Optimal. Leaf size=279 \[ \frac {(2 a d+3 b c) \log \left (\sqrt [3]{b} x-\sqrt [3]{a+b x^3}\right )}{6 b^{5/3} d^2}+\frac {(2 a d+3 b c) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} d^2}+\frac {c^{5/3} \log \left (c+d x^3\right )}{6 d^2 (b c-a d)^{2/3}}-\frac {c^{5/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^2 (b c-a d)^{2/3}}-\frac {c^{5/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2 (b c-a d)^{2/3}}+\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d} \]
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Rubi [A] time = 0.50, antiderivative size = 400, normalized size of antiderivative = 1.43, number of steps used = 16, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {494, 470, 584, 292, 31, 634, 617, 204, 628} \[ \frac {(2 a d+3 b c) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3} d^2}-\frac {(2 a d+3 b c) \log \left (\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{18 b^{5/3} d^2}+\frac {(2 a d+3 b c) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} d^2}-\frac {c^{5/3} \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 (b c-a d)^{2/3}}+\frac {c^{5/3} \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{6 d^2 (b c-a d)^{2/3}}-\frac {c^{5/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} d^2 (b c-a d)^{2/3}}+\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 470
Rule 494
Rule 584
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^3\right )^{2/3} \left (c+d x^3\right )} \, dx &=a^2 \operatorname {Subst}\left (\int \frac {x^7}{\left (1-b x^3\right )^2 \left (c-(b c-a d) x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}-\frac {a \operatorname {Subst}\left (\int \frac {x \left (2 c+(b c+2 a d) x^3\right )}{\left (1-b x^3\right ) \left (c+(-b c+a d) x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 b d}\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}-\frac {a \operatorname {Subst}\left (\int \left (\frac {(3 b c+2 a d) x}{a d \left (1-b x^3\right )}+\frac {3 b c^2 x}{a d \left (-c+(b c-a d) x^3\right )}\right ) \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 b d}\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}-\frac {c^2 \operatorname {Subst}\left (\int \frac {x}{-c+(b c-a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{d^2}-\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {x}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 b d^2}\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}-\frac {c^{5/3} \operatorname {Subst}\left (\int \frac {1}{-\sqrt [3]{c}+\sqrt [3]{b c-a d} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 \sqrt [3]{b c-a d}}+\frac {c^{5/3} \operatorname {Subst}\left (\int \frac {-\sqrt [3]{c}+\sqrt [3]{b c-a d} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 \sqrt [3]{b c-a d}}-\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt [3]{b} x} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3} d^2}+\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {1-\sqrt [3]{b} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3} d^2}\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}+\frac {(3 b c+2 a d) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3} d^2}-\frac {c^{5/3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 (b c-a d)^{2/3}}+\frac {c^{5/3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{c} \sqrt [3]{b c-a d}+2 (b c-a d)^{2/3} x}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{6 d^2 (b c-a d)^{2/3}}-\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt [3]{c} \sqrt [3]{b c-a d} x+(b c-a d)^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{2 d^2 \sqrt [3]{b c-a d}}-\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {\sqrt [3]{b}+2 b^{2/3} x}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{18 b^{5/3} d^2}+\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{6 b^{4/3} d^2}\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}+\frac {(3 b c+2 a d) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3} d^2}-\frac {(3 b c+2 a d) \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{18 b^{5/3} d^2}-\frac {c^{5/3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 (b c-a d)^{2/3}}+\frac {c^{5/3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 d^2 (b c-a d)^{2/3}}+\frac {c^{5/3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )}{d^2 (b c-a d)^{2/3}}-\frac {(3 b c+2 a d) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{3 b^{5/3} d^2}\\ &=\frac {x^2 \sqrt [3]{a+b x^3}}{3 b d}+\frac {(3 b c+2 a d) \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3} b^{5/3} d^2}-\frac {c^{5/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} d^2 (b c-a d)^{2/3}}+\frac {(3 b c+2 a d) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{5/3} d^2}-\frac {(3 b c+2 a d) \log \left (1+\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{18 b^{5/3} d^2}-\frac {c^{5/3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 (b c-a d)^{2/3}}+\frac {c^{5/3} \log \left (c^{2/3}+\frac {(b c-a d)^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} \sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{6 d^2 (b c-a d)^{2/3}}\\ \end {align*}
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Mathematica [C] time = 0.29, size = 190, normalized size = 0.68 \[ \frac {5 c x^2 \left (\left (a+b x^3\right ) \left (\frac {d x^3}{c}+1\right )^{2/3}-a \left (\frac {b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac {2}{3},\frac {2}{3};\frac {5}{3};\frac {(a d-b c) x^3}{a \left (d x^3+c\right )}\right )\right )-x^5 \left (\frac {b x^3}{a}+1\right )^{2/3} \left (\frac {d x^3}{c}+1\right )^{2/3} (2 a d+3 b c) F_1\left (\frac {5}{3};\frac {2}{3},1;\frac {8}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{15 b c d \left (a+b x^3\right )^{2/3} \left (\frac {d x^3}{c}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.14, size = 558, normalized size = 2.00 \[ \frac {6 \, \sqrt {3} b^{3} c \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} \arctan \left (-\frac {2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c - a d\right )} \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {2}{3}} + \sqrt {3} c x}{3 \, c x}\right ) + 6 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} b^{2} d x^{2} + 6 \, b^{3} c \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} \log \left (\frac {{\left (b c - a d\right )} \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} x + {\left (b x^{3} + a\right )}^{\frac {1}{3}} c}{x}\right ) - 3 \, b^{3} c \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} \log \left (\frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {2}{3}} x^{2} + {\left (b x^{3} + a\right )}^{\frac {2}{3}} c^{2} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b c^{2} - a c d\right )} \left (-\frac {c^{2}}{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}\right )^{\frac {1}{3}} x}{x^{2}}\right ) - 2 \, \sqrt {3} {\left (3 \, b^{2} c + 2 \, a b d\right )} {\left (b^{2}\right )}^{\frac {1}{6}} \arctan \left (\frac {{\left (\sqrt {3} {\left (b^{2}\right )}^{\frac {1}{3}} b x + 2 \, \sqrt {3} {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}}\right )} {\left (b^{2}\right )}^{\frac {1}{6}}}{3 \, b^{2} x}\right ) + 2 \, {\left (b^{2}\right )}^{\frac {2}{3}} {\left (3 \, b c + 2 \, a d\right )} \log \left (-\frac {{\left (b^{2}\right )}^{\frac {2}{3}} x - {\left (b x^{3} + a\right )}^{\frac {1}{3}} b}{x}\right ) - {\left (b^{2}\right )}^{\frac {2}{3}} {\left (3 \, b c + 2 \, a d\right )} \log \left (\frac {{\left (b^{2}\right )}^{\frac {1}{3}} b x^{2} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (b^{2}\right )}^{\frac {2}{3}} x + {\left (b x^{3} + a\right )}^{\frac {2}{3}} b}{x^{2}}\right )}{18 \, b^{3} d^{2}} \]
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 \[ \int \frac {x^{7}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.64, size = 0, normalized size = 0.00 \[ \int \frac {x^{7}}{\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (d \,x^{3}+c \right )}\, dx \]
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 \[ \int \frac {x^{7}}{{\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^7}{{\left (b\,x^3+a\right )}^{2/3}\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \int \frac {x^{7}}{\left (a + b x^{3}\right )^{\frac {2}{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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