Optimal. Leaf size=155 \[ \frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}+\frac {60 a x \sqrt [4]{a+b x^3}}{133 c^2 \left (c+d x^3\right )^{7/12}}+\frac {45 a^2 x \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \left (c+d x^3\right )^{5/12} \, _2F_1\left (\frac {1}{3},\frac {3}{4};\frac {4}{3};-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{133 c^3 \left (a+b x^3\right )^{3/4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {386, 388}
\begin {gather*} \frac {45 a^2 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 )}{133 c^3 \left (a+b x^3\right )^{3/4}}+\frac {60 a x \sqrt [4]{a+b x^3}}{133 c^2 \left (c+d x^3\right )^{7/12}}+\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}} \end {gather*}
Antiderivative was successfully verified.
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Rule 386
Rule 388
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx &=\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}+\frac {(15 a) \int \frac {\sqrt [4]{a+b x^3}}{\left (c+d x^3\right )^{19/12}} \, dx}{19 c}\\ &=\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}+\frac {60 a x \sqrt [4]{a+b x^3}}{133 c^2 \left (c+d x^3\right )^{7/12}}+\frac {\left (45 a^2\right ) \int \frac {1}{\left (a+b x^3\right )^{3/4} \left (c+d x^3\right )^{7/12}} \, dx}{133 c^2}\\ &=\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}+\frac {60 a x \sqrt [4]{a+b x^3}}{133 c^2 \left (c+d x^3\right )^{7/12}}+\frac {45 a^2 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 )}{133 c^3 \left (a+b x^3\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 5.54, size = 90, normalized size = 0.58 \begin {gather*} \frac {a x \sqrt [4]{a+b x^3} \sqrt [4]{1+\frac {d x^3}{c}} \, _2F_1\left (-\frac {5}{4},\frac {1}{3};\frac {4}{3};\frac {(-b c+a d) x^3}{a \left (c+d x^3\right )}\right )}{c^2 \sqrt [4]{1+\frac {b x^3}{a}} \left (c+d x^3\right )^{7/12}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {5}{4}}}{\left (d \,x^{3}+c \right )^{\frac {31}{12}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^{5/4}}{{\left (d\,x^3+c\right )}^{31/12}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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