Optimal. Leaf size=49 \[ -\frac{\, _2F_1\left (1,-\frac{1}{2}-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}} \]
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Rubi [A] time = 0.0195183, antiderivative size = 61, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {365, 364} \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Rule 365
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a+b x^n\right )^{3/2}} \, dx &=\frac{\sqrt{1+\frac{b x^n}{a}} \int \frac{1}{x^2 \left (1+\frac{b x^n}{a}\right )^{3/2}} \, dx}{a \sqrt{a+b x^n}}\\ &=-\frac{\sqrt{1+\frac{b x^n}{a}} \, _2F_1\left (\frac{3}{2},-\frac{1}{n};-\frac{1-n}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}}\\ \end{align*}
Mathematica [A] time = 0.0142702, size = 58, normalized size = 1.18 \[ -\frac{\sqrt{\frac{b x^n}{a}+1} \, _2F_1\left (\frac{3}{2},-\frac{1}{n};1-\frac{1}{n};-\frac{b x^n}{a}\right )}{a x \sqrt{a+b x^n}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.038, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b{x}^{n} \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 2.40526, size = 42, normalized size = 0.86 \begin{align*} \frac{\Gamma \left (- \frac{1}{n}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - \frac{1}{n} \\ 1 - \frac{1}{n} \end{matrix}\middle |{\frac{b x^{n} e^{i \pi }}{a}} \right )}}{a^{\frac{3}{2}} n x \Gamma \left (1 - \frac{1}{n}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{n} + a\right )}^{\frac{3}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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