Optimal. Leaf size=90 \[ \frac {(b B-A c) x}{4 c^2 \left (b+c x^2\right )^2}-\frac {(5 b B-A c) x}{8 b c^2 \left (b+c x^2\right )}+\frac {(3 b B+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{5/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 90, 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, 466, 393,
211} \begin {gather*} \frac {(A c+3 b B) \text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{5/2}}-\frac {x (5 b B-A c)}{8 b c^2 \left (b+c x^2\right )}+\frac {x (b B-A c)}{4 c^2 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 393
Rule 466
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^8 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^3} \, dx &=\int \frac {x^2 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {(b B-A c) x}{4 c^2 \left (b+c x^2\right )^2}-\frac {\int \frac {b B-A c-4 B c x^2}{\left (b+c x^2\right )^2} \, dx}{4 c^2}\\ &=\frac {(b B-A c) x}{4 c^2 \left (b+c x^2\right )^2}-\frac {(5 b B-A c) x}{8 b c^2 \left (b+c x^2\right )}+\frac {(3 b B+A c) \int \frac {1}{b+c x^2} \, dx}{8 b c^2}\\ &=\frac {(b B-A c) x}{4 c^2 \left (b+c x^2\right )^2}-\frac {(5 b B-A c) x}{8 b c^2 \left (b+c x^2\right )}+\frac {(3 b B+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{8 b^{3/2} c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 83, normalized size = 0.92 \begin {gather*} \frac {\frac {\sqrt {c} x \left (-3 b^2 B+A c^2 x^2-b c \left (A+5 B x^2\right )\right )}{b \left (b+c x^2\right )^2}+\frac {(3 b B+A c) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b}}\right )}{b^{3/2}}}{8 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 76, normalized size = 0.84
method | result | size |
default | \(\frac {\frac {\left (A c -5 B b \right ) x^{3}}{8 b c}-\frac {\left (A c +3 B b \right ) x}{8 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {\left (A c +3 B b \right ) \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 c^{2} b \sqrt {b c}}\) | \(76\) |
risch | \(\frac {\frac {\left (A c -5 B b \right ) x^{3}}{8 b c}-\frac {\left (A c +3 B b \right ) x}{8 c^{2}}}{\left (c \,x^{2}+b \right )^{2}}-\frac {\ln \left (c x +\sqrt {-b c}\right ) A}{16 \sqrt {-b c}\, c b}-\frac {3 \ln \left (c x +\sqrt {-b c}\right ) B}{16 \sqrt {-b c}\, c^{2}}+\frac {\ln \left (-c x +\sqrt {-b c}\right ) A}{16 \sqrt {-b c}\, c b}+\frac {3 \ln \left (-c x +\sqrt {-b c}\right ) B}{16 \sqrt {-b c}\, c^{2}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 92, normalized size = 1.02 \begin {gather*} -\frac {{\left (5 \, B b c - A c^{2}\right )} x^{3} + {\left (3 \, B b^{2} + A b c\right )} x}{8 \, {\left (b c^{4} x^{4} + 2 \, b^{2} c^{3} x^{2} + b^{3} c^{2}\right )}} + \frac {{\left (3 \, B b + A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.27, size = 301, normalized size = 3.34 \begin {gather*} \left [-\frac {2 \, {\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x^{3} + {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 3 \, B b^{3} + A b^{2} c + 2 \, {\left (3 \, B b^{2} c + A b c^{2}\right )} x^{2}\right )} \sqrt {-b c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-b c} x - b}{c x^{2} + b}\right ) + 2 \, {\left (3 \, B b^{3} c + A b^{2} c^{2}\right )} x}{16 \, {\left (b^{2} c^{5} x^{4} + 2 \, b^{3} c^{4} x^{2} + b^{4} c^{3}\right )}}, -\frac {{\left (5 \, B b^{2} c^{2} - A b c^{3}\right )} x^{3} - {\left ({\left (3 \, B b c^{2} + A c^{3}\right )} x^{4} + 3 \, B b^{3} + A b^{2} c + 2 \, {\left (3 \, B b^{2} c + A b c^{2}\right )} x^{2}\right )} \sqrt {b c} \arctan \left (\frac {\sqrt {b c} x}{b}\right ) + {\left (3 \, B b^{3} c + A b^{2} c^{2}\right )} x}{8 \, {\left (b^{2} c^{5} x^{4} + 2 \, b^{3} c^{4} x^{2} + b^{4} c^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.38, size = 155, normalized size = 1.72 \begin {gather*} - \frac {\sqrt {- \frac {1}{b^{3} c^{5}}} \left (A c + 3 B b\right ) \log {\left (- b^{2} c^{2} \sqrt {- \frac {1}{b^{3} c^{5}}} + x \right )}}{16} + \frac {\sqrt {- \frac {1}{b^{3} c^{5}}} \left (A c + 3 B b\right ) \log {\left (b^{2} c^{2} \sqrt {- \frac {1}{b^{3} c^{5}}} + x \right )}}{16} + \frac {x^{3} \left (A c^{2} - 5 B b c\right ) + x \left (- A b c - 3 B b^{2}\right )}{8 b^{3} c^{2} + 16 b^{2} c^{3} x^{2} + 8 b c^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.34, size = 78, normalized size = 0.87 \begin {gather*} \frac {{\left (3 \, B b + A c\right )} \arctan \left (\frac {c x}{\sqrt {b c}}\right )}{8 \, \sqrt {b c} b c^{2}} - \frac {5 \, B b c x^{3} - A c^{2} x^{3} + 3 \, B b^{2} x + A b c x}{8 \, {\left (c x^{2} + b\right )}^{2} b c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 82, normalized size = 0.91 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {b}}\right )\,\left (A\,c+3\,B\,b\right )}{8\,b^{3/2}\,c^{5/2}}-\frac {\frac {x\,\left (A\,c+3\,B\,b\right )}{8\,c^2}-\frac {x^3\,\left (A\,c-5\,B\,b\right )}{8\,b\,c}}{b^2+2\,b\,c\,x^2+c^2\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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