Optimal. Leaf size=215 \[ \frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {4 a B-7 A b}{12 a^2 b x^4}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 215, 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, 325, 292, 31, 634, 617, 204, 628} \begin {gather*} \frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 292
Rule 325
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}+\frac {(7 A b-4 a B) \int \frac {1}{x^5 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {(7 A b-4 a B) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{3 a^2}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}+\frac {(b (7 A b-4 a B)) \int \frac {x}{a+b x^3} \, dx}{3 a^3}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{10/3}}+\frac {\left (b^{2/3} (7 A b-4 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}{9 a^{10/3}}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac {\left (\sqrt [3]{b} (7 A b-4 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 a^{10/3}}+\frac {\left (b^{2/3} (7 A b-4 a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^3}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}+\frac {\left (\sqrt [3]{b} (7 A b-4 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 a^{10/3}}\\ &=-\frac {7 A b-4 a B}{12 a^2 b x^4}+\frac {7 A b-4 a B}{3 a^3 x}+\frac {A b-a B}{3 a b x^4 \left (a+b x^3\right )}-\frac {\sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{10/3}}-\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{10/3}}+\frac {\sqrt [3]{b} (7 A b-4 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{10/3}}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 185, normalized size = 0.86 \begin {gather*} \frac {2 \sqrt [3]{b} (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 {9 a^{4/3} A}{x^4}-\frac {12 \sqrt [3]{a} b x^2 (a B-A b)}{a+b x^3}-\frac {36 \sqrt [3]{a} (a B-2 A b)}{x}+4 \sqrt [3]{b} (4 a B-7 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-4 \sqrt {3} \sqrt [3]{b} (7 A b-4 a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{36 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^5 \left (a+b x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.60, size = 259, normalized size = 1.20 \begin {gather*} -\frac {12 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} x^{6} + 9 \, {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{3} + 9 \, A a^{2} + 4 \, \sqrt {3} {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) + 4 \, {\left ({\left (4 \, B a b - 7 \, A b^{2}\right )} x^{7} + {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{4}\right )} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right )}{36 \, {\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 231, normalized size = 1.07 \begin {gather*} \frac {{\left (4 \, B a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 7 \, A b^{2} \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 )}{9 \, a^{4}} + \frac {\sqrt {3} {\left (4 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 7 \, \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 )}{9 \, a^{4} b} - \frac {B a b x^{2} - A b^{2} x^{2}}{3 \, {\left (b x^{3} + a\right )} a^{3}} - \frac {{\left (4 \, \left (-a b^{2}\right )^{\frac {2}{3}} B a - 7 \, \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 )}{18 \, a^{4} b} - \frac {4 \, B a x^{3} - 8 \, A b x^{3} + A a}{4 \, a^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 257, normalized size = 1.20 \begin {gather*} \frac {A \,b^{2} x^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {B b \,x^{2}}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {7 \sqrt {3}\, A b \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 {1}{3}} a^{3}}-\frac {7 A b \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{3}}+\frac {7 A b \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 {1}{3}} a^{3}}-\frac {4 \sqrt {3}\, B \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 {1}{3}} a^{2}}+\frac {4 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \left (\frac {a}{b}\right )^{\frac {1}{3}} a^{2}}-\frac {2 B \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 {1}{3}} a^{2}}+\frac {2 A b}{a^{3} x}-\frac {B}{a^{2} x}-\frac {A}{4 a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.42, size = 186, normalized size = 0.87 \begin {gather*} -\frac {4 \, {\left (4 \, B a b - 7 \, A b^{2}\right )} x^{6} + 3 \, {\left (4 \, B a^{2} - 7 \, A a b\right )} x^{3} + 3 \, A a^{2}}{12 \, {\left (a^{3} b x^{7} + a^{4} x^{4}\right )}} - \frac {\sqrt {3} {\left (4 \, 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 )}{9 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (4 \, 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 )}{18 \, a^{3} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (4 \, B a - 7 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{3} \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 = 2.62, size = 209, normalized size = 0.97 \begin {gather*} \frac {\frac {x^3\,\left (7\,A\,b-4\,B\,a\right )}{4\,a^2}-\frac {A}{4\,a}+\frac {b\,x^6\,\left (7\,A\,b-4\,B\,a\right )}{3\,a^3}}{b\,x^7+a\,x^4}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}+b^3\,x\right )\,\left (7\,A\,b-4\,B\,a\right )}{9\,a^{10/3}}+\frac {{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}-2\,b^3\,x+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-4\,B\,a\right )}{9\,a^{10/3}}-\frac {{\left (-b\right )}^{1/3}\,\ln \left (2\,b^3\,x-a^{1/3}\,{\left (-b\right )}^{8/3}+\sqrt {3}\,a^{1/3}\,{\left (-b\right )}^{8/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (7\,A\,b-4\,B\,a\right )}{9\,a^{10/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.32, size = 153, normalized size = 0.71 \begin {gather*} \operatorname {RootSum} {\left (729 t^{3} a^{10} + 343 A^{3} b^{4} - 588 A^{2} B a b^{3} + 336 A B^{2} a^{2} b^{2} - 64 B^{3} a^{3} b, \left (t \mapsto t \log {\left (\frac {81 t^{2} a^{7}}{49 A^{2} b^{3} - 56 A B a b^{2} + 16 B^{2} a^{2} b} + x \right )} \right )\right )} + \frac {- 3 A a^{2} + x^{6} \left (28 A b^{2} - 16 B a b\right ) + x^{3} \left (21 A a b - 12 B a^{2}\right )}{12 a^{4} x^{4} + 12 a^{3} b x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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