3.2274 \(\int \frac{1}{x^6 (a+b x^{3/2})^{2/3}} \, dx\)

Optimal. Leaf size=104 \[ \frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt{x}}-\frac{27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac{9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac{\sqrt [3]{a+b x^{3/2}}}{5 a x^5} \]

[Out]

-(a + b*x^(3/2))^(1/3)/(5*a*x^5) + (9*b*(a + b*x^(3/2))^(1/3))/(35*a^2*x^(7/2)) - (27*b^2*(a + b*x^(3/2))^(1/3
))/(70*a^3*x^2) + (81*b^3*(a + b*x^(3/2))^(1/3))/(70*a^4*Sqrt[x])

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Rubi [A]  time = 0.0330675, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt{x}}-\frac{27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac{9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac{\sqrt [3]{a+b x^{3/2}}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(a + b*x^(3/2))^(2/3)),x]

[Out]

-(a + b*x^(3/2))^(1/3)/(5*a*x^5) + (9*b*(a + b*x^(3/2))^(1/3))/(35*a^2*x^(7/2)) - (27*b^2*(a + b*x^(3/2))^(1/3
))/(70*a^3*x^2) + (81*b^3*(a + b*x^(3/2))^(1/3))/(70*a^4*Sqrt[x])

Rule 271

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 264

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

Rubi steps

\begin{align*} \int \frac{1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx &=-\frac{\sqrt [3]{a+b x^{3/2}}}{5 a x^5}-\frac{(9 b) \int \frac{1}{x^{9/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{10 a}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac{9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}+\frac{\left (27 b^2\right ) \int \frac{1}{x^3 \left (a+b x^{3/2}\right )^{2/3}} \, dx}{35 a^2}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac{9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac{27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}-\frac{\left (81 b^3\right ) \int \frac{1}{x^{3/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{140 a^3}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{5 a x^5}+\frac{9 b \sqrt [3]{a+b x^{3/2}}}{35 a^2 x^{7/2}}-\frac{27 b^2 \sqrt [3]{a+b x^{3/2}}}{70 a^3 x^2}+\frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{70 a^4 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0245991, size = 59, normalized size = 0.57 \[ \frac{\sqrt [3]{a+b x^{3/2}} \left (18 a^2 b x^{3/2}-14 a^3-27 a b^2 x^3+81 b^3 x^{9/2}\right )}{70 a^4 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(a + b*x^(3/2))^(2/3)),x]

[Out]

((a + b*x^(3/2))^(1/3)*(-14*a^3 + 18*a^2*b*x^(3/2) - 27*a*b^2*x^3 + 81*b^3*x^(9/2)))/(70*a^4*x^5)

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{6}} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(a+b*x^(3/2))^(2/3),x)

[Out]

int(1/x^6/(a+b*x^(3/2))^(2/3),x)

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Maxima [A]  time = 0.96029, size = 93, normalized size = 0.89 \begin{align*} \frac{\frac{140 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} b^{3}}{\sqrt{x}} - \frac{105 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}} b^{2}}{x^{2}} + \frac{60 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{7}{3}} b}{x^{\frac{7}{2}}} - \frac{14 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{10}{3}}}{x^{5}}}{70 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(a+b*x^(3/2))^(2/3),x, algorithm="maxima")

[Out]

1/70*(140*(b*x^(3/2) + a)^(1/3)*b^3/sqrt(x) - 105*(b*x^(3/2) + a)^(4/3)*b^2/x^2 + 60*(b*x^(3/2) + a)^(7/3)*b/x
^(7/2) - 14*(b*x^(3/2) + a)^(10/3)/x^5)/a^4

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Fricas [A]  time = 4.4182, size = 134, normalized size = 1.29 \begin{align*} -\frac{{\left (27 \, a b^{2} x^{3} + 14 \, a^{3} - 9 \,{\left (9 \, b^{3} x^{4} + 2 \, a^{2} b x\right )} \sqrt{x}\right )}{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}}{70 \, a^{4} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(a+b*x^(3/2))^(2/3),x, algorithm="fricas")

[Out]

-1/70*(27*a*b^2*x^3 + 14*a^3 - 9*(9*b^3*x^4 + 2*a^2*b*x)*sqrt(x))*(b*x^(3/2) + a)^(1/3)/(a^4*x^5)

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Sympy [B]  time = 92.1808, size = 736, normalized size = 7.08 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(a+b*x**(3/2))**(2/3),x)

[Out]

-56*a**6*b**(28/3)*x**9*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6
*b**10*x**15*gamma(2/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 96*a**5*b**(
31/3)*x**(21/2)*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x
**15*gamma(2/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) - 60*a**4*b**(34/3)*x*
*12*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2
/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 160*a**3*b**(37/3)*x**(27/2)*(a/
(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 2
43*a**5*b**11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 720*a**2*b**(40/3)*x**15*(a/(b*x**(3/2)
) + 1)**(1/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b**
11*x**(33/2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 864*a*b**(43/3)*x**(33/2)*(a/(b*x**(3/2)) + 1)**(1
/3)*gamma(-10/3)/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b**11*x**(33/
2)*gamma(2/3) + 81*a**4*b**12*x**18*gamma(2/3)) + 324*b**(46/3)*x**18*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-10/3)
/(81*a**7*b**9*x**(27/2)*gamma(2/3) + 243*a**6*b**10*x**15*gamma(2/3) + 243*a**5*b**11*x**(33/2)*gamma(2/3) +
81*a**4*b**12*x**18*gamma(2/3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}} x^{6}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(a+b*x^(3/2))^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^(3/2) + a)^(2/3)*x^6), x)