Optimal. Leaf size=59 \[ \frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (a+b x^3\right )^{2/3}} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {441, 440}
\begin {gather*} \frac {x \left (\frac {b x^3}{a}+1\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (a+b x^3\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )^3} \, dx &=\frac {\left (1+\frac {b x^3}{a}\right )^{2/3} \int \frac {1}{\left (1+\frac {b x^3}{a}\right )^{2/3} \left (c+d x^3\right )^3} \, dx}{\left (a+b x^3\right )^{2/3}}\\ &=\frac {x \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {1}{3};\frac {2}{3},3;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^3 \left (a+b x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(442\) vs. \(2(59)=118\).
time = 10.47, size = 442, normalized size = 7.49 \begin {gather*} \frac {x \left (5 b d (-2 b c+a d) x^3 \left (1+\frac {b x^3}{a}\right )^{2/3} F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+\frac {4 c \left (-4 a c \left (3 a^2 d^2 \left (6 c+5 d x^3\right )+b^2 c \left (18 c^2+5 c d x^3-10 d^2 x^6\right )+a b d \left (-36 c^2-25 c d x^3+5 d^2 x^6\right )\right ) F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+d x^3 \left (a^2 d \left (8 c+5 d x^3\right )-b^2 c x^3 \left (13 c+10 d x^3\right )+a b \left (-13 c^2-2 c d x^3+5 d^2 x^6\right )\right ) \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}{\left (c+d x^3\right )^2 \left (-4 a c F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+x^3 \left (3 a d F_1\left (\frac {4}{3};\frac {2}{3},2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+2 b c F_1\left (\frac {4}{3};\frac {5}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )}\right )}{72 c^3 (b c-a d)^2 \left (a+b x^3\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )^{3}}\, dx\]
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(-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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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 = -0.02 \begin {gather*} \int \frac {1}{{\left (b\,x^3+a\right )}^{2/3}\,{\left (d\,x^3+c\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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