Optimal. Leaf size=196 \[ \frac {-5 A b+2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}+\frac {(5 A b-2 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}-\frac {(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac {(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}} \]
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Rubi [A]
time = 0.07, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {468, 331, 206,
31, 648, 631, 210, 642} \begin {gather*} \frac {(5 A b-2 a B) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}+\frac {(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac {(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}-\frac {5 A b-2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 331
Rule 468
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^3 \left (a+b x^3\right )^2} \, dx &=\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}+\frac {(5 A b-2 a B) \int \frac {1}{x^3 \left (a+b x^3\right )} \, dx}{3 a b}\\ &=-\frac {5 A b-2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac {(5 A b-2 a B) \int \frac {1}{a+b x^3} \, dx}{3 a^2}\\ &=-\frac {5 A b-2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac {(5 A b-2 a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{8/3}}-\frac {(5 A b-2 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}{9 a^{8/3}}\\ &=-\frac {5 A b-2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac {(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}-\frac {(5 A b-2 a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{7/3}}+\frac {(5 A b-2 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}{18 a^{8/3} \sqrt [3]{b}}\\ &=-\frac {5 A b-2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}-\frac {(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac {(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}-\frac {(5 A b-2 a B) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{3 a^{8/3} \sqrt [3]{b}}\\ &=-\frac {5 A b-2 a B}{6 a^2 b x^2}+\frac {A b-a B}{3 a b x^2 \left (a+b x^3\right )}+\frac {(5 A b-2 a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3} \sqrt [3]{b}}-\frac {(5 A b-2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3} \sqrt [3]{b}}+\frac {(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3} \sqrt [3]{b}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 163, normalized size = 0.83 \begin {gather*} \frac {-\frac {9 a^{2/3} A}{x^2}+\frac {6 a^{2/3} (-A b+a B) x}{a+b x^3}+\frac {2 \sqrt {3} (5 A b-2 a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}+\frac {2 (-5 A b+2 a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {(5 A b-2 a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}}{18 a^{8/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 138, normalized size = 0.70
method | result | size |
default | \(-\frac {\frac {\left (\frac {A b}{3}-\frac {B a}{3}\right ) x}{b \,x^{3}+a}+\frac {\left (5 A b -2 B a \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}}{a^{2}}-\frac {A}{2 a^{2} x^{2}}\) | \(138\) |
risch | \(\frac {-\frac {\left (5 A b -2 B a \right ) x^{3}}{6 a^{2}}-\frac {A}{2 a}}{x^{2} \left (b \,x^{3}+a \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{8} b \,\textit {\_Z}^{3}+125 A^{3} b^{3}-150 A^{2} B a \,b^{2}+60 A \,B^{2} a^{2} b -8 B^{3} a^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8} b -375 A^{3} b^{3}+450 A^{2} B a \,b^{2}-180 A \,B^{2} a^{2} b +24 B^{3} a^{3}\right ) x +\left (-25 A^{2} b^{2} a^{3}+20 A B b \,a^{4}-4 B^{2} a^{5}\right ) \textit {\_R} \right )\right )}{9}\) | \(172\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.53, size = 172, normalized size = 0.88 \begin {gather*} \frac {{\left (2 \, B a - 5 \, A b\right )} x^{3} - 3 \, A a}{6 \, {\left (a^{2} b x^{5} + a^{3} x^{2}\right )}} + \frac {\sqrt {3} {\left (2 \, B a - 5 \, 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^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (2 \, B a - 5 \, 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^{2} b \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (2 \, B a - 5 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} 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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Fricas [A]
time = 3.28, size = 618, normalized size = 3.15 \begin {gather*} \left [-\frac {9 \, A a^{3} b - 3 \, {\left (2 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{5} + {\left (2 \, B a^{3} b - 5 \, A a^{2} 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 ) + {\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (2 \, B a^{2} - 5 \, A a 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 ) - 2 \, {\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (2 \, B a^{2} - 5 \, A a 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 )}{18 \, {\left (a^{4} b^{2} x^{5} + a^{5} b x^{2}\right )}}, -\frac {9 \, A a^{3} b - 3 \, {\left (2 \, B a^{3} b - 5 \, A a^{2} b^{2}\right )} x^{3} - 6 \, \sqrt {\frac {1}{3}} {\left ({\left (2 \, B a^{2} b^{2} - 5 \, A a b^{3}\right )} x^{5} + {\left (2 \, B a^{3} b - 5 \, A a^{2} 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 ) + {\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (2 \, B a^{2} - 5 \, A a 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 ) - 2 \, {\left ({\left (2 \, B a b - 5 \, A b^{2}\right )} x^{5} + {\left (2 \, B a^{2} - 5 \, A a 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 )}{18 \, {\left (a^{4} b^{2} x^{5} + a^{5} b x^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.37, size = 109, normalized size = 0.56 \begin {gather*} \frac {- 3 A a + x^{3} \left (- 5 A b + 2 B a\right )}{6 a^{3} x^{2} + 6 a^{2} b x^{5}} + \operatorname {RootSum} {\left (729 t^{3} a^{8} b + 125 A^{3} b^{3} - 150 A^{2} B a b^{2} + 60 A B^{2} a^{2} b - 8 B^{3} a^{3}, \left ( t \mapsto t \log {\left (\frac {9 t a^{3}}{- 5 A b + 2 B a} + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.80, size = 188, normalized size = 0.96 \begin {gather*} -\frac {{\left (2 \, B a - 5 \, 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^{3}} + \frac {\sqrt {3} {\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 5 \, \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 \, a^{3} b} + \frac {B a x - A b x}{3 \, {\left (b x^{3} + a\right )} a^{2}} + \frac {{\left (2 \, \left (-a b^{2}\right )^{\frac {1}{3}} B a - 5 \, \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 \, a^{3} b} - \frac {A}{2 \, a^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.57, size = 159, normalized size = 0.81 \begin {gather*} -\frac {\frac {A}{2\,a}+\frac {x^3\,\left (5\,A\,b-2\,B\,a\right )}{6\,a^2}}{b\,x^5+a\,x^2}-\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (5\,A\,b-2\,B\,a\right )}{9\,a^{8/3}\,b^{1/3}}+\frac {\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 (5\,A\,b-2\,B\,a\right )}{9\,a^{8/3}\,b^{1/3}}-\frac {\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 (5\,A\,b-2\,B\,a\right )}{9\,a^{8/3}\,b^{1/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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