\(\int \frac {x^5 (a+b \text {arcsinh}(c x))}{(\pi +c^2 \pi x^2)^{3/2}} \, dx\) [90]

Optimal result

Integrand size = 26, antiderivative size = 137 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}+\frac {b \arctan (c x)}{c^6 \pi ^{3/2}} \]

[Out]

5/3*b*x/c^5/Pi^(3/2)-1/9*b*x^3/c^3/Pi^(3/2)+1/3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/c^6/Pi^3+b*arctan(c*x
)/c^6/Pi^(3/2)+(-a-b*arcsinh(c*x))/c^6/Pi/(Pi*c^2*x^2+Pi)^(1/2)-2*(a+b*arcsinh(c*x))*(Pi*c^2*x^2+Pi)^(1/2)/c^6
/Pi^2

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {272, 45, 5804, 12, 1167, 209} \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi ^3 c^6}-\frac {2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{\pi ^2 c^6}-\frac {a+b \text {arcsinh}(c x)}{\pi c^6 \sqrt {\pi c^2 x^2+\pi }}+\frac {b \arctan (c x)}{\pi ^{3/2} c^6}+\frac {5 b x}{3 \pi ^{3/2} c^5}-\frac {b x^3}{9 \pi ^{3/2} c^3} \]

[In]

Int[(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]

[Out]

(5*b*x)/(3*c^5*Pi^(3/2)) - (b*x^3)/(9*c^3*Pi^(3/2)) - (a + b*ArcSinh[c*x])/(c^6*Pi*Sqrt[Pi + c^2*Pi*x^2]) - (2
*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]))/(c^6*Pi^2) + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^
6*Pi^3) + (b*ArcTan[c*x])/(c^6*Pi^(3/2))

Rule 12

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

Rule 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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 1167

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -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)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}-\left (b c \sqrt {\pi }\right ) \int \frac {-8-4 c^2 x^2+c^4 x^4}{3 c^6 \pi ^2 \left (1+c^2 x^2\right )} \, dx \\ & = -\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}-\frac {b \int \frac {-8-4 c^2 x^2+c^4 x^4}{1+c^2 x^2} \, dx}{3 c^5 \pi ^{3/2}} \\ & = -\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}-\frac {b \int \left (-5+c^2 x^2-\frac {3}{1+c^2 x^2}\right ) \, dx}{3 c^5 \pi ^{3/2}} \\ & = \frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}+\frac {b \int \frac {1}{1+c^2 x^2} \, dx}{c^5 \pi ^{3/2}} \\ & = \frac {5 b x}{3 c^5 \pi ^{3/2}}-\frac {b x^3}{9 c^3 \pi ^{3/2}}-\frac {a+b \text {arcsinh}(c x)}{c^6 \pi \sqrt {\pi +c^2 \pi x^2}}-\frac {2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi ^3}+\frac {b \arctan (c x)}{c^6 \pi ^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.96 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {-24 a-12 a c^2 x^2+3 a c^4 x^4+15 b c x \sqrt {1+c^2 x^2}-b c^3 x^3 \sqrt {1+c^2 x^2}+3 b \left (-8-4 c^2 x^2+c^4 x^4\right ) \text {arcsinh}(c x)+9 b \sqrt {1+c^2 x^2} \arctan (c x)}{9 c^6 \pi ^{3/2} \sqrt {1+c^2 x^2}} \]

[In]

Integrate[(x^5*(a + b*ArcSinh[c*x]))/(Pi + c^2*Pi*x^2)^(3/2),x]

[Out]

(-24*a - 12*a*c^2*x^2 + 3*a*c^4*x^4 + 15*b*c*x*Sqrt[1 + c^2*x^2] - b*c^3*x^3*Sqrt[1 + c^2*x^2] + 3*b*(-8 - 4*c
^2*x^2 + c^4*x^4)*ArcSinh[c*x] + 9*b*Sqrt[1 + c^2*x^2]*ArcTan[c*x])/(9*c^6*Pi^(3/2)*Sqrt[1 + c^2*x^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.04

method	result	size
default	\(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )-\frac {i b \left (3 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-i x^{5} c^{5}-12 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+14 i x^{3} c^{3}-9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}-24 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+15 i c x -9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )\right )}{9 \pi ^{\frac {3}{2}} c^{6} \left (c^{2} x^{2}+1\right )}\)	\(279\)
parts	\(a \left (\frac {x^{4}}{3 \pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}-\frac {4 \left (\frac {x^{2}}{\pi \,c^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {2}{\pi \,c^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}\right )}{3 c^{2}}\right )-\frac {i b \left (3 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-i x^{5} c^{5}-12 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+14 i x^{3} c^{3}-9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right ) x^{2} c^{2}+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right ) x^{2} c^{2}-24 i \operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+15 i c x -9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )+9 \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )\right )}{9 \pi ^{\frac {3}{2}} c^{6} \left (c^{2} x^{2}+1\right )}\)	\(279\)

[In]

int(x^5*(a+b*arcsinh(c*x))/(Pi*c^2*x^2+Pi)^(3/2),x,method=_RETURNVERBOSE)

[Out]

a*(1/3*x^4/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)-4/3/c^2*(x^2/Pi/c^2/(Pi*c^2*x^2+Pi)^(1/2)+2/Pi/c^4/(Pi*c^2*x^2+Pi)^(1/
2)))-1/9*I*b*(3*I*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^4*c^4-I*x^5*c^5-12*I*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2
+14*I*x^3*c^3-9*ln(c*x+(c^2*x^2+1)^(1/2)+I)*x^2*c^2+9*ln(c*x+(c^2*x^2+1)^(1/2)-I)*x^2*c^2-24*I*arcsinh(c*x)*(c
^2*x^2+1)^(1/2)+15*I*c*x-9*ln(c*x+(c^2*x^2+1)^(1/2)+I)+9*ln(c*x+(c^2*x^2+1)^(1/2)-I))/Pi^(3/2)/c^6/(c^2*x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.43 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=-\frac {9 \, \sqrt {\pi } {\left (b c^{2} x^{2} + b\right )} \arctan \left (-\frac {2 \, \sqrt {\pi } \sqrt {\pi + \pi c^{2} x^{2}} \sqrt {c^{2} x^{2} + 1} c x}{\pi - \pi c^{4} x^{4}}\right ) - 6 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (b c^{4} x^{4} - 4 \, b c^{2} x^{2} - 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, a c^{4} x^{4} - 12 \, a c^{2} x^{2} - {\left (b c^{3} x^{3} - 15 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} - 24 \, a\right )}}{18 \, {\left (\pi ^{2} c^{8} x^{2} + \pi ^{2} c^{6}\right )}} \]

[In]

integrate(x^5*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="fricas")

[Out]

-1/18*(9*sqrt(pi)*(b*c^2*x^2 + b)*arctan(-2*sqrt(pi)*sqrt(pi + pi*c^2*x^2)*sqrt(c^2*x^2 + 1)*c*x/(pi - pi*c^4*
x^4)) - 6*sqrt(pi + pi*c^2*x^2)*(b*c^4*x^4 - 4*b*c^2*x^2 - 8*b)*log(c*x + sqrt(c^2*x^2 + 1)) - 2*sqrt(pi + pi*
c^2*x^2)*(3*a*c^4*x^4 - 12*a*c^2*x^2 - (b*c^3*x^3 - 15*b*c*x)*sqrt(c^2*x^2 + 1) - 24*a))/(pi^2*c^8*x^2 + pi^2*
c^6)

Sympy [F]

\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\frac {\int \frac {a x^{5}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx + \int \frac {b x^{5} \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{2} \sqrt {c^{2} x^{2} + 1} + \sqrt {c^{2} x^{2} + 1}}\, dx}{\pi ^{\frac {3}{2}}} \]

[In]

integrate(x**5*(a+b*asinh(c*x))/(pi*c**2*x**2+pi)**(3/2),x)

[Out]

(Integral(a*x**5/(c**2*x**2*sqrt(c**2*x**2 + 1) + sqrt(c**2*x**2 + 1)), x) + Integral(b*x**5*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]

\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{5}}{{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(x^5*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),x, algorithm="maxima")

[Out]

1/3*a*(x^4/(pi*sqrt(pi + pi*c^2*x^2)*c^2) - 4*x^2/(pi*sqrt(pi + pi*c^2*x^2)*c^4) - 8/(pi*sqrt(pi + pi*c^2*x^2)
*c^6)) + 1/3*b*((sqrt(pi)*c^4*x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt(pi))*log(c*x + sqrt(c^2*x^2 + 1))/(pi^2*sqrt(c
^2*x^2 + 1)*c^6) - integrate((sqrt(pi)*c^4*x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt(pi))/(sqrt(c^2*x^2 + 1)*x), x)/(p
i^2*c^6) + 3*integrate(1/3*(sqrt(pi)*c^4*x^4 - 4*sqrt(pi)*c^2*x^2 - 8*sqrt(pi))/(pi^2*c^9*x^4 + pi^2*c^7*x^2 +
(pi^2*c^8*x^3 + pi^2*c^6*x)*sqrt(c^2*x^2 + 1)), x))

Giac [F(-2)]

Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(x^5*(a+b*arcsinh(c*x))/(pi*c^2*x^2+pi)^(3/2),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. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\left (\pi +c^2 \pi x^2\right )^{3/2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2}} \,d x \]

[In]

int((x^5*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2),x)

[Out]

int((x^5*(a + b*asinh(c*x)))/(Pi + Pi*c^2*x^2)^(3/2), x)