Optimal. Leaf size=87 \[ \frac {x \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]
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Rubi [A] time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, 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} \[ \frac {x \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin {align*} \int \frac {1}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \, dx &=\frac {x \sqrt [4]{\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 86, normalized size = 0.99 \[ \frac {x \sqrt [4]{\frac {b x^3}{a}+1} \left (\frac {d x^3}{c}+1\right )^{3/4} \, _2F_1\left (\frac {1}{4},\frac {1}{3};\frac {4}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 51.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {3}{4}} {\left (d x^{3} + c\right )}^{\frac {11}{12}}}{b d^{2} x^{9} + {\left (2 \, b c d + a d^{2}\right )} x^{6} + {\left (b c^{2} + 2 \, a c d\right )} x^{3} + a c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{4}} {\left (d x^{3} + c\right )}^{\frac {13}{12}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.54, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {1}{4}} \left (d \,x^{3}+c \right )^{\frac {13}{12}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{4}} {\left (d x^{3} + c\right )}^{\frac {13}{12}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{1/4}\,{\left (d\,x^3+c\right )}^{13/12}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt [4]{a + b x^{3}} \left (c + d x^{3}\right )^{\frac {13}{12}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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