Optimal. Leaf size=269 \[ \frac {b \left (a+b x^3\right )^{2/3}}{3 a^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}+\frac {5 b \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} d}-\frac {2^{2/3} b \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}-\frac {5 b \log (x)}{6 a^{4/3} d}+\frac {b \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{4/3} d}+\frac {5 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3} d}-\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{4/3} d} \]
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Rubi [A]
time = 0.18, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {457, 105, 162,
52, 57, 631, 210, 31} \begin {gather*} \frac {5 b \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} d}-\frac {2^{2/3} b \text {ArcTan}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} d}+\frac {b \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{4/3} d}+\frac {5 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3} d}-\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{4/3} d}-\frac {5 b \log (x)}{6 a^{4/3} d}-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}+\frac {b \left (a+b x^3\right )^{2/3}}{3 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 52
Rule 57
Rule 105
Rule 162
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^4 \left (a d-b d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x^2 (a d-b d x)} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}-\frac {\text {Subst}\left (\int \frac {(a+b x)^{2/3} \left (-\frac {5}{3} a b d+\frac {2}{3} b^2 d x\right )}{x (a d-b d x)} \, dx,x,x^3\right )}{3 a^2 d}\\ &=-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}+\frac {b^2 \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{a d-b d x} \, dx,x,x^3\right )}{3 a^2}+\frac {(5 b) \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x} \, dx,x,x^3\right )}{9 a^2 d}\\ &=\frac {b \left (a+b x^3\right )^{2/3}}{3 a^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}+\frac {\left (2 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 a}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{9 a d}\\ &=\frac {b \left (a+b x^3\right )^{2/3}}{3 a^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}-\frac {5 b \log (x)}{6 a^{4/3} d}+\frac {b \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{4/3} d}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a^{4/3} d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{4/3} d}+\frac {(5 b) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{6 a d}-\frac {b \text {Subst}\left (\int \frac {1}{2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{a d}\\ &=\frac {b \left (a+b x^3\right )^{2/3}}{3 a^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}-\frac {5 b \log (x)}{6 a^{4/3} d}+\frac {b \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{4/3} d}+\frac {5 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3} d}-\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{4/3} d}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{3 a^{4/3} d}+\frac {\left (2^{2/3} b\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{a^{4/3} d}\\ &=\frac {b \left (a+b x^3\right )^{2/3}}{3 a^2 d}-\frac {\left (a+b x^3\right )^{5/3}}{3 a^2 d x^3}+\frac {5 b \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{3 \sqrt {3} a^{4/3} d}-\frac {2^{2/3} b \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{4/3} d}-\frac {5 b \log (x)}{6 a^{4/3} d}+\frac {b \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{4/3} d}+\frac {5 b \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{6 a^{4/3} d}-\frac {b \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{4/3} d}\\ \end {align*}
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Mathematica [A]
time = 0.40, size = 282, normalized size = 1.05 \begin {gather*} \frac {-6 \sqrt [3]{a} \left (a+b x^3\right )^{2/3}+10 \sqrt {3} b x^3 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-6\ 2^{2/3} \sqrt {3} b x^3 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+10 b x^3 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-6\ 2^{2/3} b x^3 \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-5 b x^3 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+3\ 2^{2/3} b x^3 \log \left (2 a^{2/3}+2^{2/3} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\sqrt [3]{2} \left (a+b x^3\right )^{2/3}\right )}{18 a^{4/3} d x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{4} \left (-b d \,x^{3}+a d \right )}\, dx\]
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 [A]
time = 5.85, size = 612, normalized size = 2.28 \begin {gather*} \left [-\frac {6 \cdot 4^{\frac {1}{3}} \sqrt {3} a b x^{3} \left (-\frac {1}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 15 \, \sqrt {\frac {1}{3}} a b x^{3} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} a - a^{\frac {4}{3}}\right )} \sqrt {-\frac {1}{a^{\frac {2}{3}}}} - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {2}{3}} + 3 \, a}{x^{3}}\right ) + 3 \cdot 4^{\frac {1}{3}} a b x^{3} \left (-\frac {1}{a}\right )^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a \left (-\frac {1}{a}\right )^{\frac {2}{3}} - 2 \cdot 4^{\frac {1}{3}} a \left (-\frac {1}{a}\right )^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 6 \cdot 4^{\frac {1}{3}} a b x^{3} \left (-\frac {1}{a}\right )^{\frac {1}{3}} \log \left (-4^{\frac {2}{3}} a \left (-\frac {1}{a}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 5 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 10 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{18 \, a^{2} d x^{3}}, -\frac {6 \cdot 4^{\frac {1}{3}} \sqrt {3} a b x^{3} \left (-\frac {1}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {1}{a}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + 3 \cdot 4^{\frac {1}{3}} a b x^{3} \left (-\frac {1}{a}\right )^{\frac {1}{3}} \log \left (4^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a \left (-\frac {1}{a}\right )^{\frac {2}{3}} - 2 \cdot 4^{\frac {1}{3}} a \left (-\frac {1}{a}\right )^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 6 \cdot 4^{\frac {1}{3}} a b x^{3} \left (-\frac {1}{a}\right )^{\frac {1}{3}} \log \left (-4^{\frac {2}{3}} a \left (-\frac {1}{a}\right )^{\frac {2}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) - 30 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b x^{3} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + a^{\frac {1}{3}}\right )}}{a^{\frac {1}{3}}}\right ) + 5 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + a^{\frac {2}{3}}\right ) - 10 \, a^{\frac {2}{3}} b x^{3} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{18 \, a^{2} d x^{3}}\right ] \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^{4} + b x^{7}}\, dx}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.56, size = 490, normalized size = 1.82 \begin {gather*} \ln \left (2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}-2\,2^{1/3}\,a^3\,d^2\,{\left (-\frac {b^3}{a^4\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {4\,b^3}{27\,a^4\,d^3}\right )}^{1/3}+\frac {5\,\ln \left (b^2\,{\left (b\,x^3+a\right )}^{1/3}-a^3\,d^2\,{\left (\frac {b^3}{a^4\,d^3}\right )}^{2/3}\right )\,{\left (\frac {b^3}{a^4\,d^3}\right )}^{1/3}}{9}-\ln \left (4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a^3\,d^2\,{\left (-\frac {b^3}{a^4\,d^3}\right )}^{2/3}-2^{1/3}\,\sqrt {3}\,a^3\,d^2\,{\left (-\frac {b^3}{a^4\,d^3}\right )}^{2/3}\,2{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {4\,b^3}{27\,a^4\,d^3}\right )}^{1/3}+\ln \left (4\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a^3\,d^2\,{\left (-\frac {b^3}{a^4\,d^3}\right )}^{2/3}+2^{1/3}\,\sqrt {3}\,a^3\,d^2\,{\left (-\frac {b^3}{a^4\,d^3}\right )}^{2/3}\,2{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (-\frac {4\,b^3}{27\,a^4\,d^3}\right )}^{1/3}-\ln \left (2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+a^3\,d^2\,{\left (\frac {b^3}{a^4\,d^3}\right )}^{2/3}-\sqrt {3}\,a^3\,d^2\,{\left (\frac {b^3}{a^4\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {125\,b^3}{729\,a^4\,d^3}\right )}^{1/3}+\ln \left (2\,b^2\,{\left (b\,x^3+a\right )}^{1/3}+a^3\,d^2\,{\left (\frac {b^3}{a^4\,d^3}\right )}^{2/3}+\sqrt {3}\,a^3\,d^2\,{\left (\frac {b^3}{a^4\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (\frac {125\,b^3}{729\,a^4\,d^3}\right )}^{1/3}-\frac {b\,{\left (b\,x^3+a\right )}^{2/3}}{3\,a\,\left (d\,\left (b\,x^3+a\right )-a\,d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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