3.2042 \(\int \frac{1}{(a+\frac{b}{x^3})^{3/2} x^{10}} \, dx\)

Optimal. Leaf size=59 \[ \frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

[Out]

(2*a^2)/(3*b^3*Sqrt[a + b/x^3]) + (4*a*Sqrt[a + b/x^3])/(3*b^3) - (2*(a + b/x^3)^(3/2))/(9*b^3)

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Rubi [A]  time = 0.0309743, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b/x^3)^(3/2)*x^10),x]

[Out]

(2*a^2)/(3*b^3*Sqrt[a + b/x^3]) + (4*a*Sqrt[a + b/x^3])/(3*b^3) - (2*(a + b/x^3)^(3/2))/(9*b^3)

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 43

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])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^3}\right )^{3/2} x^{10}} \, dx &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{3/2}} \, dx,x,\frac{1}{x^3}\right )\right )\\ &=-\left (\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{3/2}}-\frac{2 a}{b^2 \sqrt{a+b x}}+\frac{\sqrt{a+b x}}{b^2}\right ) \, dx,x,\frac{1}{x^3}\right )\right )\\ &=\frac{2 a^2}{3 b^3 \sqrt{a+\frac{b}{x^3}}}+\frac{4 a \sqrt{a+\frac{b}{x^3}}}{3 b^3}-\frac{2 \left (a+\frac{b}{x^3}\right )^{3/2}}{9 b^3}\\ \end{align*}

Mathematica [A]  time = 0.0145023, size = 42, normalized size = 0.71 \[ \frac{2 \left (8 a^2 x^6+4 a b x^3-b^2\right )}{9 b^3 x^6 \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b/x^3)^(3/2)*x^10),x]

[Out]

(2*(-b^2 + 4*a*b*x^3 + 8*a^2*x^6))/(9*b^3*Sqrt[a + b/x^3]*x^6)

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Maple [A]  time = 0.006, size = 50, normalized size = 0.9 \begin{align*}{\frac{ \left ( 2\,a{x}^{3}+2\,b \right ) \left ( 8\,{a}^{2}{x}^{6}+4\,{x}^{3}ab-{b}^{2} \right ) }{9\,{b}^{3}{x}^{9}} \left ({\frac{a{x}^{3}+b}{{x}^{3}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b/x^3)^(3/2)/x^10,x)

[Out]

2/9*(a*x^3+b)*(8*a^2*x^6+4*a*b*x^3-b^2)/x^9/b^3/((a*x^3+b)/x^3)^(3/2)

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Maxima [A]  time = 0.986495, size = 63, normalized size = 1.07 \begin{align*} -\frac{2 \,{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}{9 \, b^{3}} + \frac{4 \, \sqrt{a + \frac{b}{x^{3}}} a}{3 \, b^{3}} + \frac{2 \, a^{2}}{3 \, \sqrt{a + \frac{b}{x^{3}}} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^(3/2)/x^10,x, algorithm="maxima")

[Out]

-2/9*(a + b/x^3)^(3/2)/b^3 + 4/3*sqrt(a + b/x^3)*a/b^3 + 2/3*a^2/(sqrt(a + b/x^3)*b^3)

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Fricas [A]  time = 1.52856, size = 107, normalized size = 1.81 \begin{align*} \frac{2 \,{\left (8 \, a^{2} x^{6} + 4 \, a b x^{3} - b^{2}\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{9 \,{\left (a b^{3} x^{6} + b^{4} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^(3/2)/x^10,x, algorithm="fricas")

[Out]

2/9*(8*a^2*x^6 + 4*a*b*x^3 - b^2)*sqrt((a*x^3 + b)/x^3)/(a*b^3*x^6 + b^4*x^3)

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Sympy [B]  time = 4.41568, size = 466, normalized size = 7.9 \begin{align*} \frac{16 a^{\frac{9}{2}} b^{\frac{7}{2}} x^{9} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} + \frac{24 a^{\frac{7}{2}} b^{\frac{9}{2}} x^{6} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} + \frac{6 a^{\frac{5}{2}} b^{\frac{11}{2}} x^{3} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{2 a^{\frac{3}{2}} b^{\frac{13}{2}} \sqrt{\frac{a x^{3}}{b} + 1}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{16 a^{5} b^{3} x^{\frac{21}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{32 a^{4} b^{4} x^{\frac{15}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} - \frac{16 a^{3} b^{5} x^{\frac{9}{2}}}{9 a^{\frac{7}{2}} b^{6} x^{\frac{21}{2}} + 18 a^{\frac{5}{2}} b^{7} x^{\frac{15}{2}} + 9 a^{\frac{3}{2}} b^{8} x^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x**3)**(3/2)/x**10,x)

[Out]

16*a**(9/2)*b**(7/2)*x**9*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3
/2)*b**8*x**(9/2)) + 24*a**(7/2)*b**(9/2)*x**6*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**
7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) + 6*a**(5/2)*b**(11/2)*x**3*sqrt(a*x**3/b + 1)/(9*a**(7/2)*b**6*x**(21
/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 2*a**(3/2)*b**(13/2)*sqrt(a*x**3/b + 1)/(9*a**(
7/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 16*a**5*b**3*x**(21/2)/(9*a**(7
/2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 32*a**4*b**4*x**(15/2)/(9*a**(7/
2)*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2)) - 16*a**3*b**5*x**(9/2)/(9*a**(7/2)
*b**6*x**(21/2) + 18*a**(5/2)*b**7*x**(15/2) + 9*a**(3/2)*b**8*x**(9/2))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}} x^{10}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^3)^(3/2)/x^10,x, algorithm="giac")

[Out]

integrate(1/((a + b/x^3)^(3/2)*x^10), x)