Optimal. Leaf size=213 \[ \frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac {\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{10/3}}+\frac {x (4 A b-7 a B)}{3 b^3}-\frac {x^4 (4 A b-7 a B)}{12 a b^2}+\frac {x^7 (A b-a B)}{3 a b \left (a+b x^3\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 213, 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, 302, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac {x^4 (4 A b-7 a B)}{12 a b^2}+\frac {x (4 A b-7 a B)}{3 b^3}-\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac {\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{10/3}}+\frac {x^7 (A b-a B)}{3 a b \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 302
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 )^2} \, dx &=\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}+\frac {(-4 A b+7 a B) \int \frac {x^6}{a+b x^3} \, dx}{3 a b}\\ &=\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}+\frac {(-4 A b+7 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}{3 a b}\\ &=\frac {(4 A b-7 a B) x}{3 b^3}-\frac {(4 A b-7 a B) x^4}{12 a b^2}+\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac {(a (4 A b-7 a B)) \int \frac {1}{a+b x^3} \, dx}{3 b^3}\\ &=\frac {(4 A b-7 a B) x}{3 b^3}-\frac {(4 A b-7 a B) x^4}{12 a b^2}+\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac {\left (\sqrt [3]{a} (4 A b-7 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 b^3}-\frac {\left (\sqrt [3]{a} (4 A b-7 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}{9 b^3}\\ &=\frac {(4 A b-7 a B) x}{3 b^3}-\frac {(4 A b-7 a B) x^4}{12 a b^2}+\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac {\left (\sqrt [3]{a} (4 A b-7 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 b^{10/3}}-\frac {\left (a^{2/3} (4 A b-7 a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 b^3}\\ &=\frac {(4 A b-7 a B) x}{3 b^3}-\frac {(4 A b-7 a B) x^4}{12 a b^2}+\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}-\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}-\frac {\left (\sqrt [3]{a} (4 A b-7 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 b^{10/3}}\\ &=\frac {(4 A b-7 a B) x}{3 b^3}-\frac {(4 A b-7 a B) x^4}{12 a b^2}+\frac {(A b-a B) x^7}{3 a b \left (a+b x^3\right )}+\frac {\sqrt [3]{a} (4 A b-7 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} b^{10/3}}-\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 b^{10/3}}+\frac {\sqrt [3]{a} (4 A b-7 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 b^{10/3}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 181, normalized size = 0.85 \begin {gather*} \frac {-2 \sqrt [3]{a} (7 a B-4 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\frac {12 a \sqrt [3]{b} x (A b-a B)}{a+b x^3}+36 \sqrt [3]{b} x (A b-2 a B)+4 \sqrt [3]{a} (7 a B-4 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt {3} \sqrt [3]{a} (7 a B-4 A b) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+9 b^{4/3} B x^4}{36 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 )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.85, size = 240, normalized size = 1.13 \begin {gather*} \frac {9 \, B b^{2} x^{7} - 9 \, {\left (7 \, B a b - 4 \, A b^{2}\right )} x^{4} - 4 \, \sqrt {3} {\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\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 ) + 2 \, {\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\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 ) - 4 \, {\left ({\left (7 \, B a b - 4 \, A b^{2}\right )} x^{3} + 7 \, B a^{2} - 4 \, A a b\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right ) - 12 \, {\left (7 \, B a^{2} - 4 \, A a b\right )} x}{36 \, {\left (b^{4} x^{3} + a b^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 211, normalized size = 0.99 \begin {gather*} \frac {\sqrt {3} {\left (7 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 4 \, \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 \, b^{4}} - \frac {{\left (7 \, B a^{2} - 4 \, 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 )}{9 \, a b^{3}} + \frac {{\left (7 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 4 \, \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 \, b^{4}} - \frac {B a^{2} x - A a b x}{3 \, {\left (b x^{3} + a\right )} b^{3}} + \frac {B b^{6} x^{4} - 8 \, B a b^{5} x + 4 \, A b^{6} x}{4 \, b^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 257, normalized size = 1.21 \begin {gather*} \frac {B \,x^{4}}{4 b^{2}}+\frac {A a x}{3 \left (b \,x^{3}+a \right ) b^{2}}-\frac {B \,a^{2} x}{3 \left (b \,x^{3}+a \right ) b^{3}}-\frac {4 \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 )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}-\frac {4 A a \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{3}}+\frac {2 A a \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}} b^{3}}+\frac {A x}{b^{2}}+\frac {7 \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 )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}+\frac {7 B \,a^{2} \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {7 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 )}{18 \left (\frac {a}{b}\right )^{\frac {2}{3}} b^{4}}-\frac {2 B a x}{b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 187, normalized size = 0.88 \begin {gather*} -\frac {{\left (B a^{2} - A a b\right )} x}{3 \, {\left (b^{4} x^{3} + a b^{3}\right )}} + \frac {B b x^{4} - 4 \, {\left (2 \, B a - A b\right )} x}{4 \, b^{3}} + \frac {\sqrt {3} {\left (7 \, B a^{2} - 4 \, 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 )}{9 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (7 \, B a^{2} - 4 \, 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 )}{18 \, b^{4} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (7 \, B a^{2} - 4 \, A a b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, 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 = 2.62, size = 193, normalized size = 0.91 \begin {gather*} x\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )-\frac {x\,\left (\frac {B\,a^2}{3}-\frac {A\,a\,b}{3}\right )}{b^4\,x^3+a\,b^3}+\frac {B\,x^4}{4\,b^2}+\frac {{\left (-a\right )}^{1/3}\,\ln \left ({\left (-a\right )}^{4/3}+a\,b^{1/3}\,x\right )\,\left (4\,A\,b-7\,B\,a\right )}{9\,b^{10/3}}-\frac {{\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 (4\,A\,b-7\,B\,a\right )}{9\,b^{10/3}}+\frac {{\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 (4\,A\,b-7\,B\,a\right )}{9\,b^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.54, size = 126, normalized size = 0.59 \begin {gather*} \frac {B x^{4}}{4 b^{2}} + x \left (\frac {A}{b^{2}} - \frac {2 B a}{b^{3}}\right ) + \frac {x \left (A a b - B a^{2}\right )}{3 a b^{3} + 3 b^{4} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} b^{10} + 64 A^{3} a b^{3} - 336 A^{2} B a^{2} b^{2} + 588 A B^{2} a^{3} b - 343 B^{3} a^{4}, \left (t \mapsto t \log {\left (\frac {9 t b^{3}}{- 4 A b + 7 B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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