Optimal. Leaf size=227 \[ \frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac {5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {5 (4 A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{9 \sqrt {3} a^{11/3} \sqrt [3]{b}}-\frac {5 (4 A b-a B)}{18 a^3 b x^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.13, 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, 200, 31, 634, 617, 204, 628} \begin {gather*} \frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac {5 (4 A b-a B)}{18 a^3 b x^2}-\frac {5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {5 (4 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^{11/3} \sqrt [3]{b}}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 290
Rule 325
Rule 457
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^3} \, dx &=\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {(8 A b-2 a B) \int \frac {1}{x^3 \left (a+b x^3\right )^2} \, dx}{6 a b}\\ &=\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac {(5 (4 A b-a B)) \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx}{9 a^2 b}\\ &=-\frac {5 (4 A b-a B)}{18 a^3 b x^2}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac {(5 (4 A b-a B)) \int \frac {1}{a+b x^3} \, dx}{9 a^3}\\ &=-\frac {5 (4 A b-a B)}{18 a^3 b x^2}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac {(5 (4 A b-a B)) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{27 a^{11/3}}-\frac {(5 (4 A b-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^{11/3}}\\ &=-\frac {5 (4 A b-a B)}{18 a^3 b x^2}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac {5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}-\frac {(5 (4 A b-a B)) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{18 a^{10/3}}+\frac {(5 (4 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}{54 a^{11/3} \sqrt [3]{b}}\\ &=-\frac {5 (4 A b-a B)}{18 a^3 b x^2}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}-\frac {5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}-\frac {(5 (4 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^{11/3} \sqrt [3]{b}}\\ &=-\frac {5 (4 A b-a B)}{18 a^3 b x^2}+\frac {A b-a B}{6 a b x^2 \left (a+b x^3\right )^2}+\frac {4 A b-a B}{9 a^2 b x^2 \left (a+b x^3\right )}+\frac {5 (4 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^{11/3} \sqrt [3]{b}}-\frac {5 (4 A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{27 a^{11/3} \sqrt [3]{b}}+\frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{54 a^{11/3} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 189, normalized size = 0.83 \begin {gather*} \frac {\frac {5 (4 A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}+\frac {9 a^{5/3} x (a B-A b)}{\left (a+b x^3\right )^2}+\frac {3 a^{2/3} x (5 a B-11 A b)}{a+b x^3}-\frac {27 a^{2/3} A}{x^2}+\frac {10 (a B-4 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {10 \sqrt {3} (4 A b-a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{54 a^{11/3}} \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^3 \left (a+b x^3\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.52, size = 812, normalized size = 3.58 \begin {gather*} \left [\frac {15 \, {\left (B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{6} - 27 \, A a^{4} b + 24 \, {\left (B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{3} - 15 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{8} + 2 \, {\left (B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{5} + {\left (B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}\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 ) - 5 \, {\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \, {\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} + {\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\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 ) + 10 \, {\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \, {\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} + {\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{5} b^{3} x^{8} + 2 \, a^{6} b^{2} x^{5} + a^{7} b x^{2}\right )}}, \frac {15 \, {\left (B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{6} - 27 \, A a^{4} b + 24 \, {\left (B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{3} + 30 \, \sqrt {\frac {1}{3}} {\left ({\left (B a^{2} b^{3} - 4 \, A a b^{4}\right )} x^{8} + 2 \, {\left (B a^{3} b^{2} - 4 \, A a^{2} b^{3}\right )} x^{5} + {\left (B a^{4} b - 4 \, A a^{3} b^{2}\right )} x^{2}\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 ) - 5 \, {\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \, {\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} + {\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\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 ) + 10 \, {\left ({\left (B a b^{2} - 4 \, A b^{3}\right )} x^{8} + 2 \, {\left (B a^{2} b - 4 \, A a b^{2}\right )} x^{5} + {\left (B a^{3} - 4 \, A a^{2} b\right )} x^{2}\right )} \left (-a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (-a^{2} b\right )^{\frac {2}{3}}\right )}{54 \, {\left (a^{5} b^{3} x^{8} + 2 \, a^{6} b^{2} x^{5} + a^{7} b x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 209, normalized size = 0.92 \begin {gather*} -\frac {5 \, {\left (B a - 4 \, 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^{4}} + \frac {5 \, \sqrt {3} {\left (\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 )}{27 \, a^{4} b} + \frac {5 \, {\left (\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 )}{54 \, a^{4} b} + \frac {5 \, B a b x^{6} - 20 \, A b^{2} x^{6} + 8 \, B a^{2} x^{3} - 32 \, A a b x^{3} - 9 \, A a^{2}}{18 \, {\left (b x^{4} + a x\right )}^{2} a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 277, normalized size = 1.22 \begin {gather*} -\frac {11 A \,b^{2} x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{3}}+\frac {5 B b \,x^{4}}{18 \left (b \,x^{3}+a \right )^{2} a^{2}}-\frac {7 A b x}{9 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {4 B x}{9 \left (b \,x^{3}+a \right )^{2} a}-\frac {20 \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}} a^{3}}-\frac {20 A \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{3}}+\frac {10 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}} a^{3}}+\frac {5 \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 {2}{3}} a^{2} b}+\frac {5 B \ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{27 \left (\frac {a}{b}\right )^{\frac {2}{3}} a^{2} b}-\frac {5 B \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}} a^{2} b}-\frac {A}{2 a^{3} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 201, normalized size = 0.89 \begin {gather*} \frac {5 \, {\left (B a b - 4 \, A b^{2}\right )} x^{6} + 8 \, {\left (B a^{2} - 4 \, A a b\right )} x^{3} - 9 \, A a^{2}}{18 \, {\left (a^{3} b^{2} x^{8} + 2 \, a^{4} b x^{5} + a^{5} x^{2}\right )}} + \frac {5 \, \sqrt {3} {\left (B a - 4 \, 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 {2}{3}}} - \frac {5 \, {\left (B a - 4 \, 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 \, a^{3} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {5 \, {\left (B a - 4 \, 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 {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.58, size = 188, normalized size = 0.83 \begin {gather*} -\frac {\frac {A}{2\,a}+\frac {4\,x^3\,\left (4\,A\,b-B\,a\right )}{9\,a^2}+\frac {5\,b\,x^6\,\left (4\,A\,b-B\,a\right )}{18\,a^3}}{a^2\,x^2+2\,a\,b\,x^5+b^2\,x^8}-\frac {5\,\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (4\,A\,b-B\,a\right )}{27\,a^{11/3}\,b^{1/3}}+\frac {5\,\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 (4\,A\,b-B\,a\right )}{27\,a^{11/3}\,b^{1/3}}-\frac {5\,\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 (4\,A\,b-B\,a\right )}{27\,a^{11/3}\,b^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.73, size = 143, normalized size = 0.63 \begin {gather*} \frac {- 9 A a^{2} + x^{6} \left (- 20 A b^{2} + 5 B a b\right ) + x^{3} \left (- 32 A a b + 8 B a^{2}\right )}{18 a^{5} x^{2} + 36 a^{4} b x^{5} + 18 a^{3} b^{2} x^{8}} + \operatorname {RootSum} {\left (19683 t^{3} a^{11} b + 8000 A^{3} b^{3} - 6000 A^{2} B a b^{2} + 1500 A B^{2} a^{2} b - 125 B^{3} a^{3}, \left (t \mapsto t \log {\left (\frac {27 t a^{4}}{- 20 A b + 5 B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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