Optimal. Leaf size=246 \[ -\frac {2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}+\frac {4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}+\frac {4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{14/3}}+\frac {2 x^2 (5 A b-11 a B)}{9 b^4}-\frac {4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac {x^8 (5 A b-11 a B)}{18 a b^2 \left (a+b x^3\right )}+\frac {x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.16, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {457, 288, 302, 292, 31, 634, 617, 204, 628} \[ -\frac {2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}+\frac {4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}+\frac {4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{14/3}}+\frac {x^8 (5 A b-11 a B)}{18 a b^2 \left (a+b x^3\right )}-\frac {4 x^5 (5 A b-11 a B)}{45 a b^3}+\frac {2 x^2 (5 A b-11 a B)}{9 b^4}+\frac {x^{11} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 288
Rule 292
Rule 302
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^{10} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(-5 A b+11 a B) \int \frac {x^{10}}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}-\frac {(4 (5 A b-11 a B)) \int \frac {x^7}{a+b x^3} \, dx}{9 a b^2}\\ &=\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}-\frac {(4 (5 A b-11 a B)) \int \left (-\frac {a x}{b^2}+\frac {x^4}{b}+\frac {a^2 x}{b^2 \left (a+b x^3\right )}\right ) \, dx}{9 a b^2}\\ &=\frac {2 (5 A b-11 a B) x^2}{9 b^4}-\frac {4 (5 A b-11 a B) x^5}{45 a b^3}+\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}-\frac {(4 a (5 A b-11 a B)) \int \frac {x}{a+b x^3} \, dx}{9 b^4}\\ &=\frac {2 (5 A b-11 a B) x^2}{9 b^4}-\frac {4 (5 A b-11 a B) x^5}{45 a b^3}+\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac {\left (4 a^{2/3} (5 A b-11 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^{13/3}}-\frac {\left (4 a^{2/3} (5 A b-11 a B)\right ) \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 b^{13/3}}\\ &=\frac {2 (5 A b-11 a B) x^2}{9 b^4}-\frac {4 (5 A b-11 a B) x^5}{45 a b^3}+\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac {4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac {\left (2 a^{2/3} (5 A b-11 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}{27 b^{14/3}}-\frac {(2 a (5 A b-11 a B)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 b^{13/3}}\\ &=\frac {2 (5 A b-11 a B) x^2}{9 b^4}-\frac {4 (5 A b-11 a B) x^5}{45 a b^3}+\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac {4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac {2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}-\frac {\left (4 a^{2/3} (5 A b-11 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 )}{9 b^{14/3}}\\ &=\frac {2 (5 A b-11 a B) x^2}{9 b^4}-\frac {4 (5 A b-11 a B) x^5}{45 a b^3}+\frac {(A b-a B) x^{11}}{6 a b \left (a+b x^3\right )^2}+\frac {(5 A b-11 a B) x^8}{18 a b^2 \left (a+b x^3\right )}+\frac {4 a^{2/3} (5 A b-11 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{14/3}}+\frac {4 a^{2/3} (5 A b-11 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{14/3}}-\frac {2 a^{2/3} (5 A b-11 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 b^{14/3}}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 216, normalized size = 0.88 \[ \frac {20 a^{2/3} (11 a B-5 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-40 a^{2/3} (11 a B-5 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-40 \sqrt {3} a^{2/3} (11 a B-5 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+\frac {45 a^2 b^{2/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+135 b^{2/3} x^2 (A b-3 a B)+\frac {30 a b^{2/3} x^2 (7 A b-10 a B)}{a+b x^3}+54 b^{5/3} B x^5}{270 b^{14/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 364, normalized size = 1.48 \[ \frac {54 \, B b^{3} x^{11} - 27 \, {\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{8} - 96 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{5} - 60 \, {\left (11 \, B a^{3} - 5 \, A a^{2} b\right )} x^{2} + 40 \, \sqrt {3} {\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) + 20 \, {\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - 40 \, {\left ({\left (11 \, B a b^{2} - 5 \, A b^{3}\right )} x^{6} + 11 \, B a^{3} - 5 \, A a^{2} b + 2 \, {\left (11 \, B a^{2} b - 5 \, A a b^{2}\right )} x^{3}\right )} \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right )}{270 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 259, normalized size = 1.05 \[ -\frac {4 \, {\left (11 \, B a^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, A a b \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{27 \, a b^{4}} - \frac {4 \, \sqrt {3} {\left (11 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{6}} + \frac {2 \, {\left (11 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 5 \, \left (-a b^{2}\right )^{\frac {2}{3}} A b\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, b^{6}} - \frac {20 \, B a^{2} b x^{5} - 14 \, A a b^{2} x^{5} + 17 \, B a^{3} x^{2} - 11 \, A a^{2} b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} b^{4}} + \frac {2 \, B b^{12} x^{5} - 15 \, B a b^{11} x^{2} + 5 \, A b^{12} x^{2}}{10 \, b^{15}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 308, normalized size = 1.25 \[ \frac {7 A a \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {10 B \,a^{2} x^{5}}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {B \,x^{5}}{5 b^{3}}+\frac {11 A \,a^{2} x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {17 B \,a^{3} x^{2}}{18 \left (b \,x^{3}+a \right )^{2} b^{4}}+\frac {A \,x^{2}}{2 b^{3}}-\frac {3 B a \,x^{2}}{2 b^{4}}-\frac {20 \sqrt {3}\, A a \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {20 A a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}-\frac {10 A a \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{4}}+\frac {44 \sqrt {3}\, B \,a^{2} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}-\frac {44 B \,a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}}+\frac {22 B \,a^{2} \ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} b^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 228, normalized size = 0.93 \[ -\frac {2 \, {\left (10 \, B a^{2} b - 7 \, A a b^{2}\right )} x^{5} + {\left (17 \, B a^{3} - 11 \, A a^{2} b\right )} x^{2}}{18 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {4 \, \sqrt {3} {\left (11 \, B a^{2} - 5 \, A a b\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {2 \, B b x^{5} - 5 \, {\left (3 \, B a - A b\right )} x^{2}}{10 \, b^{4}} + \frac {2 \, {\left (11 \, B a^{2} - 5 \, A a b\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {4 \, {\left (11 \, B a^{2} - 5 \, A a b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{5} \left (\frac {a}{b}\right )^{\frac {1}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.58, size = 213, normalized size = 0.87 \[ \frac {x^5\,\left (\frac {7\,A\,a\,b^2}{9}-\frac {10\,B\,a^2\,b}{9}\right )-x^2\,\left (\frac {17\,B\,a^3}{18}-\frac {11\,A\,a^2\,b}{18}\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+x^2\,\left (\frac {A}{2\,b^3}-\frac {3\,B\,a}{2\,b^4}\right )+\frac {B\,x^5}{5\,b^3}+\frac {4\,a^{2/3}\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,A\,b-11\,B\,a\right )}{27\,b^{14/3}}+\frac {4\,a^{2/3}\,\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-11\,B\,a\right )}{27\,b^{14/3}}-\frac {4\,a^{2/3}\,\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (5\,A\,b-11\,B\,a\right )}{27\,b^{14/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.24, size = 192, normalized size = 0.78 \[ \frac {B x^{5}}{5 b^{3}} + x^{2} \left (\frac {A}{2 b^{3}} - \frac {3 B a}{2 b^{4}}\right ) + \frac {x^{5} \left (14 A a b^{2} - 20 B a^{2} b\right ) + x^{2} \left (11 A a^{2} b - 17 B a^{3}\right )}{18 a^{2} b^{4} + 36 a b^{5} x^{3} + 18 b^{6} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} b^{14} - 8000 A^{3} a^{2} b^{3} + 52800 A^{2} B a^{3} b^{2} - 116160 A B^{2} a^{4} b + 85184 B^{3} a^{5}, \left (t \mapsto t \log {\left (\frac {729 t^{2} b^{9}}{400 A^{2} a b^{2} - 1760 A B a^{2} b + 1936 B^{2} a^{3}} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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