Optimal. Leaf size=199 \[ -\frac {(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{7/3}}+\frac {(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\frac {(2 a B+A b) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{7/3}}-\frac {x (2 a B+A b)}{9 a b^2 \left (a+b x^3\right )}+\frac {x^4 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.11, antiderivative size = 199, 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, 288, 200, 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 )}{54 a^{5/3} b^{7/3}}+\frac {(2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\frac {(2 a B+A b) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{5/3} b^{7/3}}-\frac {x (2 a B+A b)}{9 a b^2 \left (a+b x^3\right )}+\frac {x^4 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 288
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^4}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b+4 a B) \int \frac {x^3}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^4}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+2 a B) x}{9 a b^2 \left (a+b x^3\right )}+\frac {(A b+2 a B) \int \frac {1}{a+b x^3} \, dx}{9 a b^2}\\ &=\frac {(A b-a B) x^4}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+2 a B) x}{9 a b^2 \left (a+b x^3\right )}+\frac {(A b+2 a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{5/3} b^2}+\frac {(A b+2 a B) \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}{27 a^{5/3} b^2}\\ &=\frac {(A b-a B) x^4}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+2 a B) x}{9 a b^2 \left (a+b x^3\right )}+\frac {(A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\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}{54 a^{5/3} b^{7/3}}+\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}{18 a^{4/3} b^2}\\ &=\frac {(A b-a B) x^4}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+2 a B) x}{9 a b^2 \left (a+b x^3\right )}+\frac {(A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\frac {(A b+2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{7/3}}+\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 )}{9 a^{5/3} b^{7/3}}\\ &=\frac {(A b-a B) x^4}{6 a b \left (a+b x^3\right )^2}-\frac {(A b+2 a B) x}{9 a b^2 \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 )}{9 \sqrt {3} a^{5/3} b^{7/3}}+\frac {(A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{5/3} b^{7/3}}-\frac {(A b+2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{5/3} b^{7/3}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 178, normalized size = 0.89 \begin {gather*} \frac {-\frac {(2 a B+A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac {2 (2 a B+A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {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 )}{a^{5/3}}+\frac {3 \sqrt [3]{b} x (A b-7 a B)}{a \left (a+b x^3\right )}-\frac {9 \sqrt [3]{b} x (A b-a B)}{\left (a+b x^3\right )^2}}{54 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 )}{\left (a+b x^3\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.72, size = 743, normalized size = 3.73 \begin {gather*} \left [-\frac {3 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{4} - 3 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \, B a^{4} b + A a^{3} b^{2} + 2 \, {\left (2 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) + {\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, {\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 6 \, {\left (2 \, B a^{4} b + A a^{3} b^{2}\right )} x}{54 \, {\left (a^{3} b^{5} x^{6} + 2 \, a^{4} b^{4} x^{3} + a^{5} b^{3}\right )}}, -\frac {3 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{4} - 6 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 2 \, B a^{4} b + A a^{3} b^{2} + 2 \, {\left (2 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) + {\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) - 2 \, {\left ({\left (2 \, B a b^{2} + A b^{3}\right )} x^{6} + 2 \, B a^{3} + A a^{2} b + 2 \, {\left (2 \, B a^{2} b + A a b^{2}\right )} x^{3}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 6 \, {\left (2 \, B a^{4} b + A a^{3} b^{2}\right )} x}{54 \, {\left (a^{3} b^{5} x^{6} + 2 \, a^{4} b^{4} x^{3} + a^{5} b^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 187, normalized size = 0.94 \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 )}{27 \, \left (-a b^{2}\right )^{\frac {2}{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 )}{54 \, \left (-a b^{2}\right )^{\frac {2}{3}} a b} - \frac {{\left (2 \, B a + A b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a^{2} b^{2}} - \frac {7 \, B a b x^{4} - A b^{2} x^{4} + 4 \, B a^{2} x + 2 \, A a b x}{18 \, {\left (b x^{3} + a\right )}^{2} a b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 239, normalized size = 1.20 \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 )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\frac {A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,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 )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} a \,b^{2}}+\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 )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {\frac {\left (A b -7 B a \right ) x^{4}}{18 a b}-\frac {\left (A b +2 B a \right ) x}{9 b^{2}}}{\left (b \,x^{3}+a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.25, size = 193, normalized size = 0.97 \begin {gather*} -\frac {{\left (7 \, B a b - A b^{2}\right )} x^{4} + 2 \, {\left (2 \, B a^{2} + A a b\right )} x}{18 \, {\left (a b^{4} x^{6} + 2 \, a^{2} b^{3} x^{3} + a^{3} b^{2}\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 )}{27 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{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 )}{54 \, a b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, B a + A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a 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 = 2.56, size = 173, normalized size = 0.87 \begin {gather*} \frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b+2\,B\,a\right )}{27\,a^{5/3}\,b^{7/3}}-\frac {\frac {x\,\left (A\,b+2\,B\,a\right )}{9\,b^2}-\frac {x^4\,\left (A\,b-7\,B\,a\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}-\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 )}{27\,a^{5/3}\,b^{7/3}}+\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 )}{27\,a^{5/3}\,b^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.25, size = 136, normalized size = 0.68 \begin {gather*} \frac {x^{4} \left (A b^{2} - 7 B a b\right ) + x \left (- 2 A a b - 4 B a^{2}\right )}{18 a^{3} b^{2} + 36 a^{2} b^{3} x^{3} + 18 a b^{4} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{5} b^{7} - 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 {27 t a^{2} b^{2}}{A b + 2 B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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