Optimal. Leaf size=183 \[ -\frac {a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac {a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{4/3} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}-\frac {a x (A b-a B)}{b^3}+\frac {x^4 (A b-a B)}{4 b^2}+\frac {B x^7}{7 b} \]
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Rubi [A] time = 0.15, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {459, 302, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac {a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{4/3} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}+\frac {x^4 (A b-a B)}{4 b^2}-\frac {a x (A b-a B)}{b^3}+\frac {B x^7}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 302
Rule 459
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {B x^7}{7 b}-\frac {(-7 A b+7 a B) \int \frac {x^6}{a+b x^3} \, dx}{7 b}\\ &=\frac {B x^7}{7 b}-\frac {(-7 A b+7 a B) \int \left (-\frac {a}{b^2}+\frac {x^3}{b}+\frac {a^2}{b^2 \left (a+b x^3\right )}\right ) \, dx}{7 b}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^7}{7 b}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {1}{a+b x^3} \, dx}{b^3}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^7}{7 b}+\frac {\left (a^{4/3} (A b-a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^3}+\frac {\left (a^{4/3} (A b-a B)\right ) \int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 b^3}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^7}{7 b}+\frac {a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {\left (a^{4/3} (A b-a B)\right ) \int \frac {-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{10/3}}+\frac {\left (a^{5/3} (A b-a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^3}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^7}{7 b}+\frac {a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}+\frac {\left (a^{4/3} (A b-a B)\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{b^{10/3}}\\ &=-\frac {a (A b-a B) x}{b^3}+\frac {(A b-a B) x^4}{4 b^2}+\frac {B x^7}{7 b}-\frac {a^{4/3} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{10/3}}+\frac {a^{4/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{10/3}}-\frac {a^{4/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{10/3}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 171, normalized size = 0.93 \begin {gather*} \frac {14 a^{4/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-28 a^{4/3} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+28 \sqrt {3} a^{4/3} (a B-A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+21 b^{4/3} x^4 (A b-a B)+84 a \sqrt [3]{b} x (a B-A b)+12 b^{7/3} B x^7}{84 b^{10/3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^6 \left (A+B x^3\right )}{a+b x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.68, size = 167, normalized size = 0.91 \begin {gather*} \frac {12 \, B b^{2} x^{7} - 21 \, {\left (B a b - A b^{2}\right )} x^{4} - 28 \, \sqrt {3} {\left (B a^{2} - A a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 14 \, {\left (B a^{2} - A a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 28 \, {\left (B a^{2} - A a b\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 84 \, {\left (B a^{2} - A a b\right )} x}{84 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 217, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} A a b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4}} - \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a^{2} - \left (-a b^{2}\right )^{\frac {1}{3}} A a b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4}} + \frac {{\left (B a^{3} b^{4} - A a^{2} b^{5}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{7}} + \frac {4 \, B b^{6} x^{7} - 7 \, B a b^{5} x^{4} + 7 \, A b^{6} x^{4} + 28 \, B a^{2} b^{4} x - 28 \, A a b^{5} x}{28 \, b^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 249, normalized size = 1.36 \begin {gather*} \frac {B \,x^{7}}{7 b}+\frac {A \,x^{4}}{4 b}-\frac {B a \,x^{4}}{4 b^{2}}+\frac {\sqrt {3}\, A \,a^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {A \,a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {A \,a^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {A a x}{b^{2}}-\frac {\sqrt {3}\, B \,a^{3} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {B \,a^{3} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {B \,a^{3} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {B \,a^{2} x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.43, size = 182, normalized size = 0.99 \begin {gather*} \frac {4 \, B b^{2} x^{7} - 7 \, {\left (B a b - A b^{2}\right )} x^{4} + 28 \, {\left (B a^{2} - A a b\right )} x}{28 \, b^{3}} - \frac {\sqrt {3} {\left (B a^{3} - A a^{2} b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (B a^{3} - A a^{2} b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 164, normalized size = 0.90 \begin {gather*} x^4\,\left (\frac {A}{4\,b}-\frac {B\,a}{4\,b^2}\right )+\frac {B\,x^7}{7\,b}+\frac {a^{4/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b-B\,a\right )}{3\,b^{10/3}}-\frac {a\,x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )}{b}-\frac {a^{4/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-B\,a\right )}{3\,b^{10/3}}+\frac {a^{4/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-B\,a\right )}{3\,b^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.37, size = 114, normalized size = 0.62 \begin {gather*} \frac {B x^{7}}{7 b} + x^{4} \left (\frac {A}{4 b} - \frac {B a}{4 b^{2}}\right ) + x \left (- \frac {A a}{b^{2}} + \frac {B a^{2}}{b^{3}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{10} - A^{3} a^{4} b^{3} + 3 A^{2} B a^{5} b^{2} - 3 A B^{2} a^{6} b + B^{3} a^{7}, \left (t \mapsto t \log {\left (- \frac {3 t b^{3}}{- A a b + B a^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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