Optimal. Leaf size=59 \[ \frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \sqrt [3]{1+\frac {b x^3}{a}}} \]
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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 \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \sqrt [3]{\frac {b x^3}{a}+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^3}}{\left (c+d x^3\right )^2} \, dx &=\frac {\sqrt [3]{a+b x^3} \int \frac {\sqrt [3]{1+\frac {b x^3}{a}}}{\left (c+d x^3\right )^2} \, dx}{\sqrt [3]{1+\frac {b x^3}{a}}}\\ &=\frac {x \sqrt [3]{a+b x^3} F_1\left (\frac {1}{3};-\frac {1}{3},2;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c^2 \sqrt [3]{1+\frac {b x^3}{a}}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(232\) vs. \(2(59)=118\).
time = 10.16, size = 232, normalized size = 3.93 \begin {gather*} \frac {x \left (\frac {b 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 )}{c^2}+\frac {4 \left (\frac {a+b x^3}{c}-\frac {8 a^2 F_1\left (\frac {1}{3};\frac {2}{3},1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{-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 )}{c+d x^3}\right )}{12 \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 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{\left (d \,x^{3}+c \right )^{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{a + b x^{3}}}{\left (c + d x^{3}\right )^{2}}\, dx \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 {{\left (b\,x^3+a\right )}^{1/3}}{{\left (d\,x^3+c\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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