Optimal. Leaf size=165 \[ -\frac {A}{4 a x^4}+\frac {A b-a B}{a^2 x}-\frac {\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}-\frac {\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}+\frac {\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}} \]
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Rubi [A]
time = 0.07, antiderivative size = 165, 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 = {464, 331, 298,
31, 648, 631, 210, 642} \begin {gather*} -\frac {\sqrt [3]{b} (A b-a B) \text {ArcTan}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}+\frac {\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}-\frac {\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}+\frac {A b-a B}{a^2 x}-\frac {A}{4 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 298
Rule 331
Rule 464
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^5 \left (a+b x^3\right )} \, dx &=-\frac {A}{4 a x^4}-\frac {(4 A b-4 a B) \int \frac {1}{x^2 \left (a+b x^3\right )} \, dx}{4 a}\\ &=-\frac {A}{4 a x^4}+\frac {A b-a B}{a^2 x}+\frac {(b (A b-a B)) \int \frac {x}{a+b x^3} \, dx}{a^2}\\ &=-\frac {A}{4 a x^4}+\frac {A b-a B}{a^2 x}-\frac {\left (b^{2/3} (A b-a B)\right ) \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{7/3}}+\frac {\left (b^{2/3} (A b-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}{3 a^{7/3}}\\ &=-\frac {A}{4 a x^4}+\frac {A b-a B}{a^2 x}-\frac {\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}+\frac {\left (\sqrt [3]{b} (A b-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}{6 a^{7/3}}+\frac {\left (b^{2/3} (A b-a B)\right ) \int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 a^2}\\ &=-\frac {A}{4 a x^4}+\frac {A b-a B}{a^2 x}-\frac {\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}+\frac {\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}+\frac {\left (\sqrt [3]{b} (A b-a B)\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{7/3}}\\ &=-\frac {A}{4 a x^4}+\frac {A b-a B}{a^2 x}-\frac {\sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{7/3}}-\frac {\sqrt [3]{b} (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{7/3}}+\frac {\sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{7/3}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 154, normalized size = 0.93 \begin {gather*} \frac {-\frac {3 a^{4/3} A}{x^4}+\frac {12 \sqrt [3]{a} (A b-a B)}{x}-4 \sqrt {3} \sqrt [3]{b} (A b-a B) \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+4 \sqrt [3]{b} (-A b+a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+2 \sqrt [3]{b} (A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{12 a^{7/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 130, normalized size = 0.79
method | result | size |
default | \(\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b \left (A b -B a \right )}{a^{2}}-\frac {A}{4 a \,x^{4}}-\frac {-A b +B a}{a^{2} x}\) | \(130\) |
risch | \(\frac {\frac {\left (A b -B a \right ) x^{3}}{a^{2}}-\frac {A}{4 a}}{x^{4}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{7} \textit {\_Z}^{3}+A^{3} b^{4}-3 A^{2} B a \,b^{3}+3 A \,B^{2} a^{2} b^{2}-B^{3} a^{3} b \right )}{\sum }\textit {\_R} \ln \left (\left (-4 a^{7} \textit {\_R}^{3}-3 A^{3} b^{4}+9 A^{2} B a \,b^{3}-9 A \,B^{2} a^{2} b^{2}+3 B^{3} a^{3} b \right ) x +\left (A \,a^{5} b -B \,a^{6}\right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 147, normalized size = 0.89 \begin {gather*} -\frac {\sqrt {3} {\left (B a - 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 )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {{\left (B a - 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 )}{6 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (B a - A b\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, a^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {4 \, {\left (B a - A b\right )} x^{3} + A a}{4 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.93, size = 158, normalized size = 0.96 \begin {gather*} -\frac {4 \, \sqrt {3} {\left (B a - A b\right )} x^{4} \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 (B a - A b\right )} x^{4} \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 (B a - A b\right )} x^{4} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 12 \, {\left (B a - A b\right )} x^{3} + 3 \, A a}{12 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.69, size = 112, normalized size = 0.68 \begin {gather*} \operatorname {RootSum} {\left (27 t^{3} a^{7} + A^{3} b^{4} - 3 A^{2} B a b^{3} + 3 A B^{2} a^{2} b^{2} - B^{3} a^{3} b, \left ( t \mapsto t \log {\left (\frac {9 t^{2} a^{5}}{A^{2} b^{3} - 2 A B a b^{2} + B^{2} a^{2} b} + x \right )} \right )\right )} + \frac {- A a + x^{3} \cdot \left (4 A b - 4 B a\right )}{4 a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.86, size = 197, normalized size = 1.19 \begin {gather*} \frac {{\left (B a b \left (-\frac {a}{b}\right )^{\frac {1}{3}} - 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 )}{3 \, a^{3}} + \frac {\sqrt {3} {\left (\left (-a b^{2}\right )^{\frac {2}{3}} B a - \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 )}{3 \, a^{3} b} - \frac {{\left (\left (-a b^{2}\right )^{\frac {2}{3}} B a - \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 )}{6 \, a^{3} b} - \frac {4 \, B a x^{3} - 4 \, A b x^{3} + A a}{4 \, a^{2} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.59, size = 178, normalized size = 1.08 \begin {gather*} \frac {{\left (-b\right )}^{1/3}\,\ln \left (a^{1/3}\,{\left (-b\right )}^{8/3}+b^3\,x\right )\,\left (A\,b-B\,a\right )}{3\,a^{7/3}}-\frac {\frac {A}{4\,a}-\frac {x^3\,\left (A\,b-B\,a\right )}{a^2}}{x^4}+\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 (A\,b-B\,a\right )}{3\,a^{7/3}}-\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 (A\,b-B\,a\right )}{3\,a^{7/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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