Optimal. Leaf size=40 \[ \frac {B x}{c}-\frac {(b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 396, 211}
\begin {gather*} \frac {B x}{c}-\frac {(b B-A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 396
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {A+B x^2}{b+c x^2} \, dx\\ &=\frac {B x}{c}-\frac {(b B-A c) \int \frac {1}{b+c x^2} \, dx}{c}\\ &=\frac {B x}{c}-\frac {(b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 40, normalized size = 1.00 \begin {gather*} \frac {B x}{c}-\frac {(b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{\sqrt {b} c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 34, normalized size = 0.85
method | result | size |
default | \(\frac {B x}{c}+\frac {\left (A c -B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{c \sqrt {b c}}\) | \(34\) |
risch | \(\frac {B x}{c}-\frac {\ln \left (c x +\sqrt {-b c}\right ) A}{2 \sqrt {-b c}}+\frac {\ln \left (c x +\sqrt {-b c}\right ) B b}{2 c \sqrt {-b c}}+\frac {\ln \left (-c x +\sqrt {-b c}\right ) A}{2 \sqrt {-b c}}-\frac {\ln \left (-c x +\sqrt {-b c}\right ) B b}{2 c \sqrt {-b c}}\) | \(98\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 34, normalized size = 0.85 \begin {gather*} \frac {B x}{c} - \frac {{\left (B b - A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.84, size = 99, normalized size = 2.48 \begin {gather*} \left [\frac {2 \, B b c x + {\left (B b - A c\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right )}{2 \, b c^{2}}, \frac {B b c x - {\left (B b - A c\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right )}{b c^{2}}\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. 82 vs.
\(2 (34) = 68\).
time = 0.14, size = 82, normalized size = 2.05 \begin {gather*} \frac {B x}{c} + \frac {\sqrt {- \frac {1}{b c^{3}}} \left (- A c + B b\right ) \log {\left (- b c \sqrt {- \frac {1}{b c^{3}}} + x \right )}}{2} - \frac {\sqrt {- \frac {1}{b c^{3}}} \left (- A c + B b\right ) \log {\left (b c \sqrt {- \frac {1}{b c^{3}}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, size = 34, normalized size = 0.85 \begin {gather*} \frac {B x}{c} - \frac {{\left (B b - A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 31, normalized size = 0.78 \begin {gather*} \frac {B\,x}{c}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c-B\,b\right )}{\sqrt {b}\,c^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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