Optimal. Leaf size=167 \[ -\frac {a^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac {a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {a^{2/3} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{8/3}}+\frac {x^2 (A b-a B)}{2 b^2}+\frac {B x^5}{5 b} \]
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Rubi [A] time = 0.12, antiderivative size = 167, 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, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {a^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}+\frac {a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}+\frac {a^{2/3} (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^{8/3}}+\frac {x^2 (A b-a B)}{2 b^2}+\frac {B x^5}{5 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 321
Rule 459
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^3\right )}{a+b x^3} \, dx &=\frac {B x^5}{5 b}-\frac {(-5 A b+5 a B) \int \frac {x^4}{a+b x^3} \, dx}{5 b}\\ &=\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^5}{5 b}-\frac {(a (A b-a B)) \int \frac {x}{a+b x^3} \, dx}{b^2}\\ &=\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^5}{5 b}+\frac {\left (a^{2/3} (A b-a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{7/3}}-\frac {\left (a^{2/3} (A b-a B)\right ) \int \frac {\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^{7/3}}\\ &=\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^5}{5 b}+\frac {a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}-\frac {\left (a^{2/3} (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^{8/3}}-\frac {(a (A b-a B)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 b^{7/3}}\\ &=\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^5}{5 b}+\frac {a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}-\frac {a^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}-\frac {\left (a^{2/3} (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^{8/3}}\\ &=\frac {(A b-a B) x^2}{2 b^2}+\frac {B x^5}{5 b}+\frac {a^{2/3} (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^{8/3}}+\frac {a^{2/3} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{8/3}}-\frac {a^{2/3} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{8/3}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 154, normalized size = 0.92 \begin {gather*} \frac {5 a^{2/3} (a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-10 a^{2/3} (a B-A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt {3} a^{2/3} (a B-A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+15 b^{2/3} x^2 (A b-a B)+6 b^{5/3} B x^5}{30 b^{8/3}} \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^4 \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 = 0.53, size = 162, normalized size = 0.97 \begin {gather*} \frac {6 \, B b x^{5} - 15 \, {\left (B a - A b\right )} x^{2} + 10 \, \sqrt {3} {\left (B a - A b\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 5 \, {\left (B a - A b\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 10 \, {\left (B a - A b\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{30 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 207, normalized size = 1.24 \begin {gather*} -\frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} B a - \left (-a b^{2}\right )^{\frac {2}{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^{4}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} B a - \left (-a b^{2}\right )^{\frac {2}{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^{4}} - \frac {{\left (B a^{2} b^{3} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - A a b^{4} \left (-\frac {a}{b}\right )^{\frac {1}{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^{5}} + \frac {2 \, B b^{4} x^{5} - 5 \, B a b^{3} x^{2} + 5 \, A b^{4} x^{2}}{10 \, b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 226, normalized size = 1.35 \begin {gather*} \frac {B \,x^{5}}{5 b}+\frac {A \,x^{2}}{2 b}-\frac {B a \,x^{2}}{2 b^{2}}-\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 {1}{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 {1}{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 {1}{3}} b^{2}}+\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 {1}{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 {1}{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 {1}{3}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.10, size = 157, normalized size = 0.94 \begin {gather*} \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 {1}{3}}} + \frac {2 \, B b x^{5} - 5 \, {\left (B a - A b\right )} x^{2}}{10 \, b^{2}} + \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 {1}{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 {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.55, size = 144, normalized size = 0.86 \begin {gather*} x^2\,\left (\frac {A}{2\,b}-\frac {B\,a}{2\,b^2}\right )+\frac {B\,x^5}{5\,b}+\frac {a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b-B\,a\right )}{3\,b^{8/3}}+\frac {a^{2/3}\,\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^{8/3}}-\frac {a^{2/3}\,\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^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 114, normalized size = 0.68 \begin {gather*} \frac {B x^{5}}{5 b} + x^{2} \left (\frac {A}{2 b} - \frac {B a}{2 b^{2}}\right ) + \operatorname {RootSum} {\left (27 t^{3} b^{8} - A^{3} a^{2} b^{3} + 3 A^{2} B a^{3} b^{2} - 3 A B^{2} a^{4} b + B^{3} a^{5}, \left (t \mapsto t \log {\left (\frac {9 t^{2} b^{5}}{A^{2} a b^{2} - 2 A B a^{2} b + B^{2} a^{3}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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