Optimal. Leaf size=87 \[ \frac{x \left (c+d x^3\right )^{5/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{3/4}} \]
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Rubi [A] time = 0.0173785, 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} \[ \frac{x \left (c+d x^3\right )^{5/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b x^3\right )^{3/4} \left (c+d x^3\right )^{7/12}} \, dx &=\frac{x \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \left (c+d x^3\right )^{5/12} \, _2F_1\left (\frac{1}{3},\frac{3}{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 )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0212314, size = 86, normalized size = 0.99 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{3/4} \sqrt [4]{\frac{d x^3}{c}+1} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{\left (a+b x^3\right )^{3/4} \left (c+d x^3\right )^{7/12}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.433, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{4}}} \left ( d{x}^{3}+c \right ) ^{-{\frac{7}{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{3}{4}}{\left (d x^{3} + c\right )}^{\frac{7}{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{1}{4}}{\left (d x^{3} + c\right )}^{\frac{5}{12}}}{b d x^{6} +{\left (b c + a d\right )} x^{3} + a c}, 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 + b x^{3}\right )^{\frac{3}{4}} \left (c + d x^{3}\right )^{\frac{7}{12}}}\, 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 x^{3} + a\right )}^{\frac{3}{4}}{\left (d x^{3} + c\right )}^{\frac{7}{12}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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