Optimal. Leaf size=70 \[ -\frac {A}{b^2 x}+\frac {(b B-A c) x}{2 b^2 \left (b+c x^2\right )}+\frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{5/2} \sqrt {c}} \]
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Rubi [A]
time = 0.05, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1598, 467, 464,
211} \begin {gather*} \frac {(b B-3 A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{5/2} \sqrt {c}}+\frac {x (b B-A c)}{2 b^2 \left (b+c x^2\right )}-\frac {A}{b^2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 464
Rule 467
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {A+B x^2}{x^2 \left (b+c x^2\right )^2} \, dx\\ &=\frac {(b B-A c) x}{2 b^2 \left (b+c x^2\right )}-\frac {1}{2} \int \frac {-\frac {2 A}{b}-\frac {(b B-A c) x^2}{b^2}}{x^2 \left (b+c x^2\right )} \, dx\\ &=-\frac {A}{b^2 x}+\frac {(b B-A c) x}{2 b^2 \left (b+c x^2\right )}+\frac {(b B-3 A c) \int \frac {1}{b+c x^2} \, dx}{2 b^2}\\ &=-\frac {A}{b^2 x}+\frac {(b B-A c) x}{2 b^2 \left (b+c x^2\right )}+\frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{5/2} \sqrt {c}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 70, normalized size = 1.00 \begin {gather*} -\frac {A}{b^2 x}+\frac {(b B-A c) x}{2 b^2 \left (b+c x^2\right )}+\frac {(b B-3 A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{2 b^{5/2} \sqrt {c}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 62, normalized size = 0.89
method | result | size |
default | \(-\frac {\frac {\left (\frac {A c}{2}-\frac {B b}{2}\right ) x}{c \,x^{2}+b}+\frac {\left (3 A c -B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \sqrt {b c}}}{b^{2}}-\frac {A}{b^{2} x}\) | \(62\) |
risch | \(\frac {-\frac {\left (3 A c -B b \right ) x^{2}}{2 b^{2}}-\frac {A}{b}}{x \left (c \,x^{2}+b \right )}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (b^{5} c \,\textit {\_Z}^{2}+9 A^{2} c^{2}-6 A B b c +B^{2} b^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (3 \textit {\_R}^{2} b^{5} c +18 A^{2} c^{2}-12 A B b c +2 B^{2} b^{2}\right ) x +\left (3 A \,b^{3} c -B \,b^{4}\right ) \textit {\_R} \right )\right )}{4}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 63, normalized size = 0.90 \begin {gather*} \frac {{\left (B b - 3 \, A c\right )} x^{2} - 2 \, A b}{2 \, {\left (b^{2} c x^{3} + b^{3} x\right )}} + \frac {{\left (B b - 3 \, A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.87, size = 210, normalized size = 3.00 \begin {gather*} \left [-\frac {4 \, A b^{2} c - 2 \, {\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{2} - {\left ({\left (B b c - 3 \, A c^{2}\right )} x^{3} + {\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {-b c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{4 \, {\left (b^{3} c^{2} x^{3} + b^{4} c x\right )}}, -\frac {2 \, A b^{2} c - {\left (B b^{2} c - 3 \, A b c^{2}\right )} x^{2} - {\left ({\left (B b c - 3 \, A c^{2}\right )} x^{3} + {\left (B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{2 \, {\left (b^{3} c^{2} x^{3} + b^{4} c x\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.25, size = 114, normalized size = 1.63 \begin {gather*} - \frac {\sqrt {- \frac {1}{b^{5} c}} \left (- 3 A c + B b\right ) \log {\left (- b^{3} \sqrt {- \frac {1}{b^{5} c}} + x \right )}}{4} + \frac {\sqrt {- \frac {1}{b^{5} c}} \left (- 3 A c + B b\right ) \log {\left (b^{3} \sqrt {- \frac {1}{b^{5} c}} + x \right )}}{4} + \frac {- 2 A b + x^{2} \left (- 3 A c + B b\right )}{2 b^{3} x + 2 b^{2} c x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.63, size = 62, normalized size = 0.89 \begin {gather*} \frac {{\left (B b - 3 \, A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{2 \, \sqrt {b c} b^{2}} + \frac {B b x^{2} - 3 \, A c x^{2} - 2 \, A b}{2 \, {\left (c x^{3} + b x\right )} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 63, normalized size = 0.90 \begin {gather*} -\frac {\frac {A}{b}+\frac {x^2\,\left (3\,A\,c-B\,b\right )}{2\,b^2}}{c\,x^3+b\,x}-\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (3\,A\,c-B\,b\right )}{2\,b^{5/2}\,\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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