Optimal. Leaf size=67 \[ -\frac {b (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac {2 b B-A c}{2 c^3 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^3} \]
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Rubi [A]
time = 0.05, antiderivative size = 67, 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 (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac {2 b B-A c}{2 c^3 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^3} \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 )^3} \, dx &=\int \frac {x^3 \left (A+B x^2\right )}{\left (b+c x^2\right )^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {x (A+B x)}{(b+c x)^3} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {b (b B-A c)}{c^2 (b+c x)^3}+\frac {-2 b B+A c}{c^2 (b+c x)^2}+\frac {B}{c^2 (b+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {b (b B-A c)}{4 c^3 \left (b+c x^2\right )^2}+\frac {2 b B-A c}{2 c^3 \left (b+c x^2\right )}+\frac {B \log \left (b+c x^2\right )}{2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 64, normalized size = 0.96 \begin {gather*} \frac {3 b^2 B-2 A c^2 x^2-b c \left (A-4 B x^2\right )+2 B \left (b+c x^2\right )^2 \log \left (b+c x^2\right )}{4 c^3 \left (b+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.39, size = 61, normalized size = 0.91
method | result | size |
risch | \(\frac {-\frac {\left (A c -2 B b \right ) x^{2}}{2 c^{2}}-\frac {b \left (A c -3 B b \right )}{4 c^{3}}}{\left (c \,x^{2}+b \right )^{2}}+\frac {B \ln \left (c \,x^{2}+b \right )}{2 c^{3}}\) | \(57\) |
default | \(\frac {B \ln \left (c \,x^{2}+b \right )}{2 c^{3}}+\frac {b \left (A c -B b \right )}{4 c^{3} \left (c \,x^{2}+b \right )^{2}}-\frac {A c -2 B b}{2 c^{3} \left (c \,x^{2}+b \right )}\) | \(61\) |
norman | \(\frac {-\frac {\left (A c -2 B b \right ) x^{7}}{2 c^{2}}-\frac {b \left (A c -3 B b \right ) x^{5}}{4 c^{3}}}{x^{5} \left (c \,x^{2}+b \right )^{2}}+\frac {B \ln \left (c \,x^{2}+b \right )}{2 c^{3}}\) | \(63\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 72, normalized size = 1.07 \begin {gather*} \frac {3 \, B b^{2} - A b c + 2 \, {\left (2 \, B b c - A c^{2}\right )} x^{2}}{4 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} + \frac {B \log \left (c x^{2} + b\right )}{2 \, c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.63, size = 89, normalized size = 1.33 \begin {gather*} \frac {3 \, B b^{2} - A b c + 2 \, {\left (2 \, B b c - A c^{2}\right )} x^{2} + 2 \, {\left (B c^{2} x^{4} + 2 \, B b c x^{2} + B b^{2}\right )} \log \left (c x^{2} + b\right )}{4 \, {\left (c^{5} x^{4} + 2 \, b c^{4} x^{2} + b^{2} c^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.49, size = 70, normalized size = 1.04 \begin {gather*} \frac {B \log {\left (b + c x^{2} \right )}}{2 c^{3}} + \frac {- A b c + 3 B b^{2} + x^{2} \left (- 2 A c^{2} + 4 B b c\right )}{4 b^{2} c^{3} + 8 b c^{4} x^{2} + 4 c^{5} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.39, size = 55, normalized size = 0.82 \begin {gather*} \frac {B \log \left ({\left | c x^{2} + b \right |}\right )}{2 \, c^{3}} - \frac {3 \, B c x^{4} + 2 \, B b x^{2} + 2 \, A c x^{2} + A b}{4 \, {\left (c x^{2} + b\right )}^{2} c^{2}} \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 = 1.04 \begin {gather*} \frac {\frac {3\,B\,b^2-A\,b\,c}{4\,c^3}-\frac {x^2\,\left (A\,c-2\,B\,b\right )}{2\,c^2}}{b^2+2\,b\,c\,x^2+c^2\,x^4}+\frac {B\,\ln \left (c\,x^2+b\right )}{2\,c^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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