3.2273 \(\int \frac {1}{x^3 (a+b x^{3/2})^{2/3}} \, dx\)

Optimal. Leaf size=50 \[ \frac {3 b \sqrt [3]{a+b x^{3/2}}}{2 a^2 \sqrt {x}}-\frac {\sqrt [3]{a+b x^{3/2}}}{2 a x^2} \]

[Out]

-1/2*(a+b*x^(3/2))^(1/3)/a/x^2+3/2*b*(a+b*x^(3/2))^(1/3)/a^2/x^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {271, 264} \[ \frac {3 b \sqrt [3]{a+b x^{3/2}}}{2 a^2 \sqrt {x}}-\frac {\sqrt [3]{a+b x^{3/2}}}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^(3/2))^(2/3)),x]

[Out]

-(a + b*x^(3/2))^(1/3)/(2*a*x^2) + (3*b*(a + b*x^(3/2))^(1/3))/(2*a^2*Sqrt[x])

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]

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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 33, normalized size = 0.66 \[ -\frac {\left (a-3 b x^{3/2}\right ) \sqrt [3]{a+b x^{3/2}}}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a - 3*b*x^(3/2))*(a + b*x^(3/2))^(1/3))/(a^2*x^2)

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fricas [A]  time = 1.61, size = 27, normalized size = 0.54 \[ \frac {{\left (3 \, b x^{\frac {3}{2}} - a\right )} {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}}}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(3*b*x^(3/2) - a)*(b*x^(3/2) + a)^(1/3)/(a^2*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{\frac {3}{2}}+a \right )^{\frac {2}{3}} x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.82, size = 35, normalized size = 0.70 \[ \frac {\frac {4 \, {\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} b}{\sqrt {x}} - \frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {4}{3}}}{x^{2}}}{2 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*(b*x^(3/2) + a)^(1/3)*b/sqrt(x) - (b*x^(3/2) + a)^(4/3)/x^2)/a^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^3\,{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(a + b*x^(3/2))^(2/3)),x)

[Out]

int(1/(x^3*(a + b*x^(3/2))^(2/3)), x)

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sympy [A]  time = 2.68, size = 76, normalized size = 1.52 \[ - \frac {2 \sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {4}{3}\right )}{9 a x^{\frac {3}{2}} \Gamma \left (\frac {2}{3}\right )} + \frac {2 b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b x^{\frac {3}{2}}} + 1} \Gamma \left (- \frac {4}{3}\right )}{3 a^{2} \Gamma \left (\frac {2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b**(1/3)*(a/(b*x**(3/2)) + 1)**(1/3)*gamma(-4/3)/(9*a*x**(3/2)*gamma(2/3)) + 2*b**(4/3)*(a/(b*x**(3/2)) + 1
)**(1/3)*gamma(-4/3)/(3*a**2*gamma(2/3))

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