Optimal. Leaf size=83 \[ -\frac {(2 b B-A c) x^2}{2 c^3}+\frac {B x^4}{4 c^2}+\frac {b^2 (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac {b (3 b B-2 A c) \log \left (b+c x^2\right )}{2 c^4} \]
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Rubi [A]
time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1598, 457, 78}
\begin {gather*} \frac {b^2 (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac {b (3 b B-2 A c) \log \left (b+c x^2\right )}{2 c^4}-\frac {x^2 (2 b B-A c)}{2 c^3}+\frac {B x^4}{4 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^9 \left (A+B x^2\right )}{\left (b x^2+c x^4\right )^2} \, dx &=\int \frac {x^5 \left (A+B x^2\right )}{\left (b+c x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x^2 (A+B x)}{(b+c x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {-2 b B+A c}{c^3}+\frac {B x}{c^2}-\frac {b^2 (b B-A c)}{c^3 (b+c x)^2}+\frac {b (3 b B-2 A c)}{c^3 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {(2 b B-A c) x^2}{2 c^3}+\frac {B x^4}{4 c^2}+\frac {b^2 (b B-A c)}{2 c^4 \left (b+c x^2\right )}+\frac {b (3 b B-2 A c) \log \left (b+c x^2\right )}{2 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 72, normalized size = 0.87 \begin {gather*} \frac {2 c (-2 b B+A c) x^2+B c^2 x^4+\frac {2 b^2 (b B-A c)}{b+c x^2}+2 b (3 b B-2 A c) \log \left (b+c x^2\right )}{4 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 76, normalized size = 0.92
method | result | size |
default | \(\frac {\left (B c \,x^{2}+A c -2 B b \right )^{2}}{4 c^{4} B}-\frac {b \left (\frac {\left (2 A c -3 B b \right ) \ln \left (c \,x^{2}+b \right )}{c}+\frac {b \left (A c -B b \right )}{c \left (c \,x^{2}+b \right )}\right )}{2 c^{3}}\) | \(76\) |
norman | \(\frac {\frac {B \,x^{9}}{4 c}+\frac {\left (2 A c -3 B b \right ) x^{7}}{4 c^{2}}-\frac {b \left (2 A b c -3 b^{2} B \right ) x^{3}}{2 c^{4}}}{x^{3} \left (c \,x^{2}+b \right )}-\frac {b \left (2 A c -3 B b \right ) \ln \left (c \,x^{2}+b \right )}{2 c^{4}}\) | \(86\) |
risch | \(\frac {B \,x^{4}}{4 c^{2}}+\frac {A \,x^{2}}{2 c^{2}}-\frac {B b \,x^{2}}{c^{3}}+\frac {A^{2}}{4 c^{2} B}-\frac {A b}{c^{3}}+\frac {B \,b^{2}}{c^{4}}-\frac {b^{2} A}{2 c^{3} \left (c \,x^{2}+b \right )}+\frac {b^{3} B}{2 c^{4} \left (c \,x^{2}+b \right )}-\frac {b \ln \left (c \,x^{2}+b \right ) A}{c^{3}}+\frac {3 b^{2} \ln \left (c \,x^{2}+b \right ) B}{2 c^{4}}\) | \(124\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 82, normalized size = 0.99 \begin {gather*} \frac {B b^{3} - A b^{2} c}{2 \, {\left (c^{5} x^{2} + b c^{4}\right )}} + \frac {B c x^{4} - 2 \, {\left (2 \, B b - A c\right )} x^{2}}{4 \, c^{3}} + \frac {{\left (3 \, B b^{2} - 2 \, A b c\right )} \log \left (c x^{2} + b\right )}{2 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.31, size = 121, normalized size = 1.46 \begin {gather*} \frac {B c^{3} x^{6} - {\left (3 \, B b c^{2} - 2 \, A c^{3}\right )} x^{4} + 2 \, B b^{3} - 2 \, A b^{2} c - 2 \, {\left (2 \, B b^{2} c - A b c^{2}\right )} x^{2} + 2 \, {\left (3 \, B b^{3} - 2 \, A b^{2} c + {\left (3 \, B b^{2} c - 2 \, A b c^{2}\right )} x^{2}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{5} x^{2} + b c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 78, normalized size = 0.94 \begin {gather*} \frac {B x^{4}}{4 c^{2}} + \frac {b \left (- 2 A c + 3 B b\right ) \log {\left (b + c x^{2} \right )}}{2 c^{4}} + x^{2} \left (\frac {A}{2 c^{2}} - \frac {B b}{c^{3}}\right ) + \frac {- A b^{2} c + B b^{3}}{2 b c^{4} + 2 c^{5} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.82, size = 106, normalized size = 1.28 \begin {gather*} \frac {{\left (3 \, B b^{2} - 2 \, A b c\right )} \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{4}} + \frac {B c^{2} x^{4} - 4 \, B b c x^{2} + 2 \, A c^{2} x^{2}}{4 \, c^{4}} - \frac {3 \, B b^{2} c x^{2} - 2 \, A b c^{2} x^{2} + 2 \, B b^{3} - A b^{2} c}{2 \, {\left (c x^{2} + b\right )} c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 86, normalized size = 1.04 \begin {gather*} x^2\,\left (\frac {A}{2\,c^2}-\frac {B\,b}{c^3}\right )+\frac {\ln \left (c\,x^2+b\right )\,\left (3\,B\,b^2-2\,A\,b\,c\right )}{2\,c^4}+\frac {B\,x^4}{4\,c^2}+\frac {B\,b^3-A\,b^2\,c}{2\,c\,\left (c^4\,x^2+b\,c^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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