\(\int \frac {a+b \text {arcsinh}(c x)}{x^2 (\pi +c^2 \pi x^2)^{5/2}} \, dx\) [108]

Optimal result

Integrand size = 26, antiderivative size = 150 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {b c}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b c \log (x)}{\pi ^{5/2}}+\frac {5 b c \log \left (1+c^2 x^2\right )}{6 \pi ^{5/2}} \]

[Out]

-1/6*b*c/Pi^(5/2)/(c^2*x^2+1)+(-a-b*arcsinh(c*x))/Pi/x/(Pi*c^2*x^2+Pi)^(3/2)-4/3*c^2*x*(a+b*arcsinh(c*x))/Pi/(
Pi*c^2*x^2+Pi)^(3/2)+b*c*ln(x)/Pi^(5/2)+5/6*b*c*ln(c^2*x^2+1)/Pi^(5/2)-8/3*c^2*x*(a+b*arcsinh(c*x))/Pi^2/(Pi*c
^2*x^2+Pi)^(1/2)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {277, 198, 197, 5804, 12, 1265, 907} \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi c^2 x^2+\pi }}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi c^2 x^2+\pi \right )^{3/2}}-\frac {b c}{6 \pi ^{5/2} \left (c^2 x^2+1\right )}+\frac {5 b c \log \left (c^2 x^2+1\right )}{6 \pi ^{5/2}}+\frac {b c \log (x)}{\pi ^{5/2}} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 907

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1265

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 5804

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x
^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[
SimplifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegerQ[p -
1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])

Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\left (b c \sqrt {\pi }\right ) \int \frac {-3-12 c^2 x^2-8 c^4 x^4}{3 \pi ^3 x \left (1+c^2 x^2\right )^2} \, dx \\ & = -\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {(b c) \int \frac {-3-12 c^2 x^2-8 c^4 x^4}{x \left (1+c^2 x^2\right )^2} \, dx}{3 \pi ^{5/2}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {(b c) \text {Subst}\left (\int \frac {-3-12 c^2 x-8 c^4 x^2}{x \left (1+c^2 x\right )^2} \, dx,x,x^2\right )}{6 \pi ^{5/2}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}-\frac {(b c) \text {Subst}\left (\int \left (-\frac {3}{x}-\frac {c^2}{\left (1+c^2 x\right )^2}-\frac {5 c^2}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 \pi ^{5/2}} \\ & = -\frac {b c}{6 \pi ^{5/2} \left (1+c^2 x^2\right )}-\frac {a+b \text {arcsinh}(c x)}{\pi x \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {4 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi \left (\pi +c^2 \pi x^2\right )^{3/2}}-\frac {8 c^2 x (a+b \text {arcsinh}(c x))}{3 \pi ^2 \sqrt {\pi +c^2 \pi x^2}}+\frac {b c \log (x)}{\pi ^{5/2}}+\frac {5 b c \log \left (1+c^2 x^2\right )}{6 \pi ^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.83 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {-b c x \sqrt {1+c^2 x^2}-2 a \left (3+12 c^2 x^2+8 c^4 x^4\right )-2 b \left (3+12 c^2 x^2+8 c^4 x^4\right ) \text {arcsinh}(c x)+b c x \left (1+c^2 x^2\right )^{3/2} \left (16+6 \log (x)+5 \log \left (1+c^2 x^2\right )\right )}{6 \pi ^{5/2} x \left (1+c^2 x^2\right )^{3/2}} \]

[In]

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

[Out]

(-(b*c*x*Sqrt[1 + c^2*x^2]) - 2*a*(3 + 12*c^2*x^2 + 8*c^4*x^4) - 2*b*(3 + 12*c^2*x^2 + 8*c^4*x^4)*ArcSinh[c*x]
+ b*c*x*(1 + c^2*x^2)^(3/2)*(16 + 6*Log[x] + 5*Log[1 + c^2*x^2]))/(6*Pi^(5/2)*x*(1 + c^2*x^2)^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(777\) vs. \(2(132)=264\).

Time = 0.18 (sec) , antiderivative size = 778, normalized size of antiderivative = 5.19

method	result	size
default	\(a \left (-\frac {1}{\pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-4 c^{2} \left (\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )-\frac {16 b c \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}}}+\frac {32 b \,x^{10} c^{11}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {128 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{6} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {64 b \,x^{5} \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {64 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{4} c^{5}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {200 b \,x^{4} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {56 b \,x^{3} \operatorname {arcsinh}\left (c x \right ) c^{4}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {128 b \,x^{4} c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {12 b \,x^{2} c^{3}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {208 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {44 b x \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b \,x^{2} c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c}{2 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {24 b \,\operatorname {arcsinh}\left (c x \right ) c}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 b \,\operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x}+\frac {5 b c \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{\pi ^{\frac {5}{2}}}\)	\(778\)
parts	\(a \left (-\frac {1}{\pi x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}-4 c^{2} \left (\frac {x}{3 \pi \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}+\frac {2 x}{3 \pi ^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )-\frac {16 b c \,\operatorname {arcsinh}\left (c x \right )}{3 \pi ^{\frac {5}{2}}}+\frac {32 b \,x^{10} c^{11}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {128 b \,x^{8} c^{9}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {64 b \,x^{6} \operatorname {arcsinh}\left (c x \right ) c^{7}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {64 b \,x^{5} \operatorname {arcsinh}\left (c x \right ) c^{6}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {64 b \,x^{6} c^{7}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {32 b \,x^{4} c^{5}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {200 b \,x^{4} \operatorname {arcsinh}\left (c x \right ) c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {56 b \,x^{3} \operatorname {arcsinh}\left (c x \right ) c^{4}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {128 b \,x^{4} c^{5}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {12 b \,x^{2} c^{3}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {208 b \,x^{2} \operatorname {arcsinh}\left (c x \right ) c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {44 b x \,\operatorname {arcsinh}\left (c x \right ) c^{2}}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {32 b \,x^{2} c^{3}}{3 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {3 b c}{2 \pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )}+\frac {24 b \,\operatorname {arcsinh}\left (c x \right ) c}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 b \,\operatorname {arcsinh}\left (c x \right )}{\pi ^{\frac {5}{2}} \left (8 c^{2} x^{2}+9\right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} x}+\frac {5 b c \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{3 \pi ^{\frac {5}{2}}}+\frac {b c \ln \left (\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}-1\right )}{\pi ^{\frac {5}{2}}}\)	\(778\)

[In]

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

[Out]

a*(-1/Pi/x/(Pi*c^2*x^2+Pi)^(3/2)-4*c^2*(1/3/Pi*x/(Pi*c^2*x^2+Pi)^(3/2)+2/3/Pi^2*x/(Pi*c^2*x^2+Pi)^(1/2)))-16/3
*b*c/Pi^(5/2)*arcsinh(c*x)+32/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^10*c^11-32/3*b/Pi^(5/2)/(8*c^2*x^2+9)
/(c^2*x^2+1)*x^8*c^9+128/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^8*c^9-32*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2
+1)*x^6*c^7+64/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^6*arcsinh(c*x)*c^7-64/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^
2*x^2+1)^(3/2)*x^5*arcsinh(c*x)*c^6+64*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^6*c^7-32*b/Pi^(5/2)/(8*c^2*x^2
+9)/(c^2*x^2+1)*x^4*c^5+200/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^4*arcsinh(c*x)*c^5-56*b/Pi^(5/2)/(8*c^2
*x^2+9)/(c^2*x^2+1)^(3/2)*x^3*arcsinh(c*x)*c^4+128/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^4*c^5-12*b/Pi^(5
/2)/(8*c^2*x^2+9)/(c^2*x^2+1)*x^2*c^3+208/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^2*arcsinh(c*x)*c^3-44*b/P
i^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^(3/2)*x*arcsinh(c*x)*c^2+32/3*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*x^2*c^3
-3/2*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)*c+24*b/Pi^(5/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^2*arcsinh(c*x)*c-9*b/Pi^(5
/2)/(8*c^2*x^2+9)/(c^2*x^2+1)^(3/2)/x*arcsinh(c*x)+5/3*b*c/Pi^(5/2)*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)+b*c/Pi^(5/
2)*ln((c*x+(c^2*x^2+1)^(1/2))^2-1)

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(pi + pi*c^2*x^2)*(b*arcsinh(c*x) + a)/(pi^3*c^6*x^8 + 3*pi^3*c^4*x^6 + 3*pi^3*c^2*x^4 + pi^3*x^2
), x)

Sympy [F]

\[ \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (\pi +c^2 \pi x^2\right )^{5/2}} \, dx=\frac {\int \frac {a}{c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} \sqrt {c^{2} x^{2} + 1} + 2 c^{2} x^{4} \sqrt {c^{2} x^{2} + 1} + x^{2} \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {5}{2}}} \]

[In]

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

[Out]

(Integral(a/(c**4*x**6*sqrt(c**2*x**2 + 1) + 2*c**2*x**4*sqrt(c**2*x**2 + 1) + x**2*sqrt(c**2*x**2 + 1)), x) +
Integral(b*asinh(c*x)/(c**4*x**6*sqrt(c**2*x**2 + 1) + 2*c**2*x**4*sqrt(c**2*x**2 + 1) + x**2*sqrt(c**2*x**2
+ 1)), x))/pi**(5/2)

Maxima [F]

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

[In]

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

[Out]

-1/3*a*(4*c^2*x/(pi*(pi + pi*c^2*x^2)^(3/2)) + 8*c^2*x/(pi^2*sqrt(pi + pi*c^2*x^2)) + 3/(pi*(pi + pi*c^2*x^2)^
(3/2)*x)) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/((pi + pi*c^2*x^2)^(5/2)*x^2), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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