Optimal. Leaf size=40 \[ -\frac {\sqrt [3]{a+b x^{3/2}} \, _2F_1\left (-\frac {1}{3},1;\frac {1}{3};-\frac {b x^{3/2}}{a}\right )}{a x} \]
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Rubi [A] time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {341, 365, 364} \[ -\frac {\left (\frac {b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^{3/2}}{a}\right )}{x \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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Rule 341
Rule 364
Rule 365
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (a+b x^{3/2}\right )^{2/3}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{x^3 \left (a+b x^3\right )^{2/3}} \, dx,x,\sqrt {x}\right )\\ &=\frac {\left (2 \left (1+\frac {b x^{3/2}}{a}\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 \left (1+\frac {b x^3}{a}\right )^{2/3}} \, dx,x,\sqrt {x}\right )}{\left (a+b x^{3/2}\right )^{2/3}}\\ &=-\frac {\left (1+\frac {b x^{3/2}}{a}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^{3/2}}{a}\right )}{x \left (a+b x^{3/2}\right )^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 55, normalized size = 1.38 \[ -\frac {\left (\frac {b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (-\frac {2}{3},\frac {2}{3};\frac {1}{3};-\frac {b x^{3/2}}{a}\right )}{x \left (a+b x^{3/2}\right )^{2/3}} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {1}{3}} {\left (b x^{\frac {3}{2}} - a\right )}}{b^{2} x^{5} - a^{2} x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{\frac {3}{2}}+a \right )^{\frac {2}{3}} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{\frac {3}{2}} + a\right )}^{\frac {2}{3}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x^2\,{\left (a+b\,x^{3/2}\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.73, size = 42, normalized size = 1.05 \[ \frac {2 \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {b x^{\frac {3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac {2}{3}} x \Gamma \left (\frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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