Optimal. Leaf size=223 \[ -\frac {\left (a+b x^3\right )^{5/3} (a d+b c)}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}-\frac {c^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{11/3}}+\frac {c^2 (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3}}+\frac {c^2 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{11/3}}+\frac {c^2 \left (a+b x^3\right )^{2/3}}{2 d^3} \]
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Rubi [A] time = 0.26, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {446, 88, 50, 56, 617, 204, 31} \[ -\frac {\left (a+b x^3\right )^{5/3} (a d+b c)}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}+\frac {c^2 \left (a+b x^3\right )^{2/3}}{2 d^3}-\frac {c^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{11/3}}+\frac {c^2 (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3}}+\frac {c^2 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{11/3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 56
Rule 88
Rule 204
Rule 446
Rule 617
Rubi steps
\begin {align*} \int \frac {x^8 \left (a+b x^3\right )^{2/3}}{c+d x^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 (a+b x)^{2/3}}{c+d x} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {(-b c-a d) (a+b x)^{2/3}}{b d^2}+\frac {(a+b x)^{5/3}}{b d}+\frac {c^2 (a+b x)^{2/3}}{d^2 (c+d x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}+\frac {c^2 \operatorname {Subst}\left (\int \frac {(a+b x)^{2/3}}{c+d x} \, dx,x,x^3\right )}{3 d^2}\\ &=\frac {c^2 \left (a+b x^3\right )^{2/3}}{2 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}-\frac {\left (c^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{a+b x} (c+d x)} \, dx,x,x^3\right )}{3 d^3}\\ &=\frac {c^2 \left (a+b x^3\right )^{2/3}}{2 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}-\frac {c^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{11/3}}+\frac {\left (c^2 (b c-a d)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {\sqrt [3]{b c-a d}}{\sqrt [3]{d}}+x} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^{11/3}}-\frac {\left (c^2 (b c-a d)\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {(b c-a d)^{2/3}}{d^{2/3}}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{d}}+x^2} \, dx,x,\sqrt [3]{a+b x^3}\right )}{2 d^4}\\ &=\frac {c^2 \left (a+b x^3\right )^{2/3}}{2 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}-\frac {c^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{11/3}}+\frac {c^2 (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3}}-\frac {\left (c^2 (b c-a d)^{2/3}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}\right )}{d^{11/3}}\\ &=\frac {c^2 \left (a+b x^3\right )^{2/3}}{2 d^3}-\frac {(b c+a d) \left (a+b x^3\right )^{5/3}}{5 b^2 d^2}+\frac {\left (a+b x^3\right )^{8/3}}{8 b^2 d}+\frac {c^2 (b c-a d)^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d}}}{\sqrt {3}}\right )}{\sqrt {3} d^{11/3}}-\frac {c^2 (b c-a d)^{2/3} \log \left (c+d x^3\right )}{6 d^{11/3}}+\frac {c^2 (b c-a d)^{2/3} \log \left (\sqrt [3]{b c-a d}+\sqrt [3]{d} \sqrt [3]{a+b x^3}\right )}{2 d^{11/3}}\\ \end {align*}
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Mathematica [C] time = 0.10, size = 104, normalized size = 0.47 \[ \frac {\left (a+b x^3\right )^{2/3} \left (-3 a^2 d^2-20 b^2 c^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {d \left (b x^3+a\right )}{a d-b c}\right )+2 a b d \left (d x^3-4 c\right )+b^2 \left (20 c^2-8 c d x^3+5 d^2 x^6\right )\right )}{40 b^2 d^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.10, size = 398, normalized size = 1.78 \[ \frac {40 \, \sqrt {3} b^{2} c^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} - \sqrt {3} {\left (b c - a d\right )}}{3 \, {\left (b c - a d\right )}}\right ) - 20 \, b^{2} c^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} d \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (b c - a d\right )} - {\left (b c - a d\right )} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}}\right ) + 40 \, b^{2} c^{2} \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left (-d \left (\frac {b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{2}}\right )^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )}\right ) + 3 \, {\left (5 \, b^{2} d^{2} x^{6} + 20 \, b^{2} c^{2} - 8 \, a b c d - 3 \, a^{2} d^{2} - 2 \, {\left (4 \, b^{2} c d - a b d^{2}\right )} x^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{120 \, b^{2} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 350, normalized size = 1.57 \[ \frac {{\left (b^{19} c^{3} d^{5} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} - a b^{18} c^{2} d^{6} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b^{19} c d^{8} - a b^{18} d^{9}\right )}} + \frac {\sqrt {3} {\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} c^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d^{5}} - \frac {{\left (-b c d^{2} + a d^{3}\right )}^{\frac {2}{3}} c^{2} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} \left (-\frac {b c - a d}{d}\right )^{\frac {1}{3}} + \left (-\frac {b c - a d}{d}\right )^{\frac {2}{3}}\right )}{6 \, d^{5}} + \frac {20 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} b^{16} c^{2} d^{5} - 8 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} b^{15} c d^{6} + 5 \, {\left (b x^{3} + a\right )}^{\frac {8}{3}} b^{14} d^{7} - 8 \, {\left (b x^{3} + a\right )}^{\frac {5}{3}} a b^{14} d^{7}}{40 \, b^{16} d^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} x^{8}}{d \,x^{3}+c}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.11, size = 385, normalized size = 1.73 \[ \left (\frac {a^2}{2\,b^2\,d}+\frac {\left (\frac {2\,a}{b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{b^4\,d^2}\right )\,\left (b^3\,c-a\,b^2\,d\right )}{2\,b^2\,d}\right )\,{\left (b\,x^3+a\right )}^{2/3}-\left (\frac {2\,a}{5\,b^2\,d}+\frac {b^3\,c-a\,b^2\,d}{5\,b^4\,d^2}\right )\,{\left (b\,x^3+a\right )}^{5/3}+\frac {{\left (b\,x^3+a\right )}^{8/3}}{8\,b^2\,d}+\frac {c^2\,\ln \left (\frac {{\left (b\,x^3+a\right )}^{1/3}\,\left (a^2\,c^4\,d^2-2\,a\,b\,c^5\,d+b^2\,c^6\right )}{d^5}-\frac {c^4\,{\left (a\,d-b\,c\right )}^{4/3}\,\left (9\,a\,d^3-9\,b\,c\,d^2\right )}{9\,d^{22/3}}\right )\,{\left (a\,d-b\,c\right )}^{2/3}}{3\,d^{11/3}}-\frac {c^2\,\ln \left (\frac {c^4\,{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d^5}-\frac {c^4\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{7/3}}{d^{16/3}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^{2/3}}{3\,d^{11/3}}+\frac {c^2\,\ln \left (\frac {c^4\,{\left (b\,x^3+a\right )}^{1/3}\,{\left (a\,d-b\,c\right )}^2}{d^5}-\frac {c^4\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,{\left (a\,d-b\,c\right )}^{7/3}}{4\,d^{16/3}}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^{2/3}}{d^{11/3}} \]
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 \[ \int \frac {x^{8} \left (a + b x^{3}\right )^{\frac {2}{3}}}{c + d x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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