Optimal. Leaf size=227 \[ -\frac {(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}}+\frac {2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}+\frac {2 (7 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} b^{2/3}}-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.12, antiderivative size = 227, 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, 290, 325, 292, 31, 634, 617, 204, 628} \begin {gather*} -\frac {(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}}+\frac {2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}+\frac {2 (7 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} b^{2/3}}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 290
Rule 292
Rule 325
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx &=\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {(7 A b-a B) \int \frac {1}{x^2 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac {(2 (7 A b-a B)) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{9 a^2 b}\\ &=-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}-\frac {(2 (7 A b-a B)) \int \frac {x}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac {(2 (7 A b-a B)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{10/3} \sqrt [3]{b}}-\frac {(2 (7 A b-a B)) \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 a^{10/3} \sqrt [3]{b}}\\ &=-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac {2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}-\frac {(7 A b-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^{10/3} b^{2/3}}-\frac {(7 A b-a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{9 a^3 \sqrt [3]{b}}\\ &=-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac {2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}-\frac {(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}}-\frac {(2 (7 A b-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^{10/3} b^{2/3}}\\ &=-\frac {2 (7 A b-a B)}{9 a^3 b x}+\frac {A b-a B}{6 a b x \left (a+b x^3\right )^2}+\frac {7 A b-a B}{18 a^2 b x \left (a+b x^3\right )}+\frac {2 (7 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{10/3} b^{2/3}}+\frac {2 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{10/3} b^{2/3}}-\frac {(7 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{27 a^{10/3} b^{2/3}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 193, normalized size = 0.85 \begin {gather*} \frac {\frac {2 (a B-7 A b) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {9 a^{4/3} x^2 (a B-A b)}{\left (a+b x^3\right )^2}+\frac {4 (7 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{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 )}{b^{2/3}}+\frac {6 \sqrt [3]{a} x^2 (2 a B-5 A b)}{a+b x^3}-\frac {54 \sqrt [3]{a} A}{x}}{54 a^{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 {A+B x^3}{x^2 \left (a+b x^3\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.69, size = 776, normalized size = 3.42 \begin {gather*} \left [\frac {12 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{6} - 54 \, A a^{3} b^{2} + 21 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} - 6 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} + 2 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, b^{2} x^{3} - a b - 3 \, \sqrt {\frac {1}{3}} {\left (a b x + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} - 3 \, \left (a b^{2}\right )^{\frac {2}{3}} x}{b x^{3} + a}\right ) + 2 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{4} b^{4} x^{7} + 2 \, a^{5} b^{3} x^{4} + a^{6} b^{2} x\right )}}, \frac {12 \, {\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{6} - 54 \, A a^{3} b^{2} + 21 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{3} - 12 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{3} - 7 \, A a b^{4}\right )} x^{7} + 2 \, {\left (B a^{3} b^{2} - 7 \, A a^{2} b^{3}\right )} x^{4} + {\left (B a^{4} b - 7 \, A a^{3} b^{2}\right )} x\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (-\frac {\sqrt {\frac {1}{3}} {\left (2 \, b x - \left (a b^{2}\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b}\right ) + 2 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b^{2} x^{2} - \left (a b^{2}\right )^{\frac {1}{3}} b x + \left (a b^{2}\right )^{\frac {2}{3}}\right ) - 4 \, {\left ({\left (B a b^{2} - 7 \, A b^{3}\right )} x^{7} + 2 \, {\left (B a^{2} b - 7 \, A a b^{2}\right )} x^{4} + {\left (B a^{3} - 7 \, A a^{2} b\right )} x\right )} \left (a b^{2}\right )^{\frac {2}{3}} \log \left (b x + \left (a b^{2}\right )^{\frac {1}{3}}\right )}{54 \, {\left (a^{4} b^{4} x^{7} + 2 \, a^{5} b^{3} x^{4} + a^{6} b^{2} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 204, normalized size = 0.90 \begin {gather*} \frac {2 \, \sqrt {3} {\left (B a - 7 \, 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 {1}{3}} a^{3}} - \frac {{\left (B a - 7 \, 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 {1}{3}} a^{3}} - \frac {2 \, {\left (B a \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, 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^{4}} - \frac {A}{a^{3} x} + \frac {4 \, B a b x^{5} - 10 \, A b^{2} x^{5} + 7 \, B a^{2} x^{2} - 13 \, A a b x^{2}}{18 \, {\left (b x^{3} + a\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 281, normalized size = 1.24 \begin {gather*} -\frac {5 A \,b^{2} x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {2 B b \,x^{5}}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {13 A b \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {7 B \,x^{2}}{18 \left (b \,x^{3}+a \right )^{2} a}-\frac {14 \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 {1}{3}} a^{3}}+\frac {14 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}-\frac {7 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}} a^{3}}+\frac {2 \sqrt {3}\, B \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}} a^{2} b}-\frac {2 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2} b}+\frac {B \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}} a^{2} b}-\frac {A}{a^{3} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.41, size = 199, normalized size = 0.88 \begin {gather*} \frac {4 \, {\left (B a b - 7 \, A b^{2}\right )} x^{6} + 7 \, {\left (B a^{2} - 7 \, A a b\right )} x^{3} - 18 \, A a^{2}}{18 \, {\left (a^{3} b^{2} x^{7} + 2 \, a^{4} b x^{4} + a^{5} x\right )}} + \frac {2 \, \sqrt {3} {\left (B a - 7 \, 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 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (B a - 7 \, 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 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {2 \, {\left (B a - 7 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 185, normalized size = 0.81 \begin {gather*} \frac {2\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (7\,A\,b-B\,a\right )}{27\,a^{10/3}\,b^{2/3}}-\frac {\frac {A}{a}+\frac {7\,x^3\,\left (7\,A\,b-B\,a\right )}{18\,a^2}+\frac {2\,b\,x^6\,\left (7\,A\,b-B\,a\right )}{9\,a^3}}{a^2\,x+2\,a\,b\,x^4+b^2\,x^7}+\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 (7\,A\,b-B\,a\right )}{27\,a^{10/3}\,b^{2/3}}-\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 (7\,A\,b-B\,a\right )}{27\,a^{10/3}\,b^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.90, size = 162, normalized size = 0.71 \begin {gather*} \frac {- 18 A a^{2} + x^{6} \left (- 28 A b^{2} + 4 B a b\right ) + x^{3} \left (- 49 A a b + 7 B a^{2}\right )}{18 a^{5} x + 36 a^{4} b x^{4} + 18 a^{3} b^{2} x^{7}} + \operatorname {RootSum} {\left (19683 t^{3} a^{10} b^{2} - 2744 A^{3} b^{3} + 1176 A^{2} B a b^{2} - 168 A B^{2} a^{2} b + 8 B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {729 t^{2} a^{7} b}{196 A^{2} b^{2} - 56 A B a b + 4 B^{2} a^{2}} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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