Optimal. Leaf size=87 \[ \frac{x \left (c+d x^3\right )^{11/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]
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Rubi [A] time = 0.0163963, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {380} \[ \frac{x \left (c+d x^3\right )^{11/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx &=\frac{x \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \left (c+d x^3\right )^{11/12} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \left (a+b x^3\right )^{5/4}}\\ \end{align*}
Mathematica [A] time = 0.0220741, size = 89, normalized size = 1.02 \[ \frac{x \sqrt [4]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{a \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3} \sqrt [4]{\frac{d x^3}{c}+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.449, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{3}+a \right ) ^{-{\frac{5}{4}}} \left ( d{x}^{3}+c \right ) ^{-{\frac{1}{12}}}}\, 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^{3} + a\right )}^{\frac{5}{4}}{\left (d x^{3} + c\right )}^{\frac{1}{12}}}\,{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{{\left (b x^{3} + a\right )}^{\frac{3}{4}}{\left (d x^{3} + c\right )}^{\frac{11}{12}}}{b^{2} d x^{9} +{\left (b^{2} c + 2 \, a b d\right )} x^{6} +{\left (2 \, a b c + a^{2} d\right )} x^{3} + a^{2} c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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^{3} + a\right )}^{\frac{5}{4}}{\left (d x^{3} + c\right )}^{\frac{1}{12}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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