Optimal. Leaf size=215 \[ -\frac {b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac {b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac {b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{11/3}}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {5 a B-8 A b}{15 a^2 b x^5}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {457, 325, 200, 31, 634, 617, 204, 628} \[ -\frac {b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac {b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac {b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{11/3}}+\frac {8 A b-5 a B}{6 a^3 x^2}-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 325
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^6 \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {(8 A b-5 a B) \int \frac {1}{x^6 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}-\frac {(8 A b-5 a B) \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx}{3 a^2}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {(b (8 A b-5 a B)) \int \frac {1}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {(b (8 A b-5 a B)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{11/3}}+\frac {(b (8 A b-5 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}{9 a^{11/3}}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac {\left (b^{2/3} (8 A b-5 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}{18 a^{11/3}}+\frac {(b (8 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 a^{10/3}}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}+\frac {b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac {b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}+\frac {\left (b^{2/3} (8 A b-5 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 )}{3 a^{11/3}}\\ &=-\frac {8 A b-5 a B}{15 a^2 b x^5}+\frac {8 A b-5 a B}{6 a^3 x^2}+\frac {A b-a B}{3 a b x^5 \left (a+b x^3\right )}-\frac {b^{2/3} (8 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} a^{11/3}}+\frac {b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{11/3}}-\frac {b^{2/3} (8 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{11/3}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 183, normalized size = 0.85 \[ \frac {5 b^{2/3} (5 a B-8 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac {30 a^{2/3} b x (a B-A b)}{a+b x^3}-\frac {45 a^{2/3} (a B-2 A b)}{x^2}-\frac {18 a^{5/3} A}{x^5}+10 b^{2/3} (8 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-10 \sqrt {3} b^{2/3} (8 A b-5 a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{90 a^{11/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 277, normalized size = 1.29 \[ -\frac {15 \, {\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 9 \, {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 18 \, A a^{2} + 10 \, \sqrt {3} {\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} + {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) - 5 \, {\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} + {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) + 10 \, {\left ({\left (5 \, B a b - 8 \, A b^{2}\right )} x^{8} + {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{5}\right )} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right )}{90 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 206, normalized size = 0.96 \[ -\frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{4}} + \frac {{\left (5 \, B a b - 8 \, A b^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{4}} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 8 \, \left (-a b^{2}\right )^{\frac {1}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{4}} - \frac {B a b x - A b^{2} x}{3 \, {\left (b x^{3} + a\right )} a^{3}} - \frac {5 \, B a x^{3} - 10 \, A b x^{3} + 2 \, A a}{10 \, a^{3} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 252, normalized size = 1.17 \[ \frac {A \,b^{2} x}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {B b x}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {8 \sqrt {3}\, A b \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {8 A b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {4 A b \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}-\frac {5 \sqrt {3}\, B \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}-\frac {5 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {5 B \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2}}+\frac {A b}{a^{3} x^{2}}-\frac {B}{2 a^{2} x^{2}}-\frac {A}{5 a^{2} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 186, normalized size = 0.87 \[ -\frac {5 \, {\left (5 \, B a b - 8 \, A b^{2}\right )} x^{6} + 3 \, {\left (5 \, B a^{2} - 8 \, A a b\right )} x^{3} + 6 \, A a^{2}}{30 \, {\left (a^{3} b x^{8} + a^{4} x^{5}\right )}} - \frac {\sqrt {3} {\left (5 \, B a - 8 \, A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (5 \, B a - 8 \, A b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (5 \, B a - 8 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.57, size = 176, normalized size = 0.82 \[ \frac {\frac {x^3\,\left (8\,A\,b-5\,B\,a\right )}{10\,a^2}-\frac {A}{5\,a}+\frac {b\,x^6\,\left (8\,A\,b-5\,B\,a\right )}{6\,a^3}}{b\,x^8+a\,x^5}+\frac {b^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}}-\frac {b^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}}+\frac {b^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (8\,A\,b-5\,B\,a\right )}{9\,a^{11/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.84, size = 138, normalized size = 0.64 \[ \operatorname {RootSum} {\left (729 t^{3} a^{11} - 512 A^{3} b^{5} + 960 A^{2} B a b^{4} - 600 A B^{2} a^{2} b^{3} + 125 B^{3} a^{3} b^{2}, \left (t \mapsto t \log {\left (- \frac {9 t a^{4}}{- 8 A b^{2} + 5 B a b} + x \right )} \right )\right )} + \frac {- 6 A a^{2} + x^{6} \left (40 A b^{2} - 25 B a b\right ) + x^{3} \left (24 A a b - 15 B a^{2}\right )}{30 a^{4} x^{5} + 30 a^{3} b x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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