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 = 0.02, 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} \[ \frac {x \left (c+d x^3\right )^{11/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac {1}{3},\frac {5}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]
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*}
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Mathematica [A] time = 0.04, 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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fricas [F] time = 64.29, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {3}{4}} {\left (d x^{3} + c\right )}^{\frac {11}{12}}}{b^{2} d x^{9} + {\left (b^{2} c + 2 \, a b d\right )} x^{6} + {\left (2 \, a b c + a^{2} d\right )} x^{3} + a^{2} c}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{4}} {\left (d x^{3} + c\right )}^{\frac {1}{12}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.56, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {5}{4}} \left (d \,x^{3}+c \right )^{\frac {1}{12}}}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{3} + a\right )}^{\frac {5}{4}} {\left (d x^{3} + c\right )}^{\frac {1}{12}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (b\,x^3+a\right )}^{5/4}\,{\left (d\,x^3+c\right )}^{1/12}} \,d x \]
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 \[ \int \frac {1}{\left (a + b x^{3}\right )^{\frac {5}{4}} \sqrt [12]{c + d x^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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