Optimal. Leaf size=196 \[ \frac {(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}-\frac {(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}-\frac {(2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {x^2 (2 A b-5 a B)}{6 a b^2}+\frac {x^5 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.11, antiderivative size = 196, 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 = {457, 321, 292, 31, 634, 617, 204, 628} \begin {gather*} \frac {(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}-\frac {x^2 (2 A b-5 a B)}{6 a b^2}-\frac {(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}-\frac {(2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{8/3}}+\frac {x^5 (A b-a B)}{3 a b \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 321
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x^5}{3 a b \left (a+b x^3\right )}+\frac {(-2 A b+5 a B) \int \frac {x^4}{a+b x^3} \, dx}{3 a b}\\ &=-\frac {(2 A b-5 a B) x^2}{6 a b^2}+\frac {(A b-a B) x^5}{3 a b \left (a+b x^3\right )}+\frac {(2 A b-5 a B) \int \frac {x}{a+b x^3} \, dx}{3 b^2}\\ &=-\frac {(2 A b-5 a B) x^2}{6 a b^2}+\frac {(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac {(2 A b-5 a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 \sqrt [3]{a} b^{7/3}}+\frac {(2 A b-5 a B) \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}{9 \sqrt [3]{a} b^{7/3}}\\ &=-\frac {(2 A b-5 a B) x^2}{6 a b^2}+\frac {(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac {(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}+\frac {(2 A b-5 a B) \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}{18 \sqrt [3]{a} b^{8/3}}+\frac {(2 A b-5 a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^{7/3}}\\ &=-\frac {(2 A b-5 a B) x^2}{6 a b^2}+\frac {(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac {(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}+\frac {(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}+\frac {(2 A b-5 a B) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 \sqrt [3]{a} b^{8/3}}\\ &=-\frac {(2 A b-5 a B) x^2}{6 a b^2}+\frac {(A b-a B) x^5}{3 a b \left (a+b x^3\right )}-\frac {(2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} \sqrt [3]{a} b^{8/3}}-\frac {(2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 \sqrt [3]{a} b^{8/3}}+\frac {(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 \sqrt [3]{a} b^{8/3}}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 165, normalized size = 0.84 \begin {gather*} \frac {\frac {(2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{a}}-\frac {6 b^{2/3} x^2 (A b-a B)}{a+b x^3}+\frac {2 (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a}}+\frac {2 \sqrt {3} (5 a B-2 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{a}}+9 b^{2/3} B x^2}{18 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 )}{\left (a+b x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.79, size = 578, normalized size = 2.95 \begin {gather*} \left [\frac {9 \, B a b^{3} x^{5} + 3 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2} + {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3}\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b + 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac {2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) - {\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a b^{5} x^{3} + a^{2} b^{4}\right )}}, \frac {9 \, B a b^{3} x^{5} + 3 \, {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (5 \, B a^{3} b - 2 \, A a^{2} b^{2} + {\left (5 \, B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{3}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x + \left (-a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {-\frac {\left (-a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) - {\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac {1}{3}} b x + \left (-a b^{2}\right )^{\frac {2}{3}}\right ) + 2 \, {\left ({\left (5 \, B a b - 2 \, A b^{2}\right )} x^{3} + 5 \, B a^{2} - 2 \, A a b\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{18 \, {\left (a b^{5} x^{3} + a^{2} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 189, normalized size = 0.96 \begin {gather*} \frac {B x^{2}}{2 \, b^{2}} - \frac {\sqrt {3} {\left (5 \, B a - 2 \, 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 )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{2}} + \frac {{\left (5 \, B a - 2 \, 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 )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} b^{2}} + \frac {{\left (5 \, B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 2 \, A b \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 )}{9 \, a b^{2}} + \frac {B a x^{2} - A b x^{2}}{3 \, {\left (b x^{3} + a\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 235, normalized size = 1.20 \begin {gather*} -\frac {A \,x^{2}}{3 \left (b \,x^{3}+a \right ) b}+\frac {B a \,x^{2}}{3 \left (b \,x^{3}+a \right ) b^{2}}+\frac {B \,x^{2}}{2 b^{2}}+\frac {2 \sqrt {3}\, A \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {2 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {A \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}-\frac {5 \sqrt {3}\, B a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}+\frac {5 B a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{3}}-\frac {5 B a \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \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 = 1.17, size = 162, normalized size = 0.83 \begin {gather*} \frac {{\left (B a - A b\right )} x^{2}}{3 \, {\left (b^{3} x^{3} + a b^{2}\right )}} + \frac {B x^{2}}{2 \, b^{2}} - \frac {\sqrt {3} {\left (5 \, B a - 2 \, 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 )}{9 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (5 \, B a - 2 \, 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 )}{18 \, b^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (5 \, B a - 2 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, 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 = 2.58, size = 158, normalized size = 0.81 \begin {gather*} \frac {B\,x^2}{2\,b^2}-\frac {x^2\,\left (\frac {A\,b}{3}-\frac {B\,a}{3}\right )}{b^3\,x^3+a\,b^2}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (2\,A\,b-5\,B\,a\right )}{9\,a^{1/3}\,b^{8/3}}-\frac {\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 (2\,A\,b-5\,B\,a\right )}{9\,a^{1/3}\,b^{8/3}}+\frac {\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 (2\,A\,b-5\,B\,a\right )}{9\,a^{1/3}\,b^{8/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.09, size = 126, normalized size = 0.64 \begin {gather*} \frac {B x^{2}}{2 b^{2}} + \frac {x^{2} \left (- A b + B a\right )}{3 a b^{2} + 3 b^{3} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a b^{8} + 8 A^{3} b^{3} - 60 A^{2} B a b^{2} + 150 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a b^{5}}{4 A^{2} b^{2} - 20 A B a b + 25 B^{2} a^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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