Optimal. Leaf size=273 \[ \frac {(a d+3 b c) \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )}{6 b^{4/3} d^2}-\frac {(a d+3 b c) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3} d^2}+\frac {c^{4/3} \log \left (c+d x^3\right )}{6 d^2 \sqrt [3]{b c-a d}}-\frac {c^{4/3} \log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 d^2 \sqrt [3]{b c-a d}}+\frac {c^{4/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} d^2 \sqrt [3]{b c-a d}}+\frac {x \left (a+b x^3\right )^{2/3}}{3 b d} \]
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Rubi [A] time = 0.51, antiderivative size = 394, normalized size of antiderivative = 1.44, number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {494, 470, 522, 200, 31, 634, 617, 204, 628} \[ \frac {(a d+3 b c) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3} d^2}-\frac {(a d+3 b c) \log \left (\frac {b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )}{18 b^{4/3} d^2}-\frac {(a d+3 b c) \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} b^{4/3} d^2}-\frac {c^{4/3} \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 \sqrt [3]{b c-a d}}+\frac {c^{4/3} \log \left (\frac {x^2 (b c-a d)^{2/3}}{\left (a+b x^3\right )^{2/3}}+\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+c^{2/3}\right )}{6 d^2 \sqrt [3]{b c-a d}}+\frac {c^{4/3} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{a+b x^3}}+\sqrt [3]{c}}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} d^2 \sqrt [3]{b c-a d}}+\frac {x \left (a+b x^3\right )^{2/3}}{3 b d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 470
Rule 494
Rule 522
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx &=a^2 \operatorname {Subst}\left (\int \frac {x^6}{\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 \left (a+b x^3\right )^{2/3}}{3 b d}-\frac {a \operatorname {Subst}\left (\int \frac {c+(2 b c+a d) x^3}{\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 \left (a+b x^3\right )^{2/3}}{3 b d}+\frac {c^2 \operatorname {Subst}\left (\int \frac {1}{c+(-b c+a d) x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{d^2}-\frac {(3 b c+a d) \operatorname {Subst}\left (\int \frac {1}{1-b x^3} \, dx,x,\frac {x}{\sqrt [3]{a+b x^3}}\right )}{3 b d^2}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 b d}+\frac {c^{4/3} \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}+\frac {c^{4/3} \operatorname {Subst}\left (\int \frac {2 \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}-\frac {(3 b c+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 d^2}-\frac {(3 b c+a d) \operatorname {Subst}\left (\int \frac {2+\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 d^2}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 b d}+\frac {(3 b c+a d) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3} d^2}-\frac {c^{4/3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} 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 {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}+\frac {c^{4/3} \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 \sqrt [3]{b c-a d}}-\frac {(3 b c+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^{4/3} d^2}-\frac {(3 b c+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 d^2}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 b d}+\frac {(3 b c+a d) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3} d^2}-\frac {(3 b c+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^{4/3} d^2}-\frac {c^{4/3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 \sqrt [3]{b c-a d}}+\frac {c^{4/3} \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 \sqrt [3]{b c-a d}}-\frac {c^{4/3} \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 \sqrt [3]{b c-a d}}+\frac {(3 b c+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^{4/3} d^2}\\ &=\frac {x \left (a+b x^3\right )^{2/3}}{3 b d}-\frac {(3 b c+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^{4/3} d^2}+\frac {c^{4/3} \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 \sqrt [3]{b c-a d}}+\frac {(3 b c+a d) \log \left (1-\frac {\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )}{9 b^{4/3} d^2}-\frac {(3 b c+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^{4/3} d^2}-\frac {c^{4/3} \log \left (\sqrt [3]{c}-\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{a+b x^3}}\right )}{3 d^2 \sqrt [3]{b c-a d}}+\frac {c^{4/3} \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 \sqrt [3]{b c-a d}}\\ \end {align*}
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Mathematica [C] time = 0.82, size = 288, normalized size = 1.05 \[ \frac {\frac {2 \left (-a \sqrt [3]{c} \log \left (\frac {\sqrt [3]{c} x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}+\frac {x^2 (b c-a d)^{2/3}}{\left (a x^3+b\right )^{2/3}}+c^{2/3}\right )+6 x \left (a+b x^3\right )^{2/3} \sqrt [3]{b c-a d}+2 a \sqrt [3]{c} \log \left (\sqrt [3]{c}-\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{a x^3+b}}\right )-2 \sqrt {3} a \sqrt [3]{c} \tan ^{-1}\left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a x^3+b}}+1}{\sqrt {3}}\right )\right )}{\sqrt [3]{b c-a d}}-\frac {3 x^4 \sqrt [3]{\frac {b x^3}{a}+1} (a d+3 b c) F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{c \sqrt [3]{a+b x^3}}}{36 b d} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.84, size = 826, normalized size = 3.03 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} \left (d \,x^{3}+c \right )}\, dx \]
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 \[ \int \frac {x^{6}}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )}}\,{d x} \]
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 \[ \int \frac {x^6}{{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \]
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 \[ \int \frac {x^{6}}{\sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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