Optimal. Leaf size=121 \[ \frac {5 x \left (c+d x^3\right )^{5/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{9 a \left (a+b x^3\right )^{3/4}}+\frac {4 x \left (c+d x^3\right )^{5/12}}{9 a \left (a+b x^3\right )^{3/4}} \]
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Rubi [A] time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, 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} \[ \frac {5 x \left (c+d x^3\right )^{5/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{9 a \left (a+b x^3\right )^{3/4}}+\frac {4 x \left (c+d x^3\right )^{5/12}}{9 a \left (a+b x^3\right )^{3/4}} \]
Antiderivative was successfully verified.
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Rule 378
Rule 380
Rubi steps
\begin {align*} \int \frac {\left (c+d x^3\right )^{5/12}}{\left (a+b x^3\right )^{7/4}} \, dx &=\frac {4 x \left (c+d x^3\right )^{5/12}}{9 a \left (a+b x^3\right )^{3/4}}+\frac {(5 c) \int \frac {1}{\left (a+b x^3\right )^{3/4} \left (c+d x^3\right )^{7/12}} \, dx}{9 a}\\ &=\frac {4 x \left (c+d x^3\right )^{5/12}}{9 a \left (a+b x^3\right )^{3/4}}+\frac {5 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 )}{9 a \left (a+b x^3\right )^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 89, normalized size = 0.74 \[ \frac {x \left (\frac {b x^3}{a}+1\right )^{3/4} \left (c+d x^3\right )^{5/12} \, _2F_1\left (\frac {1}{3},\frac {7}{4};\frac {4}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{a \left (a+b x^3\right )^{3/4} \left (\frac {d x^3}{c}+1\right )^{3/4}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 2.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}} {\left (d x^{3} + c\right )}^{\frac {5}{12}}}{b^{2} x^{6} + 2 \, a b x^{3} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {5}{12}}}{{\left (b x^{3} + a\right )}^{\frac {7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.55, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{3}+c \right )^{\frac {5}{12}}}{\left (b \,x^{3}+a \right )^{\frac {7}{4}}}\, dx \]
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 \[ \int \frac {{\left (d x^{3} + c\right )}^{\frac {5}{12}}}{{\left (b x^{3} + a\right )}^{\frac {7}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (d\,x^3+c\right )}^{5/12}}{{\left (b\,x^3+a\right )}^{7/4}} \,d x \]
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 \[ \int \frac {\left (c + d x^{3}\right )^{\frac {5}{12}}}{\left (a + b x^{3}\right )^{\frac {7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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