Optimal. Leaf size=57 \[ \frac{x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \]
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Rubi [A] time = 0.0158974, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1343, 245} \[ \frac{x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 1343
Rule 245
Rubi steps
\begin{align*} \int \frac{1}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}} \, dx &=\frac{\left (2 a b+2 b^2 x^n\right )^3 \int \frac{1}{\left (2 a b+2 b^2 x^n\right )^3} \, dx}{\left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}\\ &=\frac{x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0117737, size = 46, normalized size = 0.81 \[ \frac{x \left (a+b x^n\right )^3 \, _2F_1\left (3,\frac{1}{n};1+\frac{1}{n};-\frac{b x^n}{a}\right )}{a^3 \left (\left (a+b x^n\right )^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int \left ({a}^{2}+2\,ab{x}^{n}+{b}^{2}{x}^{2\,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*}{\left (2 \, n^{2} - 3 \, n + 1\right )} \int \frac{1}{2 \,{\left (a^{2} b n^{2} x^{n} + a^{3} n^{2}\right )}}\,{d x} + \frac{b{\left (2 \, n - 1\right )} x x^{n} + a{\left (3 \, n - 1\right )} x}{2 \,{\left (a^{2} b^{2} n^{2} x^{2 \, n} + 2 \, a^{3} b n^{2} x^{n} + a^{4} n^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}}{b^{4} x^{4 \, n} + 4 \, a^{2} b^{2} x^{2 \, n} + 4 \, a^{3} b x^{n} + a^{4} + 2 \,{\left (2 \, a b^{3} x^{n} + a^{2} b^{2}\right )} x^{2 \, n}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{\frac{3}{2}}}\, dx \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^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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