Optimal. Leaf size=73 \[ \frac{\left (a+b x^3\right ) (d x)^{m+1} \, _2F_1\left (3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^3 d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
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Rubi [A] time = 0.0345446, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {1355, 364} \[ \frac{\left (a+b x^3\right ) (d x)^{m+1} \, _2F_1\left (3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^3 d (m+1) \sqrt{a^2+2 a b x^3+b^2 x^6}} \]
Antiderivative was successfully verified.
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Rule 1355
Rule 364
Rubi steps
\begin{align*} \int \frac{(d x)^m}{\left (a^2+2 a b x^3+b^2 x^6\right )^{3/2}} \, dx &=\frac{\left (b^2 \left (a b+b^2 x^3\right )\right ) \int \frac{(d x)^m}{\left (a b+b^2 x^3\right )^3} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{(d x)^{1+m} \left (a+b x^3\right ) \, _2F_1\left (3,\frac{1+m}{3};\frac{4+m}{3};-\frac{b x^3}{a}\right )}{a^3 d (1+m) \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}
Mathematica [A] time = 0.0199385, size = 60, normalized size = 0.82 \[ \frac{x \left (a+b x^3\right ) (d x)^m \, _2F_1\left (3,\frac{m+1}{3};\frac{m+4}{3};-\frac{b x^3}{a}\right )}{a^3 (m+1) \sqrt{\left (a+b x^3\right )^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.022, size = 0, normalized size = 0. \begin{align*} \int{ \left ( dx \right ) ^{m} \left ({b}^{2}{x}^{6}+2\,ab{x}^{3}+{a}^{2} \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{\left (d x\right )^{m}}{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \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^{6} + 2 \, a b x^{3} + a^{2}} \left (d x\right )^{m}}{b^{4} x^{12} + 4 \, a b^{3} x^{9} + 6 \, a^{2} b^{2} x^{6} + 4 \, a^{3} b x^{3} + a^{4}}, 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{\left (d x\right )^{m}}{\left (\left (a + b x^{3}\right )^{2}\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{\left (d x\right )^{m}}{{\left (b^{2} x^{6} + 2 \, a b x^{3} + a^{2}\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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