2}{2 b c^5 \pi ^{5/2}}+\frac {2 b \log \left (1+c^2 x^2\right )}{3 c^5 \pi ^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {b+b c^2 x^2-6 a c x \sqrt {1+c^2 x^2}-8 a c^3 x^3 \sqrt {1+c^2 x^2}+2 \left (3 a \left (1+c^2 x^2\right )^2-b c x \sqrt {1+c^2 x^2} \left (3+4 c^2 x^2\right )\right ) \text {arcsinh}(c x)+3 b \left (1+c^2 x^2\right )^2 \text {arcsinh}(c x)^2+4 b \left (1+c^2 x^2\right )^2 \log \left (1+c^2 x^2\right )}{6 c^5 \pi ^{5/2} \left (1+c^2 x^2\right )^2} \]

[In]

Integrate[(x^4*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(5/2),x]

[Out]

(b + b*c^2*x^2 - 6*a*c*x*Sqrt[1 + c^2*x^2] - 8*a*c^3*x^3*Sqrt[1 + c^2*x^2] + 2*(3*a*(1 + c^2*x^2)^2 - b*c*x*Sq
rt[1 + c^2*x^2]*(3 + 4*c^2*x^2))*ArcSinh[c*x] + 3*b*(1 + c^2*x^2)^2*ArcSinh[c*x]^2 + 4*b*(1 + c^2*x^2)^2*Log[1
+ c^2*x^2])/(6*c^5*Pi^(5/2)*(1 + c^2*x^2)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(896\) vs. \(2(121)=242\).

Time = 0.17 (sec) , antiderivative size = 897, normalized size of antiderivative = 6.45

method	result	size
default	\(-\frac {a \,x^{3}}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {a x}{\pi ^{2} c^{4} \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 ^{2} c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{5} \pi ^{\frac {5}{2}}}-\frac {8 b \,\operatorname {arcsinh}\left (c x \right )}{3 c^{5} \pi ^{\frac {5}{2}}}+\frac {32 b \,c^{3} \operatorname {arcsinh}\left (c x \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{3} x^{8}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )}+\frac {116 b c \,\operatorname {arcsinh}\left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {76 b \,\operatorname {arcsinh}\left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {4 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c}+\frac {472 b \,\operatorname {arcsinh}\left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {181 b \,\operatorname {arcsinh}\left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {16 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {3 b \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c^{3}}+\frac {284 b \,\operatorname {arcsinh}\left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {16 b \,\operatorname {arcsinh}\left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{4}}+\frac {32 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}+\frac {64 b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {8 b}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {4 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{5} \pi ^{\frac {5}{2}}}\)	\(897\)
parts	\(-\frac {a \,x^{3}}{3 \pi \,c^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-\frac {a x}{\pi ^{2} c^{4} \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 ^{2} c^{4} \sqrt {\pi \,c^{2}}}+\frac {b \operatorname {arcsinh}\left (c x \right )^{2}}{2 c^{5} \pi ^{\frac {5}{2}}}-\frac {8 b \,\operatorname {arcsinh}\left (c x \right )}{3 c^{5} \pi ^{\frac {5}{2}}}+\frac {32 b \,c^{3} \operatorname {arcsinh}\left (c x \right ) x^{8}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,c^{2} \operatorname {arcsinh}\left (c x \right ) x^{7}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {8 b \,c^{3} x^{8}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {8 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )}+\frac {116 b c \,\operatorname {arcsinh}\left (c x \right ) x^{6}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {76 b \,\operatorname {arcsinh}\left (c x \right ) x^{5}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b c \,x^{6}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {4 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c}+\frac {472 b \,\operatorname {arcsinh}\left (c x \right ) x^{4}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {181 b \,\operatorname {arcsinh}\left (c x \right ) x^{3}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{2}}+\frac {16 b \,x^{4}}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c}-\frac {3 b \,x^{2}}{2 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right ) c^{3}}+\frac {284 b \,\operatorname {arcsinh}\left (c x \right ) x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}-\frac {16 b \,\operatorname {arcsinh}\left (c x \right ) x}{\pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} c^{4}}+\frac {32 b \,x^{2}}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{3}}+\frac {64 b \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {8 b}{3 \pi ^{\frac {5}{2}} \left (24 c^{4} x^{4}+39 c^{2} x^{2}+16\right ) \left (c^{2} x^{2}+1\right )^{2} c^{5}}+\frac {4 b \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 c^{5} \pi ^{\frac {5}{2}}}\)	\(897\)

[In]

int(x^4*(a+b*arcsinh(c*x))/(Pi*c^2*x^2+Pi)^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/3*a*x^3/Pi/c^2/(Pi*c^2*x^2+Pi)^(3/2)-a/Pi^2/c^4*x/(Pi*c^2*x^2+Pi)^(1/2)+a/Pi^2/c^4*ln(Pi*c^2*x/(Pi*c^2)^(1/
2)+(Pi*c^2*x^2+Pi)^(1/2))/(Pi*c^2)^(1/2)+1/2*b/c^5/Pi^(5/2)*arcsinh(c*x)^2-8/3*b/c^5/Pi^(5/2)*arcsinh(c*x)+32*
b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c^3*arcsinh(c*x)*x^8-32*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+
16)/(c^2*x^2+1)^(3/2)*c^2*arcsinh(c*x)*x^7+8/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c^3*x^8-8/3
*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)*c*x^6+116*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)
^2*c*arcsinh(c*x)*x^6-76*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^(3/2)*arcsinh(c*x)*x^5+32/3*b/Pi^(5
/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2*c*x^6-4*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)/c*x^4+4
72/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/c*arcsinh(c*x)*x^4-181/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^
2*x^2+16)/(c^2*x^2+1)^(3/2)/c^2*arcsinh(c*x)*x^3+16*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/c*x^4-
3/2*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)/c^3*x^2+284/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2
*x^2+1)^2/c^3*arcsinh(c*x)*x^2-16*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^(3/2)/c^4*arcsinh(c*x)*x+3
2/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/c^3*x^2+64/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^
2*x^2+1)^2/c^5*arcsinh(c*x)+8/3*b/Pi^(5/2)/(24*c^4*x^4+39*c^2*x^2+16)/(c^2*x^2+1)^2/c^5+4/3*b/c^5/Pi^(5/2)*ln(
1+(c*x+(c^2*x^2+1)^(1/2))^2)

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="fricas")

[Out]

integral(sqrt(pi + pi*c^2*x^2)*(b*x^4*arcsinh(c*x) + a*x^4)/(pi^3*c^6*x^6 + 3*pi^3*c^4*x^4 + 3*pi^3*c^2*x^2 +
pi^3), x)

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a x^{4}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{4} \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{4} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]

[In]

integrate(x**4*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(5/2),x)

[Out]

(Integral(a*x**4/(c**4*x**4*sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) +
Integral(b*x**4*asinh(c*x)/(c**4*x**4*sqrt(c**2*x**2 + 1) + 2*c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2
+ 1)), x))/pi**(5/2)

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="maxima")

[Out]

-1/3*(x*(3*x^2/(pi*(pi + pi*c^2*x^2)^(3/2)*c^2) + 2/(pi*(pi + pi*c^2*x^2)^(3/2)*c^4)) + x/(pi^2*sqrt(pi + pi*c
^2*x^2)*c^4) - 3*arcsinh(c*x)/(pi^(5/2)*c^5))*a + b*integrate(x^4*log(c*x + sqrt(c^2*x^2 + 1))/(pi + pi*c^2*x^
2)^(5/2), x)

Giac [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{4}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)*x^4/(pi + pi*c^2*x^2)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}} \,d x \]

[In]

int((x^4*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(5/2),x)

[Out]

int((x^4*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(5/2), x)