Optimal. Leaf size=61 \[ -\frac {A}{3 b x^3}-\frac {b B-A c}{b^2 x}-\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 61, 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, 464, 331,
211} \begin {gather*} -\frac {\sqrt {c} (b B-A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}}-\frac {b B-A c}{b^2 x}-\frac {A}{3 b x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 331
Rule 464
Rule 1598
Rubi steps
\begin {align*} \int \frac {A+B x^2}{x^2 \left (b x^2+c x^4\right )} \, dx &=\int \frac {A+B x^2}{x^4 \left (b+c x^2\right )} \, dx\\ &=-\frac {A}{3 b x^3}-\frac {(-3 b B+3 A c) \int \frac {1}{x^2 \left (b+c x^2\right )} \, dx}{3 b}\\ &=-\frac {A}{3 b x^3}-\frac {b B-A c}{b^2 x}-\frac {(c (b B-A c)) \int \frac {1}{b+c x^2} \, dx}{b^2}\\ &=-\frac {A}{3 b x^3}-\frac {b B-A c}{b^2 x}-\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 60, normalized size = 0.98 \begin {gather*} -\frac {A}{3 b x^3}+\frac {-b B+A c}{b^2 x}-\frac {\sqrt {c} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 54, normalized size = 0.89
method | result | size |
default | \(\frac {c \left (A c -B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{b^{2} \sqrt {b c}}-\frac {A}{3 b \,x^{3}}-\frac {-A c +B b}{b^{2} x}\) | \(54\) |
risch | \(\frac {\frac {\left (A c -B b \right ) x^{2}}{b^{2}}-\frac {A}{3 b}}{x^{3}}+\frac {\sqrt {-b c}\, \ln \left (-c x -\sqrt {-b c}\right ) A c}{2 b^{3}}-\frac {\sqrt {-b c}\, \ln \left (-c x -\sqrt {-b c}\right ) B}{2 b^{2}}-\frac {\sqrt {-b c}\, \ln \left (-c x +\sqrt {-b c}\right ) A c}{2 b^{3}}+\frac {\sqrt {-b c}\, \ln \left (-c x +\sqrt {-b c}\right ) B}{2 b^{2}}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 56, normalized size = 0.92 \begin {gather*} -\frac {{\left (B b c - A c^{2}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} - \frac {3 \, {\left (B b - A c\right )} x^{2} + A b}{3 \, b^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.77, size = 135, normalized size = 2.21 \begin {gather*} \left [-\frac {3 \, {\left (B b - A c\right )} x^{3} \sqrt {-\frac {c}{b}} \log \left (\frac {c x^{2} + 2 \, b x \sqrt {-\frac {c}{b}} - b}{c x^{2} + b}\right ) + 6 \, {\left (B b - A c\right )} x^{2} + 2 \, A b}{6 \, b^{2} x^{3}}, -\frac {3 \, {\left (B b - A c\right )} x^{3} \sqrt {\frac {c}{b}} \arctan \left (x \sqrt {\frac {c}{b}}\right ) + 3 \, {\left (B b - A c\right )} x^{2} + A b}{3 \, b^{2} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs.
\(2 (49) = 98\).
time = 0.20, size = 129, normalized size = 2.11 \begin {gather*} \frac {\sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right ) \log {\left (- \frac {b^{3} \sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right )}{- A c^{2} + B b c} + x \right )}}{2} - \frac {\sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right ) \log {\left (\frac {b^{3} \sqrt {- \frac {c}{b^{5}}} \left (- A c + B b\right )}{- A c^{2} + B b c} + x \right )}}{2} + \frac {- A b + x^{2} \cdot \left (3 A c - 3 B b\right )}{3 b^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.69, size = 57, normalized size = 0.93 \begin {gather*} -\frac {{\left (B b c - A c^{2}\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} b^{2}} - \frac {3 \, B b x^{2} - 3 \, A c x^{2} + A b}{3 \, b^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 53, normalized size = 0.87 \begin {gather*} \frac {\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{b^{5/2}}-\frac {\frac {A}{3\,b}-\frac {x^2\,\left (A\,c-B\,b\right )}{b^2}}{x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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