Optimal. Leaf size=201 \[ \frac {(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{7/3} b^{5/3}}-\frac {(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}-\frac {(a B+2 A b) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{7/3} b^{5/3}}+\frac {x^2 (a B+2 A b)}{9 a^2 b \left (a+b x^3\right )}+\frac {x^2 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {457, 290, 292, 31, 634, 617, 204, 628} \begin {gather*} \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^{7/3} b^{5/3}}-\frac {(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}-\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^{7/3} b^{5/3}}+\frac {x^2 (a B+2 A b)}{9 a^2 b \left (a+b x^3\right )}+\frac {x^2 (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 204
Rule 290
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 )^3} \, dx &=\frac {(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac {(4 A b+2 a B) \int \frac {x}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}+\frac {(2 A b+a B) \int \frac {x}{a+b x^3} \, dx}{9 a^2 b}\\ &=\frac {(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac {(2 A b+a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{7/3} b^{4/3}}+\frac {(2 A b+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}{27 a^{7/3} b^{4/3}}\\ &=\frac {(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}+\frac {(2 A b+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^{7/3} b^{5/3}}+\frac {(2 A b+a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^2 b^{4/3}}\\ &=\frac {(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}+\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^{7/3} b^{5/3}}+\frac {(2 A b+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^{7/3} b^{5/3}}\\ &=\frac {(A b-a B) x^2}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b+a B) x^2}{9 a^2 b \left (a+b x^3\right )}-\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^{7/3} b^{5/3}}-\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{7/3} b^{5/3}}+\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^{7/3} b^{5/3}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 178, normalized size = 0.89 \begin {gather*} \frac {(a B+2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {9 a^{4/3} b^{2/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac {6 \sqrt [3]{a} b^{2/3} x^2 (a B+2 A b)}{a+b x^3}-2 (a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt {3} (a B+2 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{54 a^{7/3} 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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.79, size = 752, normalized size = 3.74 \begin {gather*} \left [\frac {6 \, {\left (B a^{2} b^{3} + 2 \, A a b^{4}\right )} x^{5} - 3 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{3} + 2 \, A a b^{4}\right )} x^{6} + B a^{4} b + 2 \, A a^{3} b^{2} + 2 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} 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 (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \, {\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\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 (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \, {\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{3} b^{5} x^{6} + 2 \, a^{4} b^{4} x^{3} + a^{5} b^{3}\right )}}, \frac {6 \, {\left (B a^{2} b^{3} + 2 \, A a b^{4}\right )} x^{5} - 3 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{2} + 6 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{3} + 2 \, A a b^{4}\right )} x^{6} + B a^{4} b + 2 \, A a^{3} b^{2} + 2 \, {\left (B a^{3} b^{2} + 2 \, A a^{2} 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 (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \, {\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\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 (B a b^{2} + 2 \, A b^{3}\right )} x^{6} + B a^{3} + 2 \, A a^{2} b + 2 \, {\left (B a^{2} b + 2 \, A a b^{2}\right )} x^{3}\right )} \left (-a b^{2}\right )^{\frac {2}{3}} \log \left (b x - \left (-a b^{2}\right )^{\frac {1}{3}}\right )}{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 = 0.21, size = 207, normalized size = 1.03 \begin {gather*} \frac {\sqrt {3} {\left (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 )}{27 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b} - \frac {{\left (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 )}{54 \, \left (-a b^{2}\right )^{\frac {1}{3}} a^{2} b} - \frac {{\left (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 )}{27 \, a^{3} b} + \frac {2 \, B a b x^{5} + 4 \, A b^{2} x^{5} - B a^{2} x^{2} + 7 \, A a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{2} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 251, normalized size = 1.25 \begin {gather*} \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 )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}-\frac {2 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} 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 )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {\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 {1}{3}} a \,b^{2}}-\frac {B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,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 )}{54 \left (\frac {a}{b}\right )^{\frac {1}{3}} a \,b^{2}}+\frac {\frac {\left (2 A b +B a \right ) x^{5}}{9 a^{2}}+\frac {\left (7 A b -B a \right ) x^{2}}{18 a b}}{\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 = 1.40, size = 195, normalized size = 0.97 \begin {gather*} \frac {2 \, {\left (B a b + 2 \, A b^{2}\right )} x^{5} - {\left (B a^{2} - 7 \, A a b\right )} x^{2}}{18 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} + \frac {\sqrt {3} {\left (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 )}{27 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (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 )}{54 \, a^{2} b^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (B a + 2 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{2} 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 = 0.27, size = 175, normalized size = 0.87 \begin {gather*} \frac {\frac {x^5\,\left (2\,A\,b+B\,a\right )}{9\,a^2}+\frac {x^2\,\left (7\,A\,b-B\,a\right )}{18\,a\,b}}{a^2+2\,a\,b\,x^3+b^2\,x^6}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (2\,A\,b+B\,a\right )}{27\,a^{7/3}\,b^{5/3}}-\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+B\,a\right )}{27\,a^{7/3}\,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 (2\,A\,b+B\,a\right )}{27\,a^{7/3}\,b^{5/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.03, size = 153, normalized size = 0.76 \begin {gather*} \frac {x^{5} \left (4 A b^{2} + 2 B a b\right ) + x^{2} \left (7 A a b - B a^{2}\right )}{18 a^{4} b + 36 a^{3} b^{2} x^{3} + 18 a^{2} b^{3} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{7} b^{5} + 8 A^{3} b^{3} + 12 A^{2} B a b^{2} + 6 A B^{2} a^{2} b + B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {729 t^{2} a^{5} b^{3}}{4 A^{2} b^{2} + 4 A B a b + B^{2} a^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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