Optimal. Leaf size=57 \[ -\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1607, 296, 331,
211} \begin {gather*} -\frac {3 \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 296
Rule 331
Rule 1607
Rubi steps
\begin {align*} \int \frac {1}{\left (a x+b x^3\right )^2} \, dx &=\int \frac {1}{x^2 \left (a+b x^2\right )^2} \, dx\\ &=\frac {1}{2 a x \left (a+b x^2\right )}+\frac {3 \int \frac {1}{x^2 \left (a+b x^2\right )} \, dx}{2 a}\\ &=-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )}-\frac {(3 b) \int \frac {1}{a+b x^2} \, dx}{2 a^2}\\ &=-\frac {3}{2 a^2 x}+\frac {1}{2 a x \left (a+b x^2\right )}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 54, normalized size = 0.95 \begin {gather*} -\frac {1}{a^2 x}-\frac {b x}{2 a^2 \left (a+b x^2\right )}-\frac {3 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{2 a^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 45, normalized size = 0.79
method | result | size |
default | \(-\frac {b \left (\frac {x}{2 b \,x^{2}+2 a}+\frac {3 \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {1}{a^{2} x}\) | \(45\) |
risch | \(\frac {-\frac {3 b \,x^{2}}{2 a^{2}}-\frac {1}{a}}{x \left (b \,x^{2}+a \right )}+\frac {3 \left (\munderset {\textit {\_R} =\RootOf \left (a^{5} \textit {\_Z}^{2}+b \right )}{\sum }\textit {\_R} \ln \left (\left (3 a^{5} \textit {\_R}^{2}+2 b \right ) x +a^{3} \textit {\_R} \right )\right )}{4}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 49, normalized size = 0.86 \begin {gather*} -\frac {3 \, b x^{2} + 2 \, a}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}} - \frac {3 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.82, size = 136, normalized size = 2.39 \begin {gather*} \left [-\frac {6 \, b x^{2} - 3 \, {\left (b x^{3} + a x\right )} \sqrt {-\frac {b}{a}} \log \left (\frac {b x^{2} - 2 \, a x \sqrt {-\frac {b}{a}} - a}{b x^{2} + a}\right ) + 4 \, a}{4 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}, -\frac {3 \, b x^{2} + 3 \, {\left (b x^{3} + a x\right )} \sqrt {\frac {b}{a}} \arctan \left (x \sqrt {\frac {b}{a}}\right ) + 2 \, a}{2 \, {\left (a^{2} b x^{3} + a^{3} x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 92, normalized size = 1.61 \begin {gather*} \frac {3 \sqrt {- \frac {b}{a^{5}}} \log {\left (- \frac {a^{3} \sqrt {- \frac {b}{a^{5}}}}{b} + x \right )}}{4} - \frac {3 \sqrt {- \frac {b}{a^{5}}} \log {\left (\frac {a^{3} \sqrt {- \frac {b}{a^{5}}}}{b} + x \right )}}{4} + \frac {- 2 a - 3 b x^{2}}{2 a^{3} x + 2 a^{2} b x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 47, normalized size = 0.82 \begin {gather*} -\frac {3 \, b \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} - \frac {3 \, b x^{2} + 2 \, a}{2 \, {\left (b x^{3} + a x\right )} a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.98, size = 44, normalized size = 0.77 \begin {gather*} -\frac {\frac {1}{a}+\frac {3\,b\,x^2}{2\,a^2}}{b\,x^3+a\,x}-\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,x}{\sqrt {a}}\right )}{2\,a^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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