Optimal. Leaf size=244 \[ \frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{13/3}}+\frac {7 x (2 A b-5 a B)}{9 b^4}-\frac {7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac {x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}+\frac {x^{10} (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.17, antiderivative size = 244, 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, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}+\frac {x^7 (2 A b-5 a B)}{9 a b^2 \left (a+b x^3\right )}-\frac {7 x^4 (2 A b-5 a B)}{36 a b^3}+\frac {7 x (2 A b-5 a B)}{9 b^4}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{13/3}}+\frac {x^{10} (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 302
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^9 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(-4 A b+10 a B) \int \frac {x^9}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac {(7 (2 A b-5 a B)) \int \frac {x^6}{a+b x^3} \, dx}{9 a b^2}\\ &=\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac {(7 (2 A b-5 a B)) \int \left (-\frac {a}{b^2}+\frac {x^3}{b}+\frac {a^2}{b^2 \left (a+b x^3\right )}\right ) \, dx}{9 a b^2}\\ &=\frac {7 (2 A b-5 a B) x}{9 b^4}-\frac {7 (2 A b-5 a B) x^4}{36 a b^3}+\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac {(7 a (2 A b-5 a B)) \int \frac {1}{a+b x^3} \, dx}{9 b^4}\\ &=\frac {7 (2 A b-5 a B) x}{9 b^4}-\frac {7 (2 A b-5 a B) x^4}{36 a b^3}+\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac {\left (7 \sqrt [3]{a} (2 A b-5 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 b^4}-\frac {\left (7 \sqrt [3]{a} (2 A b-5 a B)\right ) \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 b^4}\\ &=\frac {7 (2 A b-5 a B) x}{9 b^4}-\frac {7 (2 A b-5 a B) x^4}{36 a b^3}+\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {\left (7 \sqrt [3]{a} (2 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}{54 b^{13/3}}-\frac {\left (7 a^{2/3} (2 A b-5 a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 b^4}\\ &=\frac {7 (2 A b-5 a B) x}{9 b^4}-\frac {7 (2 A b-5 a B) x^4}{36 a b^3}+\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}-\frac {\left (7 \sqrt [3]{a} (2 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 )}{9 b^{13/3}}\\ &=\frac {7 (2 A b-5 a B) x}{9 b^4}-\frac {7 (2 A b-5 a B) x^4}{36 a b^3}+\frac {(A b-a B) x^{10}}{6 a b \left (a+b x^3\right )^2}+\frac {(2 A b-5 a B) x^7}{9 a b^2 \left (a+b x^3\right )}+\frac {7 \sqrt [3]{a} (2 A b-5 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} b^{13/3}}-\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 b^{13/3}}+\frac {7 \sqrt [3]{a} (2 A b-5 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 b^{13/3}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 210, normalized size = 0.86 \begin {gather*} \frac {-14 \sqrt [3]{a} (5 a B-2 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac {18 a^2 \sqrt [3]{b} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac {6 a \sqrt [3]{b} x (13 A b-19 a B)}{a+b x^3}+108 \sqrt [3]{b} x (A b-3 a B)+28 \sqrt [3]{a} (5 a B-2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-28 \sqrt {3} \sqrt [3]{a} (5 a B-2 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+27 b^{4/3} B x^4}{108 b^{13/3}} \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^9 \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 [A] time = 0.60, size = 347, normalized size = 1.42 \begin {gather*} \frac {27 \, B b^{3} x^{10} - 54 \, {\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{7} - 147 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{4} - 28 \, \sqrt {3} {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 14 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right ) - 28 \, {\left ({\left (5 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 5 \, B a^{3} - 2 \, A a^{2} b + 2 \, {\left (5 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 84 \, {\left (5 \, B a^{3} - 2 \, A a^{2} b\right )} x}{108 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 234, normalized size = 0.96 \begin {gather*} \frac {7 \, \sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 2 \, \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 )}{27 \, b^{5}} - \frac {7 \, {\left (5 \, B a^{2} - 2 \, 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 b^{4}} + \frac {7 \, {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 2 \, \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 )}{54 \, b^{5}} - \frac {19 \, B a^{2} b x^{4} - 13 \, A a b^{2} x^{4} + 16 \, B a^{3} x - 10 \, A a^{2} b x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{4}} + \frac {B b^{9} x^{4} - 12 \, B a b^{8} x + 4 \, A b^{9} x}{4 \, b^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 299, normalized size = 1.23 \begin {gather*} \frac {13 A a \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {19 B \,a^{2} x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {B \,x^{4}}{4 b^{3}}+\frac {5 A \,a^{2} x}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}-\frac {8 B \,a^{3} x}{9 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {14 \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 {2}{3}} b^{4}}-\frac {14 A a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {7 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 {2}{3}} b^{4}}+\frac {A x}{b^{3}}+\frac {35 \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 {2}{3}} b^{5}}+\frac {35 B \,a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {35 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 )}{54 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{5}}-\frac {3 B a x}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 223, normalized size = 0.91 \begin {gather*} -\frac {{\left (19 \, B a^{2} b - 13 \, A a b^{2}\right )} x^{4} + 2 \, {\left (8 \, B a^{3} - 5 \, A a^{2} b\right )} x}{18 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} + \frac {B b x^{4} - 4 \, {\left (3 \, B a - A b\right )} x}{4 \, b^{4}} + \frac {7 \, \sqrt {3} {\left (5 \, B a^{2} - 2 \, 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 {2}{3}}} - \frac {7 \, {\left (5 \, B a^{2} - 2 \, 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 \, b^{5} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {7 \, {\left (5 \, B a^{2} - 2 \, 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 {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 227, normalized size = 0.93 \begin {gather*} \frac {x^4\,\left (\frac {13\,A\,a\,b^2}{18}-\frac {19\,B\,a^2\,b}{18}\right )-x\,\left (\frac {8\,B\,a^3}{9}-\frac {5\,A\,a^2\,b}{9}\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+x\,\left (\frac {A}{b^3}-\frac {3\,B\,a}{b^4}\right )+\frac {B\,x^4}{4\,b^3}+\frac {7\,{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )\,\left (2\,A\,b-5\,B\,a\right )}{27\,b^{13/3}}-\frac {7\,{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}-2\,a\,b^{1/3}\,x+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,A\,b-5\,B\,a\right )}{27\,b^{13/3}}+\frac {7\,{\left (-a\right )}^{1/3}\,\ln \left (2\,a\,b^{1/3}\,x-{\left (-a\right )}^{4/3}+\sqrt {3}\,{\left (-a\right )}^{4/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (2\,A\,b-5\,B\,a\right )}{27\,b^{13/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.01, size = 163, normalized size = 0.67 \begin {gather*} \frac {B x^{4}}{4 b^{3}} + x \left (\frac {A}{b^{3}} - \frac {3 B a}{b^{4}}\right ) + \frac {x^{4} \left (13 A a b^{2} - 19 B a^{2} b\right ) + x \left (10 A a^{2} b - 16 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^{13} + 2744 A^{3} a b^{3} - 20580 A^{2} B a^{2} b^{2} + 51450 A B^{2} a^{3} b - 42875 B^{3} a^{4}, \left (t \mapsto t \log {\left (\frac {27 t b^{4}}{- 14 A b + 35 B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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