Optimal. Leaf size=51 \[ -\frac {A b^2}{2 x^2}+\frac {1}{2} c (2 b B+A c) x^2+\frac {1}{4} B c^2 x^4+b (b B+2 A c) \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 51, 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, 77}
\begin {gather*} -\frac {A b^2}{2 x^2}+\frac {1}{2} c x^2 (A c+2 b B)+b \log (x) (2 A c+b B)+\frac {1}{4} B c^2 x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 457
Rule 1598
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^7} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x^3} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) (b+c x)^2}{x^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (c (2 b B+A c)+\frac {A b^2}{x^2}+\frac {b (b B+2 A c)}{x}+B c^2 x\right ) \, dx,x,x^2\right )\\ &=-\frac {A b^2}{2 x^2}+\frac {1}{2} c (2 b B+A c) x^2+\frac {1}{4} B c^2 x^4+b (b B+2 A c) \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 49, normalized size = 0.96 \begin {gather*} \frac {1}{4} \left (-\frac {2 A b^2}{x^2}+2 c (2 b B+A c) x^2+B c^2 x^4+4 b (b B+2 A c) \log (x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 48, normalized size = 0.94
method | result | size |
default | \(\frac {B \,c^{2} x^{4}}{4}+\frac {A \,c^{2} x^{2}}{2}+B b c \,x^{2}-\frac {A \,b^{2}}{2 x^{2}}+b \left (2 A c +B b \right ) \ln \left (x \right )\) | \(48\) |
norman | \(\frac {\left (\frac {1}{2} A \,c^{2}+b B c \right ) x^{8}-\frac {A \,b^{2} x^{4}}{2}+\frac {B \,c^{2} x^{10}}{4}}{x^{6}}+\left (2 A b c +b^{2} B \right ) \ln \left (x \right )\) | \(54\) |
risch | \(\frac {B \,c^{2} x^{4}}{4}+\frac {A \,c^{2} x^{2}}{2}+B b c \,x^{2}+\frac {A^{2} c^{2}}{4 B}+A b c +b^{2} B -\frac {A \,b^{2}}{2 x^{2}}+2 A \ln \left (x \right ) b c +b^{2} B \ln \left (x \right )\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 52, normalized size = 1.02 \begin {gather*} \frac {1}{4} \, B c^{2} x^{4} + \frac {1}{2} \, {\left (2 \, B b c + A c^{2}\right )} x^{2} + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b c\right )} \log \left (x^{2}\right ) - \frac {A b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.04, size = 54, normalized size = 1.06 \begin {gather*} \frac {B c^{2} x^{6} + 2 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + 4 \, {\left (B b^{2} + 2 \, A b c\right )} x^{2} \log \left (x\right ) - 2 \, A b^{2}}{4 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 48, normalized size = 0.94 \begin {gather*} - \frac {A b^{2}}{2 x^{2}} + \frac {B c^{2} x^{4}}{4} + b \left (2 A c + B b\right ) \log {\left (x \right )} + x^{2} \left (\frac {A c^{2}}{2} + B b c\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.72, size = 70, normalized size = 1.37 \begin {gather*} \frac {1}{4} \, B c^{2} x^{4} + B b c x^{2} + \frac {1}{2} \, A c^{2} x^{2} + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b c\right )} \log \left (x^{2}\right ) - \frac {B b^{2} x^{2} + 2 \, A b c x^{2} + A b^{2}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 48, normalized size = 0.94 \begin {gather*} x^2\,\left (\frac {A\,c^2}{2}+B\,b\,c\right )+\ln \left (x\right )\,\left (B\,b^2+2\,A\,c\,b\right )-\frac {A\,b^2}{2\,x^2}+\frac {B\,c^2\,x^4}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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