Optimal. Leaf size=122 \[ \frac{3 a x \left (c+d x^3\right )^{5/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{7 c^2 \left (a+b x^3\right )^{3/4}}+\frac{4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}} \]
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Rubi [A] time = 0.036754, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {378, 380} \[ \frac{3 a x \left (c+d x^3\right )^{5/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{7 c^2 \left (a+b x^3\right )^{3/4}}+\frac{4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}} \]
Antiderivative was successfully verified.
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Rule 378
Rule 380
Rubi steps
\begin{align*} \int \frac{\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx &=\frac{4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}}+\frac{(3 a) \int \frac{1}{\left (a+b x^3\right )^{3/4} \left (c+d x^3\right )^{7/12}} \, dx}{7 c}\\ &=\frac{4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}}+\frac{3 a 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 )}{7 c^2 \left (a+b x^3\right )^{3/4}}\\ \end{align*}
Mathematica [A] time = 0.0237392, size = 89, normalized size = 0.73 \[ \frac{x \sqrt [4]{a+b x^3} \sqrt [4]{\frac{d x^3}{c}+1} \, _2F_1\left (-\frac{1}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{\frac{b x^3}{a}+1} \left (c+d x^3\right )^{7/12}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.453, size = 0, normalized size = 0. \begin{align*} \int{\sqrt [4]{b{x}^{3}+a} \left ( d{x}^{3}+c \right ) ^{-{\frac{19}{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{{\left (b x^{3} + a\right )}^{\frac{1}{4}}}{{\left (d x^{3} + c\right )}^{\frac{19}{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}}}{d^{2} x^{6} + 2 \, c d x^{3} + c^{2}}, 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{{\left (b x^{3} + a\right )}^{\frac{1}{4}}}{{\left (d x^{3} + c\right )}^{\frac{19}{12}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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