3.2037 \(\int \frac{x^5}{(a+\frac{b}{x^3})^{3/2}} \, dx\)

Optimal. Leaf size=95 \[ -\frac{5 b^2}{4 a^3 \sqrt{a+\frac{b}{x^3}}}+\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{7/2}}-\frac{5 b x^3}{12 a^2 \sqrt{a+\frac{b}{x^3}}}+\frac{x^6}{6 a \sqrt{a+\frac{b}{x^3}}} \]

[Out]

(-5*b^2)/(4*a^3*Sqrt[a + b/x^3]) - (5*b*x^3)/(12*a^2*Sqrt[a + b/x^3]) + x^6/(6*a*Sqrt[a + b/x^3]) + (5*b^2*Arc
Tanh[Sqrt[a + b/x^3]/Sqrt[a]])/(4*a^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.046514, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 208} \[ \frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{7/2}}+\frac{5 x^6 \sqrt{a+\frac{b}{x^3}}}{6 a^2}-\frac{5 b x^3 \sqrt{a+\frac{b}{x^3}}}{4 a^3}-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x^3)^(3/2),x]

[Out]

(-5*b*Sqrt[a + b/x^3]*x^3)/(4*a^3) - (2*x^6)/(3*a*Sqrt[a + b/x^3]) + (5*Sqrt[a + b/x^3]*x^6)/(6*a^2) + (5*b^2*
ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/(4*a^(7/2))

Rule 266

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 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{x^5}{\left (a+\frac{b}{x^3}\right )^{3/2}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^{3/2}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}}-\frac{5 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )}{3 a}\\ &=-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{5 \sqrt{a+\frac{b}{x^3}} x^6}{6 a^2}+\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )}{4 a^2}\\ &=-\frac{5 b \sqrt{a+\frac{b}{x^3}} x^3}{4 a^3}-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{5 \sqrt{a+\frac{b}{x^3}} x^6}{6 a^2}-\frac{\left (5 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x^3}\right )}{8 a^3}\\ &=-\frac{5 b \sqrt{a+\frac{b}{x^3}} x^3}{4 a^3}-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{5 \sqrt{a+\frac{b}{x^3}} x^6}{6 a^2}-\frac{(5 b) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x^3}}\right )}{4 a^3}\\ &=-\frac{5 b \sqrt{a+\frac{b}{x^3}} x^3}{4 a^3}-\frac{2 x^6}{3 a \sqrt{a+\frac{b}{x^3}}}+\frac{5 \sqrt{a+\frac{b}{x^3}} x^6}{6 a^2}+\frac{5 b^2 \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )}{4 a^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.0541821, size = 96, normalized size = 1.01 \[ \frac{\sqrt{a} x^{3/2} \left (2 a^2 x^6-5 a b x^3-15 b^2\right )+15 b^{5/2} \sqrt{\frac{a x^3}{b}+1} \sinh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{b}}\right )}{12 a^{7/2} x^{3/2} \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x^3)^(3/2),x]

[Out]

(Sqrt[a]*x^(3/2)*(-15*b^2 - 5*a*b*x^3 + 2*a^2*x^6) + 15*b^(5/2)*Sqrt[1 + (a*x^3)/b]*ArcSinh[(Sqrt[a]*x^(3/2))/
Sqrt[b]])/(12*a^(7/2)*Sqrt[a + b/x^3]*x^(3/2))

________________________________________________________________________________________

Maple [C]  time = 0.035, size = 3910, normalized size = 41.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(a+b/x^3)^(3/2),x)

[Out]

1/12/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^5*(2*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*
(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+
b))^(1/2)*x^4*a^3-7*I*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2
)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a^2*b+180*I*(-(I
*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(
1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*
x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)
+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a*b^2+90*I
*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/
3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))
/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I
*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b)
)^(1/2)*x^2*a^2*b^2-180*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)
^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a
^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b
*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1
/2))*(-b*a^2)^(1/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a*b^2+90*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)
^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I
*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1
/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/
2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*b^2+90*(-(I*3^(1/2)-3)*x*a/
(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a
*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)
))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2
))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2-90*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))
/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1
/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*Ellip
ticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2
))/(I*3^(1/2)-3))^(1/2))*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2-90*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(
-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1
/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-
(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+
I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^2-6*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-
a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(
1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x^4*a^3-180*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*(
(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2
)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*
3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^
2)^(1/3)*(x*(a*x^3+b))^(1/2)*x*a*b^2+180*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3
^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1
/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(
1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I
*3^(1/2)-3))^(1/2))*(-b*a^2)^(1/3)*(x*(a*x^3+b))^(1/2)*x*a*b^2-8*I*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(
-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*3^(1/2)*x^2*a^2*b^2
+90*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^
(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/
2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),((I
*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*(x*(a*x^3+b))^(1/2)*b^2-90*(-(I*
3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))/(1
+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3))/(-1+I*3^(1/2))/(-a*x
+(-b*a^2)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2),(-1+I*3^(1/
2))/(I*3^(1/2)-3),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*(x*(a*x^3+b
))^(1/2)*b^2-90*I*(-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)+2
*a*x+(-b*a^2)^(1/3))/(1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*((I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3
))/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(-1+I*3^(1/2))/(-a*x+(-b*a^2)^(1/
3)))^(1/2),((I*3^(1/2)+3)*(-1+I*3^(1/2))/(1+I*3^(1/2))/(I*3^(1/2)-3))^(1/2))*(-b*a^2)^(2/3)*3^(1/2)*(x*(a*x^3+
b))^(1/2)*b^2+21*(a*x^4+b*x)^(1/2)*(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/
3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x*a^2*b+24*x^2*b^2*a^2*(1/a^2*x
*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2
)^(1/3)))^(1/2))/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-b*a^2)^(1/3))*(I*3^(1/2)*(-b*a^2)^(1/3)+2*a*x+(-b*a^2)^(1/3))*
(I*3^(1/2)*(-b*a^2)^(1/3)-2*a*x-(-b*a^2)^(1/3)))^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.60585, size = 533, normalized size = 5.61 \begin{align*} \left [\frac{15 \,{\left (a b^{2} x^{3} + b^{3}\right )} \sqrt{a} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) + 4 \,{\left (2 \, a^{3} x^{9} - 5 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{48 \,{\left (a^{5} x^{3} + a^{4} b\right )}}, -\frac{15 \,{\left (a b^{2} x^{3} + b^{3}\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{2 \, a x^{3} + b}\right ) - 2 \,{\left (2 \, a^{3} x^{9} - 5 \, a^{2} b x^{6} - 15 \, a b^{2} x^{3}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{24 \,{\left (a^{5} x^{3} + a^{4} b\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/48*(15*(a*b^2*x^3 + b^3)*sqrt(a)*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sqrt((a*x^3
+ b)/x^3)) + 4*(2*a^3*x^9 - 5*a^2*b*x^6 - 15*a*b^2*x^3)*sqrt((a*x^3 + b)/x^3))/(a^5*x^3 + a^4*b), -1/24*(15*(
a*b^2*x^3 + b^3)*sqrt(-a)*arctan(2*sqrt(-a)*x^3*sqrt((a*x^3 + b)/x^3)/(2*a*x^3 + b)) - 2*(2*a^3*x^9 - 5*a^2*b*
x^6 - 15*a*b^2*x^3)*sqrt((a*x^3 + b)/x^3))/(a^5*x^3 + a^4*b)]

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Sympy [A]  time = 5.91343, size = 110, normalized size = 1.16 \begin{align*} \frac{x^{\frac{15}{2}}}{6 a \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{5 \sqrt{b} x^{\frac{9}{2}}}{12 a^{2} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{5 b^{\frac{3}{2}} x^{\frac{3}{2}}}{4 a^{3} \sqrt{\frac{a x^{3}}{b} + 1}} + \frac{5 b^{2} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{4 a^{\frac{7}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(15/2)/(6*a*sqrt(b)*sqrt(a*x**3/b + 1)) - 5*sqrt(b)*x**(9/2)/(12*a**2*sqrt(a*x**3/b + 1)) - 5*b**(3/2)*x**(
3/2)/(4*a**3*sqrt(a*x**3/b + 1)) + 5*b**2*asinh(sqrt(a)*x**(3/2)/sqrt(b))/(4*a**(7/2))

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Giac [A]  time = 1.19895, size = 158, normalized size = 1.66 \begin{align*} -\frac{1}{12} \, b^{2}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{\frac{a x^{3} + b}{x^{3}}}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{3}} + \frac{8}{a^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}} - \frac{9 \, a \sqrt{\frac{a x^{3} + b}{x^{3}}} - \frac{7 \,{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{x^{3}}}{{\left (a - \frac{a x^{3} + b}{x^{3}}\right )}^{2} a^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(a+b/x^3)^(3/2),x, algorithm="giac")

[Out]

-1/12*b^2*(15*arctan(sqrt((a*x^3 + b)/x^3)/sqrt(-a))/(sqrt(-a)*a^3) + 8/(a^3*sqrt((a*x^3 + b)/x^3)) - (9*a*sqr
t((a*x^3 + b)/x^3) - 7*(a*x^3 + b)*sqrt((a*x^3 + b)/x^3)/x^3)/((a - (a*x^3 + b)/x^3)^2*a^3))