Optimal. Leaf size=43 \[ A b c x^2+\frac {1}{4} A c^2 x^4+\frac {B \left (b+c x^2\right )^3}{6 c}+A b^2 \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1598, 457, 81,
45} \begin {gather*} A b^2 \log (x)+A b c x^2+\frac {1}{4} A c^2 x^4+\frac {B \left (b+c x^2\right )^3}{6 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 81
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^5} \, dx &=\int \frac {\left (A+B x^2\right ) \left (b+c x^2\right )^2}{x} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(A+B x) (b+c x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b+c x^2\right )^3}{6 c}+\frac {1}{2} A \text {Subst}\left (\int \frac {(b+c x)^2}{x} \, dx,x,x^2\right )\\ &=\frac {B \left (b+c x^2\right )^3}{6 c}+\frac {1}{2} A \text {Subst}\left (\int \left (2 b c+\frac {b^2}{x}+c^2 x\right ) \, dx,x,x^2\right )\\ &=A b c x^2+\frac {1}{4} A c^2 x^4+\frac {B \left (b+c x^2\right )^3}{6 c}+A b^2 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 51, normalized size = 1.19 \begin {gather*} \frac {1}{2} b (b B+2 A c) x^2+\frac {1}{4} c (2 b B+A c) x^4+\frac {1}{6} B c^2 x^6+A b^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 51, normalized size = 1.19
method | result | size |
default | \(\frac {B \,c^{2} x^{6}}{6}+\frac {A \,c^{2} x^{4}}{4}+\frac {x^{4} b B c}{2}+A b c \,x^{2}+\frac {b^{2} B \,x^{2}}{2}+A \,b^{2} \ln \left (x \right )\) | \(51\) |
risch | \(\frac {B \,c^{2} x^{6}}{6}+\frac {A \,c^{2} x^{4}}{4}+\frac {x^{4} b B c}{2}+A b c \,x^{2}+\frac {b^{2} B \,x^{2}}{2}+A \,b^{2} \ln \left (x \right )\) | \(51\) |
norman | \(\frac {\left (\frac {1}{4} A \,c^{2}+\frac {1}{2} b B c \right ) x^{8}+\left (A b c +\frac {1}{2} b^{2} B \right ) x^{6}+\frac {B \,c^{2} x^{10}}{6}}{x^{4}}+A \,b^{2} \ln \left (x \right )\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 52, normalized size = 1.21 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{4} \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + \frac {1}{2} \, A b^{2} \log \left (x^{2}\right ) + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b c\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.06, size = 49, normalized size = 1.14 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{4} \, {\left (2 \, B b c + A c^{2}\right )} x^{4} + A b^{2} \log \left (x\right ) + \frac {1}{2} \, {\left (B b^{2} + 2 \, A b c\right )} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.05, size = 49, normalized size = 1.14 \begin {gather*} A b^{2} \log {\left (x \right )} + \frac {B c^{2} x^{6}}{6} + x^{4} \left (\frac {A c^{2}}{4} + \frac {B b c}{2}\right ) + x^{2} \left (A b c + \frac {B b^{2}}{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.93, size = 53, normalized size = 1.23 \begin {gather*} \frac {1}{6} \, B c^{2} x^{6} + \frac {1}{2} \, B b c x^{4} + \frac {1}{4} \, A c^{2} x^{4} + \frac {1}{2} \, B b^{2} x^{2} + A b c x^{2} + \frac {1}{2} \, A b^{2} \log \left (x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 48, normalized size = 1.12 \begin {gather*} x^2\,\left (\frac {B\,b^2}{2}+A\,c\,b\right )+x^4\,\left (\frac {A\,c^2}{4}+\frac {B\,b\,c}{2}\right )+\frac {B\,c^2\,x^6}{6}+A\,b^2\,\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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