Optimal. Leaf size=58 \[ -\frac {(b B-A c) x}{c^2}+\frac {B x^3}{3 c}+\frac {\sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{5/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 58, 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, 470, 327,
211} \begin {gather*} \frac {\sqrt {b} (b B-A c) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{5/2}}-\frac {x (b B-A c)}{c^2}+\frac {B x^3}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 327
Rule 470
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^4 \left (A+B x^2\right )}{b x^2+c x^4} \, dx &=\int \frac {x^2 \left (A+B x^2\right )}{b+c x^2} \, dx\\ &=\frac {B x^3}{3 c}-\frac {(3 b B-3 A c) \int \frac {x^2}{b+c x^2} \, dx}{3 c}\\ &=-\frac {(b B-A c) x}{c^2}+\frac {B x^3}{3 c}+\frac {(b (b B-A c)) \int \frac {1}{b+c x^2} \, dx}{c^2}\\ &=-\frac {(b B-A c) x}{c^2}+\frac {B x^3}{3 c}+\frac {\sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.98 \begin {gather*} \frac {(-b B+A c) x}{c^2}+\frac {B x^3}{3 c}+\frac {\sqrt {b} (b B-A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 51, normalized size = 0.88
method | result | size |
default | \(\frac {\frac {1}{3} B c \,x^{3}+A c x -b B x}{c^{2}}-\frac {b \left (A c -B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{c^{2} \sqrt {b c}}\) | \(51\) |
risch | \(\frac {B \,x^{3}}{3 c}+\frac {A x}{c}-\frac {b B x}{c^{2}}+\frac {\sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right ) A}{2 c^{2}}-\frac {\sqrt {-b c}\, \ln \left (-\sqrt {-b c}\, x -b \right ) B b}{2 c^{3}}-\frac {\sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right ) A}{2 c^{2}}+\frac {\sqrt {-b c}\, \ln \left (\sqrt {-b c}\, x -b \right ) B b}{2 c^{3}}\) | \(129\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 53, normalized size = 0.91 \begin {gather*} \frac {{\left (B b^{2} - A b c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {B c x^{3} - 3 \, {\left (B b - A c\right )} x}{3 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.55, size = 129, normalized size = 2.22 \begin {gather*} \left [\frac {2 \, B c x^{3} - 3 \, {\left (B b - A c\right )} \sqrt {-\frac {b}{c}} \log \left (\frac {c x^{2} - 2 \, c x \sqrt {-\frac {b}{c}} - b}{c x^{2} + b}\right ) - 6 \, {\left (B b - A c\right )} x}{6 \, c^{2}}, \frac {B c x^{3} + 3 \, {\left (B b - A c\right )} \sqrt {\frac {b}{c}} \arctan \left (\frac {c x \sqrt {\frac {b}{c}}}{b}\right ) - 3 \, {\left (B b - A c\right )} x}{3 \, c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.17, size = 90, normalized size = 1.55 \begin {gather*} \frac {B x^{3}}{3 c} + x \left (\frac {A}{c} - \frac {B b}{c^{2}}\right ) - \frac {\sqrt {- \frac {b}{c^{5}}} \left (- A c + B b\right ) \log {\left (- c^{2} \sqrt {- \frac {b}{c^{5}}} + x \right )}}{2} + \frac {\sqrt {- \frac {b}{c^{5}}} \left (- A c + B b\right ) \log {\left (c^{2} \sqrt {- \frac {b}{c^{5}}} + x \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.66, size = 57, normalized size = 0.98 \begin {gather*} \frac {{\left (B b^{2} - A b c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{\sqrt {b c} c^{2}} + \frac {B c^{2} x^{3} - 3 \, B b c x + 3 \, A c^{2} x}{3 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 70, normalized size = 1.21 \begin {gather*} x\,\left (\frac {A}{c}-\frac {B\,b}{c^2}\right )+\frac {B\,x^3}{3\,c}+\frac {\sqrt {b}\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c}\,x\,\left (A\,c-B\,b\right )}{B\,b^2-A\,b\,c}\right )\,\left (A\,c-B\,b\right )}{c^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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