3.2275 \(\int \frac{1}{x^9 (a+b x^{3/2})^{2/3}} \, dx\)

Optimal. Leaf size=158 \[ \frac{729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt{x}}-\frac{243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac{9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8} \]

[Out]

-(a + b*x^(3/2))^(1/3)/(8*a*x^8) + (15*b*(a + b*x^(3/2))^(1/3))/(104*a^2*x^(13/2)) - (9*b^2*(a + b*x^(3/2))^(1
/3))/(52*a^3*x^5) + (81*b^3*(a + b*x^(3/2))^(1/3))/(364*a^4*x^(7/2)) - (243*b^4*(a + b*x^(3/2))^(1/3))/(728*a^
5*x^2) + (729*b^5*(a + b*x^(3/2))^(1/3))/(728*a^6*Sqrt[x])

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Rubi [A]  time = 0.060664, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac{729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt{x}}-\frac{243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac{9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^9*(a + b*x^(3/2))^(2/3)),x]

[Out]

-(a + b*x^(3/2))^(1/3)/(8*a*x^8) + (15*b*(a + b*x^(3/2))^(1/3))/(104*a^2*x^(13/2)) - (9*b^2*(a + b*x^(3/2))^(1
/3))/(52*a^3*x^5) + (81*b^3*(a + b*x^(3/2))^(1/3))/(364*a^4*x^(7/2)) - (243*b^4*(a + b*x^(3/2))^(1/3))/(728*a^
5*x^2) + (729*b^5*(a + b*x^(3/2))^(1/3))/(728*a^6*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^9 \left (a+b x^{3/2}\right )^{2/3}} \, dx &=-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8}-\frac{(15 b) \int \frac{1}{x^{15/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{16 a}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}+\frac{\left (45 b^2\right ) \int \frac{1}{x^6 \left (a+b x^{3/2}\right )^{2/3}} \, dx}{52 a^2}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac{9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}-\frac{\left (81 b^3\right ) \int \frac{1}{x^{9/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{104 a^3}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac{9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}+\frac{\left (243 b^4\right ) \int \frac{1}{x^3 \left (a+b x^{3/2}\right )^{2/3}} \, dx}{364 a^4}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac{9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac{243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}-\frac{\left (729 b^5\right ) \int \frac{1}{x^{3/2} \left (a+b x^{3/2}\right )^{2/3}} \, dx}{1456 a^5}\\ &=-\frac{\sqrt [3]{a+b x^{3/2}}}{8 a x^8}+\frac{15 b \sqrt [3]{a+b x^{3/2}}}{104 a^2 x^{13/2}}-\frac{9 b^2 \sqrt [3]{a+b x^{3/2}}}{52 a^3 x^5}+\frac{81 b^3 \sqrt [3]{a+b x^{3/2}}}{364 a^4 x^{7/2}}-\frac{243 b^4 \sqrt [3]{a+b x^{3/2}}}{728 a^5 x^2}+\frac{729 b^5 \sqrt [3]{a+b x^{3/2}}}{728 a^6 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0353163, size = 83, normalized size = 0.53 \[ \frac{\sqrt [3]{a+b x^{3/2}} \left (162 a^2 b^3 x^{9/2}-126 a^3 b^2 x^3+105 a^4 b x^{3/2}-91 a^5-243 a b^4 x^6+729 b^5 x^{15/2}\right )}{728 a^6 x^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^9*(a + b*x^(3/2))^(2/3)),x]

[Out]

((a + b*x^(3/2))^(1/3)*(-91*a^5 + 105*a^4*b*x^(3/2) - 126*a^3*b^2*x^3 + 162*a^2*b^3*x^(9/2) - 243*a*b^4*x^6 +
729*b^5*x^(15/2)))/(728*a^6*x^8)

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{9}} \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^9/(a+b*x^(3/2))^(2/3),x)

[Out]

int(1/x^9/(a+b*x^(3/2))^(2/3),x)

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Maxima [A]  time = 0.970337, size = 139, normalized size = 0.88 \begin{align*} \frac{\frac{1456 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}} b^{5}}{\sqrt{x}} - \frac{1820 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{4}{3}} b^{4}}{x^{2}} + \frac{2080 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{7}{3}} b^{3}}{x^{\frac{7}{2}}} - \frac{1456 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{10}{3}} b^{2}}{x^{5}} + \frac{560 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{13}{3}} b}{x^{\frac{13}{2}}} - \frac{91 \,{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{16}{3}}}{x^{8}}}{728 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(a+b*x^(3/2))^(2/3),x, algorithm="maxima")

[Out]

1/728*(1456*(b*x^(3/2) + a)^(1/3)*b^5/sqrt(x) - 1820*(b*x^(3/2) + a)^(4/3)*b^4/x^2 + 2080*(b*x^(3/2) + a)^(7/3
)*b^3/x^(7/2) - 1456*(b*x^(3/2) + a)^(10/3)*b^2/x^5 + 560*(b*x^(3/2) + a)^(13/3)*b/x^(13/2) - 91*(b*x^(3/2) +
a)^(16/3)/x^8)/a^6

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Fricas [A]  time = 4.27937, size = 188, normalized size = 1.19 \begin{align*} -\frac{{\left (243 \, a b^{4} x^{6} + 126 \, a^{3} b^{2} x^{3} + 91 \, a^{5} - 3 \,{\left (243 \, b^{5} x^{7} + 54 \, a^{2} b^{3} x^{4} + 35 \, a^{4} b x\right )} \sqrt{x}\right )}{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{1}{3}}}{728 \, a^{6} x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(a+b*x^(3/2))^(2/3),x, algorithm="fricas")

[Out]

-1/728*(243*a*b^4*x^6 + 126*a^3*b^2*x^3 + 91*a^5 - 3*(243*b^5*x^7 + 54*a^2*b^3*x^4 + 35*a^4*b*x)*sqrt(x))*(b*x
^(3/2) + a)^(1/3)/(a^6*x^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**9/(a+b*x**(3/2))**(2/3),x)

[Out]

Timed out

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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^{9}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^9/(a+b*x^(3/2))^(2/3),x, algorithm="giac")

[Out]

integrate(1/((b*x^(3/2) + a)^(2/3)*x^9), x)