Optimal. Leaf size=169 \[ \frac {(A b-a B) x}{3 a b \left (a+b x^3\right )}-\frac {(2 A b+a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{4/3}}+\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}-\frac {(2 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{4/3}} \]
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Rubi [A]
time = 0.06, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {393, 206, 31,
648, 631, 210, 642} \begin {gather*} -\frac {(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^{5/3} b^{4/3}}-\frac {(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^{5/3} b^{4/3}}+\frac {(a B+2 A b) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}+\frac {x (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 206
Rule 210
Rule 393
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {A+B x^3}{\left (a+b x^3\right )^2} \, dx &=\frac {(A b-a B) x}{3 a b \left (a+b x^3\right )}+\frac {(2 A b+a B) \int \frac {1}{a+b x^3} \, dx}{3 a b}\\ &=\frac {(A b-a B) x}{3 a b \left (a+b x^3\right )}+\frac {(2 A b+a B) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{9 a^{5/3} b}+\frac {(2 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}{9 a^{5/3} b}\\ &=\frac {(A b-a B) x}{3 a b \left (a+b x^3\right )}+\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}-\frac {(2 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}{18 a^{5/3} b^{4/3}}+\frac {(2 A b+a B) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 a^{4/3} b}\\ &=\frac {(A b-a B) x}{3 a b \left (a+b x^3\right )}+\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}-\frac {(2 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{4/3}}+\frac {(2 A b+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^{5/3} b^{4/3}}\\ &=\frac {(A b-a B) x}{3 a b \left (a+b x^3\right )}-\frac {(2 A b+a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} b^{4/3}}+\frac {(2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} b^{4/3}}-\frac {(2 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 145, normalized size = 0.86 \begin {gather*} \frac {-\frac {6 a^{2/3} \sqrt [3]{b} (-A b+a B) x}{a+b x^3}-2 \sqrt {3} (2 A b+a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 (2 A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-(2 A b+a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{5/3} b^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 134, normalized size = 0.79
method | result | size |
risch | \(\frac {\left (A b -B a \right ) x}{3 a b \left (b \,x^{3}+a \right )}+\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (2 A b +B a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{9 a \,b^{2}}\) | \(65\) |
default | \(\frac {\left (A b -B a \right ) x}{3 a b \left (b \,x^{3}+a \right )}+\frac {\left (2 A b +B a \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a b}\) | \(134\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 158, normalized size = 0.93 \begin {gather*} -\frac {{\left (B a - A b\right )} x}{3 \, {\left (a b^{2} x^{3} + a^{2} b\right )}} + \frac {\sqrt {3} {\left (B a + 2 \, 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 b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (B a + 2 \, 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 b^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (B a + 2 \, A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a b^{2} \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 = 1.87, size = 537, normalized size = 3.18 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (B a^{3} b + 2 \, A a^{2} b^{2} + {\left (B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3}\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 (B a b + 2 \, A b^{2}\right )} x^{3} + B a^{2} + 2 \, A a b\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 (B a b + 2 \, A b^{2}\right )} x^{3} + B a^{2} + 2 \, A a b\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) - 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x}{18 \, {\left (a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}}, \frac {6 \, \sqrt {\frac {1}{3}} {\left (B a^{3} b + 2 \, A a^{2} b^{2} + {\left (B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{3}\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 (B a b + 2 \, A b^{2}\right )} x^{3} + B a^{2} + 2 \, A a b\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 (B a b + 2 \, A b^{2}\right )} x^{3} + B a^{2} + 2 \, A a b\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) - 6 \, {\left (B a^{3} b - A a^{2} b^{2}\right )} x}{18 \, {\left (a^{3} b^{3} x^{3} + a^{4} b^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.31, size = 97, normalized size = 0.57 \begin {gather*} \frac {x \left (A b - B a\right )}{3 a^{2} b + 3 a b^{2} x^{3}} + \operatorname {RootSum} {\left (729 t^{3} a^{5} b^{4} - 8 A^{3} b^{3} - 12 A^{2} B a b^{2} - 6 A B^{2} a^{2} b - B^{3} a^{3}, \left ( t \mapsto t \log {\left (\frac {9 t a^{2} b}{2 A b + 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 = 0.64, size = 160, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {3} {\left (B a + 2 \, 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 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (B a + 2 \, 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 \, \left (-a b^{2}\right )^{\frac {2}{3}} a} - \frac {{\left (B a + 2 \, 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^{2} b} - \frac {B a x - A b x}{3 \, {\left (b x^{3} + a\right )} a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.55, size = 143, normalized size = 0.85 \begin {gather*} \frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,\left (2\,A\,b+B\,a\right )}{9\,a^{5/3}\,b^{4/3}}-\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 (2\,A\,b+B\,a\right )}{9\,a^{5/3}\,b^{4/3}}+\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 (2\,A\,b+B\,a\right )}{9\,a^{5/3}\,b^{4/3}}+\frac {x\,\left (A\,b-B\,a\right )}{3\,a\,b\,\left (b\,x^3+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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