Optimal. Leaf size=220 \[ \frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}-\frac {a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d}+\frac {\sqrt [3]{2} a^{10/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^4 d}-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d} \]
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Rubi [A] time = 0.24, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {446, 88, 50, 57, 617, 204, 31} \[ -\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}-\frac {a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d}+\frac {\sqrt [3]{2} a^{10/3} \tan ^{-1}\left (\frac {2^{2/3} \sqrt [3]{a+b x^3}+\sqrt [3]{a}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 88
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^{11} \sqrt [3]{a+b x^3}}{a d-b d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3 \sqrt [3]{a+b x}}{a d-b d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a^2 \sqrt [3]{a+b x}}{b^3 d}+\frac {a (a+b x)^{4/3}}{b^3 d}-\frac {(a+b x)^{7/3}}{b^3 d}+\frac {a^3 \sqrt [3]{a+b x}}{b^3 (a d-b d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d}+\frac {a^3 \operatorname {Subst}\left (\int \frac {\sqrt [3]{a+b x}}{a d-b d x} \, dx,x,x^3\right )}{3 b^3}\\ &=-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d}+\frac {\left (2 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a+b x)^{2/3} (a d-b d x)} \, dx,x,x^3\right )}{3 b^3}\\ &=-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d}+\frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}+\frac {a^{10/3} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{2} \sqrt [3]{a}-x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d}+\frac {a^{11/3} \operatorname {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 )}{\sqrt [3]{2} b^4 d}\\ &=-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d}+\frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}-\frac {a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d}-\frac {\left (\sqrt [3]{2} a^{10/3}\right ) \operatorname {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 )}{b^4 d}\\ &=-\frac {a^3 \sqrt [3]{a+b x^3}}{b^4 d}-\frac {a^2 \left (a+b x^3\right )^{4/3}}{4 b^4 d}+\frac {a \left (a+b x^3\right )^{7/3}}{7 b^4 d}-\frac {\left (a+b x^3\right )^{10/3}}{10 b^4 d}+\frac {\sqrt [3]{2} a^{10/3} \tan ^{-1}\left (\frac {1+\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt {3} b^4 d}+\frac {a^{10/3} \log \left (a-b x^3\right )}{3\ 2^{2/3} b^4 d}-\frac {a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )}{2^{2/3} b^4 d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 230, normalized size = 1.05 \[ -\frac {140 \sqrt [3]{2} a^{10/3} \log \left (\sqrt [3]{2} \sqrt [3]{a}-\sqrt [3]{a+b x^3}\right )-70 \sqrt [3]{2} a^{10/3} \log \left (2^{2/3} a^{2/3}+\sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{a+b x^3}+\left (a+b x^3\right )^{2/3}\right )-140 \sqrt [3]{2} \sqrt {3} a^{10/3} \tan ^{-1}\left (\frac {\frac {2^{2/3} \sqrt [3]{a+b x^3}}{\sqrt [3]{a}}+1}{\sqrt {3}}\right )+507 a^3 \sqrt [3]{a+b x^3}+111 a^2 b x^3 \sqrt [3]{a+b x^3}+42 b^3 x^9 \sqrt [3]{a+b x^3}+66 a b^2 x^6 \sqrt [3]{a+b x^3}}{420 b^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 186, normalized size = 0.85 \[ -\frac {140 \, \sqrt {3} 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{3} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {2}{3}} + \sqrt {3} a}{3 \, a}\right ) + 70 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{3} \log \left (2^{\frac {2}{3}} \left (-a\right )^{\frac {2}{3}} - 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right ) - 140 \cdot 2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} a^{3} \log \left (2^{\frac {1}{3}} \left (-a\right )^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (14 \, b^{3} x^{9} + 22 \, a b^{2} x^{6} + 37 \, a^{2} b x^{3} + 169 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {1}{3}}}{420 \, b^{4} d} \]
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 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.65, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{11}}{-b d \,x^{3}+a d}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.55, size = 183, normalized size = 0.83 \[ \frac {\frac {140 \, \sqrt {3} 2^{\frac {1}{3}} a^{\frac {10}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a^{\frac {1}{3}} + 2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}}{6 \, a^{\frac {1}{3}}}\right )}{d} + \frac {70 \cdot 2^{\frac {1}{3}} a^{\frac {10}{3}} \log \left (2^{\frac {2}{3}} a^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {2}{3}}\right )}{d} - \frac {140 \cdot 2^{\frac {1}{3}} a^{\frac {10}{3}} \log \left (-2^{\frac {1}{3}} a^{\frac {1}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}}\right )}{d} - \frac {3 \, {\left (14 \, {\left (b x^{3} + a\right )}^{\frac {10}{3}} - 20 \, {\left (b x^{3} + a\right )}^{\frac {7}{3}} a + 35 \, {\left (b x^{3} + a\right )}^{\frac {4}{3}} a^{2} + 140 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} a^{3}\right )}}{d}}{420 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.81, size = 240, normalized size = 1.09 \[ \frac {a\,{\left (b\,x^3+a\right )}^{7/3}}{7\,b^4\,d}-\frac {a^3\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d}-\frac {a^2\,{\left (b\,x^3+a\right )}^{4/3}}{4\,b^4\,d}-\frac {{\left (b\,x^3+a\right )}^{10/3}}{10\,b^4\,d}-\frac {2^{1/3}\,a^{10/3}\,\ln \left ({\left (b\,x^3+a\right )}^{1/3}-2^{1/3}\,a^{1/3}\right )}{3\,b^4\,d}-\frac {2^{1/3}\,a^{10/3}\,\ln \left (\frac {6\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d}-\frac {6\,2^{1/3}\,a^{13/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{b^4\,d}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3\,b^4\,d}+\frac {2^{1/3}\,a^{10/3}\,\ln \left (\frac {6\,a^4\,{\left (b\,x^3+a\right )}^{1/3}}{b^4\,d}+\frac {18\,2^{1/3}\,a^{13/3}\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^4\,d}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )}{b^4\,d} \]
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 \[ - \frac {\int \frac {x^{11} \sqrt [3]{a + b x^{3}}}{- a + b x^{3}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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