Optimal. Leaf size=162 \[ \frac {\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac {\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac {\sqrt [3]{a} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{7/3}}+\frac {x (A b-a B)}{b^2}+\frac {B x^4}{4 b} \]
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Rubi [A] time = 0.12, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {459, 321, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}+\frac {x (A b-a B)}{b^2}-\frac {\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac {\sqrt [3]{a} (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^{7/3}}+\frac {B x^4}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 321
Rule 459
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {B x^4}{4 b}-\frac {(-4 A b+4 a B) \int \frac {x^3}{a+b x^3} \, dx}{4 b}\\ &=\frac {(A b-a B) x}{b^2}+\frac {B x^4}{4 b}-\frac {(a (A b-a B)) \int \frac {1}{a+b x^3} \, dx}{b^2}\\ &=\frac {(A b-a B) x}{b^2}+\frac {B x^4}{4 b}-\frac {\left (\sqrt [3]{a} (A b-a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^2}-\frac {\left (\sqrt [3]{a} (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^2}\\ &=\frac {(A b-a B) x}{b^2}+\frac {B x^4}{4 b}-\frac {\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac {\left (\sqrt [3]{a} (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^{7/3}}-\frac {\left (a^{2/3} (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^2}\\ &=\frac {(A b-a B) x}{b^2}+\frac {B x^4}{4 b}-\frac {\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac {\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}-\frac {\left (\sqrt [3]{a} (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^{7/3}}\\ &=\frac {(A b-a B) x}{b^2}+\frac {B x^4}{4 b}+\frac {\sqrt [3]{a} (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^{7/3}}-\frac {\sqrt [3]{a} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{7/3}}+\frac {\sqrt [3]{a} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 152, normalized size = 0.94 \begin {gather*} \frac {-2 \sqrt [3]{a} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+12 \sqrt [3]{b} x (A b-a B)+4 \sqrt [3]{a} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt {3} \sqrt [3]{a} (a B-A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+3 b^{4/3} B x^4}{12 b^{7/3}} \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^3 \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 = 0.49, size = 145, normalized size = 0.90 \begin {gather*} \frac {3 \, B b x^{4} - 4 \, \sqrt {3} {\left (B 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 ) + 2 \, {\left (B 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 ) - 4 \, {\left (B a - A b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 12 \, {\left (B a - A b\right )} x}{12 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 186, normalized size = 1.15 \begin {gather*} \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a - \left (-a b^{2}\right )^{\frac {1}{3}} 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^{3}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} B a - \left (-a b^{2}\right )^{\frac {1}{3}} 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^{3}} - \frac {{\left (B a^{2} b^{2} - A a b^{3}\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^{4}} + \frac {B b^{3} x^{4} - 4 \, B a b^{2} x + 4 \, A b^{3} x}{4 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 221, normalized size = 1.36 \begin {gather*} \frac {B \,x^{4}}{4 b}-\frac {\sqrt {3}\, A a \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^{2}}-\frac {A a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{2}}+\frac {A a \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^{2}}+\frac {A x}{b}+\frac {\sqrt {3}\, B \,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 {B \,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 {B \,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 {B a x}{b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.08, size = 154, normalized size = 0.95 \begin {gather*} \frac {B b x^{4} - 4 \, {\left (B a - A b\right )} x}{4 \, b^{2}} + \frac {\sqrt {3} {\left (B a^{2} - 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^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (B a^{2} - 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^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (B a^{2} - A a b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \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 = 2.61, size = 162, normalized size = 1.00 \begin {gather*} x\,\left (\frac {A}{b}-\frac {B\,a}{b^2}\right )+\frac {B\,x^4}{4\,b}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )\,\left (A\,b-B\,a\right )}{3\,b^{7/3}}-\frac {{\left (-a\right )}^{1/3}\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}-\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-B\,a\right )}{3\,b^{7/3}}+\frac {{\left (-a\right )}^{1/3}\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (A\,b-B\,a\right )}{3\,b^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.15, size = 87, normalized size = 0.54 \begin {gather*} \frac {B x^{4}}{4 b} + x \left (\frac {A}{b} - \frac {B a}{b^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{7} + A^{3} a b^{3} - 3 A^{2} B a^{2} b^{2} + 3 A B^{2} a^{3} b - B^{3} a^{4}, \left (t \mapsto t \log {\left (\frac {3 t b^{2}}{- A b + B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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