Optimal. Leaf size=157 \[ -\frac {\left (a+b x^3\right )^{2/3}}{2 a d x^2}+\frac {2^{2/3} b^{2/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} a d}+\frac {b^{2/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a d}-\frac {b^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a d} \]
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Rubi [A]
time = 0.05, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {486, 12, 384}
\begin {gather*} \frac {2^{2/3} b^{2/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{2} \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} a d}+\frac {b^{2/3} \log \left (a d-b d x^3\right )}{3 \sqrt [3]{2} a d}-\frac {b^{2/3} \log \left (\sqrt [3]{2} \sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a d}-\frac {\left (a+b x^3\right )^{2/3}}{2 a d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 384
Rule 486
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^3 \left (a d-b d x^3\right )} \, dx &=\frac {\left (a+b x^3\right )^{2/3} \int \frac {\left (1+\frac {b x^3}{a}\right )^{2/3}}{x^3 \left (a d-b d x^3\right )} \, dx}{\left (1+\frac {b x^3}{a}\right )^{2/3}}\\ &=-\frac {\left (a+b x^3\right )^{2/3} \left (1-\frac {b x^3}{a}\right )^{2/3} \, _2F_1\left (-\frac {2}{3},-\frac {2}{3};\frac {1}{3};-\frac {2 b x^3}{a-b x^3}\right )}{2 a d x^2 \left (1+\frac {b x^3}{a}\right )^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 195, normalized size = 1.24 \begin {gather*} \frac {-3 \left (a+b x^3\right )^{2/3}+2\ 2^{2/3} \sqrt {3} b^{2/3} x^2 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x}{\sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}}\right )-2\ 2^{2/3} b^{2/3} x^2 \log \left (-2 \sqrt [3]{b} x+2^{2/3} \sqrt [3]{a+b x^3}\right )+2^{2/3} b^{2/3} x^2 \log \left (2 b^{2/3} x^2+2^{2/3} \sqrt [3]{b} x \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{6 a d x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{3} \left (-b d \,x^{3}+a d \right )}\, dx\]
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 434 vs.
\(2 (125) = 250\).
time = 82.54, size = 434, normalized size = 2.76 \begin {gather*} -\frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} \left (-b^{2}\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (5 \, b^{2} x^{7} - 4 \, a b x^{4} - a^{2} x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b^{2}\right )^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (19 \, b^{3} x^{8} + 16 \, a b^{2} x^{5} + a^{2} b x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}} - \sqrt {3} {\left (71 \, b^{4} x^{9} + 111 \, a b^{3} x^{6} + 33 \, a^{2} b^{2} x^{3} + a^{3} b\right )}}{3 \, {\left (109 \, b^{4} x^{9} + 105 \, a b^{3} x^{6} + 3 \, a^{2} b^{2} x^{3} - a^{3} b\right )}}\right ) - 2 \cdot 4^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}} x^{2} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {2}{3}} x^{2} - 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b x + 4^{\frac {1}{3}} {\left (b x^{3} - a\right )} \left (-b^{2}\right )^{\frac {1}{3}}}{b x^{3} - a}\right ) + 4^{\frac {1}{3}} \left (-b^{2}\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, b^{2} x^{4} + a b x\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}} \left (-b^{2}\right )^{\frac {1}{3}} - 4^{\frac {2}{3}} {\left (19 \, b^{2} x^{6} + 16 \, a b x^{3} + a^{2}\right )} \left (-b^{2}\right )^{\frac {2}{3}} - 24 \, {\left (2 \, b^{3} x^{5} + a b^{2} x^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{b^{2} x^{6} - 2 \, a b x^{3} + a^{2}}\right ) + 9 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{18 \, a d x^{2}} \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*} - \frac {\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{- a x^{3} + b x^{6}}\, dx}{d} \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 )}^{2/3}}{x^3\,\left (a\,d-b\,d\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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