Optimal. Leaf size=284 \[ -\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}+\frac {14 b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2 \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} d}-\frac {2^{2/3} b^2 \tan ^{-1}\left (\frac {\sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3} d}-\frac {7 b^2 \log (x)}{9 a^{7/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d} \]
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Rubi [A]
time = 0.19, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {457, 105, 154,
162, 57, 631, 210, 31} \begin {gather*} \frac {14 b^2 \text {ArcTan}\left (\frac {2 \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} d}-\frac {2^{2/3} b^2 \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^{7/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}-\frac {7 b^2 \log (x)}{9 a^{7/3} d}-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 105
Rule 154
Rule 162
Rule 210
Rule 457
Rule 631
Rubi steps
\begin {align*} \int \frac {\left (a+b x^3\right )^{2/3}}{x^7 \left (a d-b d x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {(a+b x)^{2/3}}{x^3 (a d-b d x)} \, dx,x,x^3\right )\\ &=-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac {\text {Subst}\left (\int \frac {(a+b x)^{2/3} \left (-\frac {5}{3} a b d-\frac {1}{3} b^2 d x\right )}{x^2 (a d-b d x)} \, dx,x,x^3\right )}{6 a^2 d}\\ &=-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac {\text {Subst}\left (\int \frac {-\frac {28}{9} a^2 b^2 d^2-\frac {8}{9} a b^3 d^2 x}{x \sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{6 a^3 d^2}\\ &=-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}+\frac {\left (2 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x} (a d-b d x)} \, dx,x,x^3\right )}{3 a^2}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt [3]{a+b x}} \, dx,x,x^3\right )}{27 a^2 d}\\ &=-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac {7 b^2 \log (x)}{9 a^{7/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}-\frac {\left (7 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}+\frac {b^2 \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^{7/3} d}+\frac {\left (7 b^2\right ) \text {Subst}\left (\int \frac {1}{a^{2/3}+\sqrt [3]{a} x+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{9 a^2 d}-\frac {b^2 \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^2 d}\\ &=-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}-\frac {7 b^2 \log (x)}{9 a^{7/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}-\frac {\left (14 b^2\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}\right )}{9 a^{7/3} d}+\frac {\left (2^{2/3} b^2\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^{7/3} d}\\ &=-\frac {5 b \left (a+b x^3\right )^{2/3}}{18 a^2 d x^3}-\frac {\left (a+b x^3\right )^{5/3}}{6 a^2 d x^6}+\frac {14 b^2 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{9 \sqrt {3} a^{7/3} d}-\frac {2^{2/3} b^2 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} a^{7/3} d}-\frac {7 b^2 \log (x)}{9 a^{7/3} d}+\frac {b^2 \log \left (a-b x^3\right )}{3 \sqrt [3]{2} a^{7/3} d}+\frac {7 b^2 \log \left (\sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{9 a^{7/3} d}-\frac {b^2 \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{\sqrt [3]{2} a^{7/3} d}\\ \end {align*}
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Mathematica [A]
time = 0.52, size = 316, normalized size = 1.11 \begin {gather*} \frac {-9 a^{4/3} \left (a+b x^3\right )^{2/3}-24 \sqrt [3]{a} b x^3 \left (a+b x^3\right )^{2/3}+28 \sqrt {3} b^2 x^6 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-18\ 2^{2/3} \sqrt {3} b^2 x^6 \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )+28 b^2 x^6 \log \left (-\sqrt [3]{a}+\sqrt [3]{a+b x^3}\right )-18\ 2^{2/3} b^2 x^6 \log \left (-2 \sqrt [3]{a}+2^{2/3} \sqrt [3]{a+b x^3}\right )-14 b^2 x^6 \log \left (a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )+9\ 2^{2/3} b^2 x^6 \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 )}{54 a^{7/3} d x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{7} \left (-b d \,x^{3}+a d \right )}\, dx\]
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 = 3.73, size = 660, normalized size = 2.32 \begin {gather*} \left [-\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a b^{2} x^{6} \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 ) - 42 \, \sqrt {\frac {1}{3}} a b^{2} x^{6} \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 ) + 9 \cdot 4^{\frac {1}{3}} a b^{2} x^{6} \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 ) - 18 \cdot 4^{\frac {1}{3}} a b^{2} x^{6} \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 ) + 14 \, a^{\frac {2}{3}} b^{2} x^{6} \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 ) - 28 \, a^{\frac {2}{3}} b^{2} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (8 \, a b x^{3} + 3 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, a^{3} d x^{6}}, -\frac {18 \cdot 4^{\frac {1}{3}} \sqrt {3} a b^{2} x^{6} \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 ) + 9 \cdot 4^{\frac {1}{3}} a b^{2} x^{6} \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 ) - 18 \cdot 4^{\frac {1}{3}} a b^{2} x^{6} \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 ) - 84 \, \sqrt {\frac {1}{3}} a^{\frac {2}{3}} b^{2} x^{6} \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 ) + 14 \, a^{\frac {2}{3}} b^{2} x^{6} \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 ) - 28 \, a^{\frac {2}{3}} b^{2} x^{6} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - a^{\frac {1}{3}}\right ) + 3 \, {\left (8 \, a b x^{3} + 3 \, a^{2}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{54 \, a^{3} d x^{6}}\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^{7} + b x^{10}}\, 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 = 5.45, size = 513, normalized size = 1.81 \begin {gather*} \frac {\frac {5\,b^2\,{\left (b\,x^3+a\right )}^{2/3}}{18\,a}-\frac {4\,b^2\,{\left (b\,x^3+a\right )}^{5/3}}{9\,a^2}}{d\,{\left (b\,x^3+a\right )}^2+a^2\,d-2\,a\,d\,\left (b\,x^3+a\right )}+\ln \left (2\,b^4\,{\left (b\,x^3+a\right )}^{1/3}-2\,2^{1/3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}\right )\,{\left (-\frac {4\,b^6}{27\,a^7\,d^3}\right )}^{1/3}+\frac {14\,\ln \left (b^4\,{\left (b\,x^3+a\right )}^{1/3}-a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}\right )\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{1/3}}{27}-\ln \left (4\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}-2^{1/3}\,\sqrt {3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,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^6}{27\,a^7\,d^3}\right )}^{1/3}+\ln \left (4\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+2\,2^{1/3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,d^3}\right )}^{2/3}+2^{1/3}\,\sqrt {3}\,a^5\,d^2\,{\left (-\frac {b^6}{a^7\,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^6}{27\,a^7\,d^3}\right )}^{1/3}-\frac {7\,\ln \left (2\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}-\sqrt {3}\,a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{1/3}}{27}+\frac {7\,\ln \left (2\,b^4\,{\left (b\,x^3+a\right )}^{1/3}+a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}+\sqrt {3}\,a^5\,d^2\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{2/3}\,1{}\mathrm {i}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {b^6}{a^7\,d^3}\right )}^{1/3}}{27} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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