Optimal. Leaf size=57 \[ \frac {x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} F_1\left (\frac {1}{3};-m,1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c} \]
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Rubi [A]
time = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {441, 440}
\begin {gather*} \frac {x \left (a+b x^3\right )^m \left (\frac {b x^3}{a}+1\right )^{-m} F_1\left (\frac {1}{3};-m,1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 440
Rule 441
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^m}{c+d x^3} \, dx &=\left (\left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m}\right ) \int \frac {\left (1+\frac {b x^3}{a}\right )^m}{c+d x^3} \, dx\\ &=\frac {x \left (a+b x^3\right )^m \left (1+\frac {b x^3}{a}\right )^{-m} F_1\left (\frac {1}{3};-m,1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(57)=114\).
time = 0.26, size = 162, normalized size = 2.84 \begin {gather*} -\frac {4 a c x \left (a+b x^3\right )^m F_1\left (\frac {1}{3};-m,1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{\left (c+d x^3\right ) \left (-4 a c F_1\left (\frac {1}{3};-m,1;\frac {4}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+3 x^3 \left (-b c m F_1\left (\frac {4}{3};1-m,1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )+a d F_1\left (\frac {4}{3};-m,2;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )\right )\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{m}}{d \,x^{3}+c}\, dx\]
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 = -0.02 \begin {gather*} \int \frac {{\left (b\,x^3+a\right )}^m}{d\,x^3+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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