Optimal. Leaf size=220 \[ -\frac {(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac {2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac {2 (A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{10/3}}-\frac {2 x (A b-7 a B)}{9 a b^3}+\frac {x^4 (A b-7 a B)}{18 a b^2 \left (a+b x^3\right )}+\frac {x^7 (A b-a B)}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.13, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {457, 288, 321, 200, 31, 634, 617, 204, 628} \begin {gather*} -\frac {(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac {2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac {2 (A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{10/3}}+\frac {x^4 (A b-7 a B)}{18 a b^2 \left (a+b x^3\right )}-\frac {2 x (A b-7 a B)}{9 a b^3}+\frac {x^7 (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 321
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^6 \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(-A b+7 a B) \int \frac {x^6}{\left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}-\frac {(2 (A b-7 a B)) \int \frac {x^3}{a+b x^3} \, dx}{9 a b^2}\\ &=-\frac {2 (A b-7 a B) x}{9 a b^3}+\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac {(2 (A b-7 a B)) \int \frac {1}{a+b x^3} \, dx}{9 b^3}\\ &=-\frac {2 (A b-7 a B) x}{9 a b^3}+\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac {(2 (A b-7 a B)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{2/3} b^3}+\frac {(2 (A b-7 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^{2/3} b^3}\\ &=-\frac {2 (A b-7 a B) x}{9 a b^3}+\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac {2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac {(A b-7 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}{27 a^{2/3} b^{10/3}}+\frac {(A b-7 a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 \sqrt [3]{a} b^3}\\ &=-\frac {2 (A b-7 a B) x}{9 a b^3}+\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}+\frac {2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac {(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}+\frac {(2 (A b-7 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^{2/3} b^{10/3}}\\ &=-\frac {2 (A b-7 a B) x}{9 a b^3}+\frac {(A b-a B) x^7}{6 a b \left (a+b x^3\right )^2}+\frac {(A b-7 a B) x^4}{18 a b^2 \left (a+b x^3\right )}-\frac {2 (A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{2/3} b^{10/3}}+\frac {2 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{2/3} b^{10/3}}-\frac {(A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{2/3} b^{10/3}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 188, normalized size = 0.85 \begin {gather*} \frac {\frac {2 (7 a B-A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{2/3}}+\frac {4 (A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{2/3}}+\frac {4 \sqrt {3} (7 a B-A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{2/3}}-\frac {3 \sqrt [3]{b} x (7 A b-13 a B)}{a+b x^3}+\frac {9 a \sqrt [3]{b} x (A b-a B)}{\left (a+b x^3\right )^2}+54 \sqrt [3]{b} B x}{54 b^{10/3}} \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^6 \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 = 0.65, size = 789, normalized size = 3.59 \begin {gather*} \left [\frac {54 \, B a^{2} b^{3} x^{7} + 21 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{4} - 6 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} x^{6} + 7 \, B a^{4} b - A a^{3} b^{2} + 2 \, {\left (7 \, 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 ) + 2 \, {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \, {\left (7 \, 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 ) - 4 \, {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \, {\left (7 \, 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 ) + 12 \, {\left (7 \, B a^{4} b - A a^{3} b^{2}\right )} x}{54 \, {\left (a^{2} b^{6} x^{6} + 2 \, a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}, \frac {54 \, B a^{2} b^{3} x^{7} + 21 \, {\left (7 \, B a^{3} b^{2} - A a^{2} b^{3}\right )} x^{4} - 12 \, \sqrt {\frac {1}{3}} {\left ({\left (7 \, B a^{2} b^{3} - A a b^{4}\right )} x^{6} + 7 \, B a^{4} b - A a^{3} b^{2} + 2 \, {\left (7 \, 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 ) + 2 \, {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \, {\left (7 \, 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 ) - 4 \, {\left ({\left (7 \, B a b^{2} - A b^{3}\right )} x^{6} + 7 \, B a^{3} - A a^{2} b + 2 \, {\left (7 \, 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 ) + 12 \, {\left (7 \, B a^{4} b - A a^{3} b^{2}\right )} x}{54 \, {\left (a^{2} b^{6} x^{6} + 2 \, a^{3} b^{5} x^{3} + a^{4} b^{4}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 187, normalized size = 0.85 \begin {gather*} \frac {2 \, \sqrt {3} {\left (7 \, 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}} b^{2}} + \frac {{\left (7 \, 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 )}{27 \, \left (-a b^{2}\right )^{\frac {2}{3}} b^{2}} + \frac {B x}{b^{3}} + \frac {2 \, {\left (7 \, 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 b^{3}} + \frac {13 \, B a b x^{4} - 7 \, A b^{2} x^{4} + 10 \, B a^{2} x - 4 \, A a b x}{18 \, {\left (b x^{3} + a\right )}^{2} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 268, normalized size = 1.22 \begin {gather*} -\frac {7 A \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b}+\frac {13 B a \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} b^{2}}-\frac {2 A a x}{9 \left (b \,x^{3}+a \right )^{2} b^{2}}+\frac {5 B \,a^{2} x}{9 \left (b \,x^{3}+a \right )^{2} b^{3}}+\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 {2}{3}} b^{3}}+\frac {2 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\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 {2}{3}} b^{3}}-\frac {14 \sqrt {3}\, B 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 B 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 B 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 {B x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 191, normalized size = 0.87 \begin {gather*} \frac {{\left (13 \, B a b - 7 \, A b^{2}\right )} x^{4} + 2 \, {\left (5 \, B a^{2} - 2 \, A a b\right )} x}{18 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\right )}} + \frac {B x}{b^{3}} - \frac {2 \, \sqrt {3} {\left (7 \, 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 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, 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 )}{27 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {2 \, {\left (7 \, B a - A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, b^{4} \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 = 2.60, size = 183, normalized size = 0.83 \begin {gather*} \frac {B\,x}{b^3}-\frac {x^4\,\left (\frac {7\,A\,b^2}{18}-\frac {13\,B\,a\,b}{18}\right )-x\,\left (\frac {5\,B\,a^2}{9}-\frac {2\,A\,a\,b}{9}\right )}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (A\,b-7\,B\,a\right )}{27\,a^{2/3}\,b^{10/3}}-\frac {2\,\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-7\,B\,a\right )}{27\,a^{2/3}\,b^{10/3}}+\frac {2\,\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-7\,B\,a\right )}{27\,a^{2/3}\,b^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.39, size = 141, normalized size = 0.64 \begin {gather*} \frac {B x}{b^{3}} + \frac {x^{4} \left (- 7 A b^{2} + 13 B a b\right ) + x \left (- 4 A a b + 10 B a^{2}\right )}{18 a^{2} b^{3} + 36 a b^{4} x^{3} + 18 b^{5} x^{6}} + \operatorname {RootSum} {\left (19683 t^{3} a^{2} b^{10} - 8 A^{3} b^{3} + 168 A^{2} B a b^{2} - 1176 A B^{2} a^{2} b + 2744 B^{3} a^{3}, \left (t \mapsto t \log {\left (- \frac {27 t a b^{3}}{- 2 A b + 14 B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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