Optimal. Leaf size=171 \[ \frac {(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}-\frac {(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}-\frac {(2 a B+A b) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{5/3}}+\frac {x^2 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.09, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {457, 292, 31, 634, 617, 204, 628} \begin {gather*} \frac {(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}-\frac {(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}-\frac {(2 a B+A b) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{5/3}}+\frac {x^2 (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 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x^2}{3 a b \left (a+b x^3\right )}+\frac {(A b+2 a B) \int \frac {x}{a+b x^3} \, dx}{3 a b}\\ &=\frac {(A b-a B) x^2}{3 a b \left (a+b x^3\right )}-\frac {(A b+2 a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{4/3} b^{4/3}}+\frac {(A b+2 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 a^{4/3} b^{4/3}}\\ &=\frac {(A b-a B) x^2}{3 a b \left (a+b x^3\right )}-\frac {(A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}+\frac {(A b+2 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 a^{4/3} b^{5/3}}+\frac {(A b+2 a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a b^{4/3}}\\ &=\frac {(A b-a B) x^2}{3 a b \left (a+b x^3\right )}-\frac {(A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}+\frac {(A b+2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}+\frac {(A b+2 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 a^{4/3} b^{5/3}}\\ &=\frac {(A b-a B) x^2}{3 a b \left (a+b x^3\right )}-\frac {(A b+2 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{4/3} b^{5/3}}-\frac {(A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{4/3} b^{5/3}}+\frac {(A b+2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{4/3} b^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 146, normalized size = 0.85 \begin {gather*} \frac {(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {6 \sqrt [3]{a} b^{2/3} x^2 (a B-A b)}{a+b x^3}-2 (2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} (2 a B+A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{18 a^{4/3} b^{5/3}} \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 \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 = 0.55, size = 548, normalized size = 3.20 \begin {gather*} \left [-\frac {6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, B a^{3} b + A a^{2} b^{2} + {\left (2 \, B a^{2} b^{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 (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{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 (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{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^{2} b^{4} x^{3} + a^{3} b^{3}\right )}}, -\frac {6 \, {\left (B a^{2} b^{2} - A a b^{3}\right )} x^{2} - 6 \, \sqrt {\frac {1}{3}} {\left (2 \, B a^{3} b + A a^{2} b^{2} + {\left (2 \, B a^{2} b^{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 (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{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 (2 \, B a b + A b^{2}\right )} x^{3} + 2 \, B a^{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^{2} b^{4} x^{3} + a^{3} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 186, normalized size = 1.09 \begin {gather*} \frac {\sqrt {3} {\left (2 \, B 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 )}{9 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b} - \frac {{\left (2 \, B 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 )}{18 \, \left (-a b^{2}\right )^{\frac {1}{3}} a b} - \frac {{\left (2 \, B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 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^{2} b} - \frac {B a x^{2} - A b x^{2}}{3 \, {\left (b x^{3} + a\right )} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 223, normalized size = 1.30 \begin {gather*} \frac {\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}} a b}-\frac {A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a b}+\frac {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}} a b}+\frac {2 \sqrt {3}\, B \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 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{2}}+\frac {B \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 {\left (A b -B a \right ) x^{2}}{3 \left (b \,x^{3}+a \right ) a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 160, normalized size = 0.94 \begin {gather*} -\frac {{\left (B a - A b\right )} x^{2}}{3 \, {\left (a b^{2} x^{3} + a^{2} b\right )}} + \frac {\sqrt {3} {\left (2 \, B 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 )}{9 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (2 \, B 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 )}{18 \, a b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (2 \, B a + A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{2} \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 = 0.25, size = 145, normalized size = 0.85 \begin {gather*} \frac {x^2\,\left (A\,b-B\,a\right )}{3\,a\,b\,\left (b\,x^3+a\right )}-\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 (A\,b+2\,B\,a\right )}{9\,a^{4/3}\,b^{5/3}}+\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 (A\,b+2\,B\,a\right )}{9\,a^{4/3}\,b^{5/3}}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b+2\,B\,a\right )}{9\,a^{4/3}\,b^{5/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 117, normalized size = 0.68 \begin {gather*} \frac {x^{2} \left (A b - B a\right )}{3 a^{2} b + 3 a b^{2} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{4} b^{5} + A^{3} b^{3} + 6 A^{2} B a b^{2} + 12 A B^{2} a^{2} b + 8 B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{3} b^{3}}{A^{2} b^{2} + 4 A B a b + 4 B^{2} a^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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