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 = 0.04, antiderivative size = 122, 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} \[ \frac {3 a x \left (c+d x^3\right )^{5/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{7 c^2 \left (a+b x^3\right )^{3/4}}+\frac {4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}} \]
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*}
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Mathematica [A] time = 0.02, 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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fricas [F] time = 2.25, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}} {\left (d x^{3} + c\right )}^{\frac {5}{12}}}{d^{2} x^{6} + 2 \, c d x^{3} + c^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}}}{{\left (d x^{3} + c\right )}^{\frac {19}{12}}}\,{d x} \]
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 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{4}}}{\left (d \,x^{3}+c \right )^{\frac {19}{12}}}\, dx \]
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 \[ \int \frac {{\left (b x^{3} + a\right )}^{\frac {1}{4}}}{{\left (d x^{3} + c\right )}^{\frac {19}{12}}}\,{d x} \]
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 \[ \int \frac {{\left (b\,x^3+a\right )}^{1/4}}{{\left (d\,x^3+c\right )}^{19/12}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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