Optimal. Leaf size=42 \[ \frac {B \left (b+c x^2\right )^4}{8 c^2}-\frac {\left (b+c x^2\right )^3 (b B-A c)}{6 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 42, 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 = {1584, 444, 43} \begin {gather*} \frac {B \left (b+c x^2\right )^4}{8 c^2}-\frac {\left (b+c x^2\right )^3 (b B-A c)}{6 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 444
Rule 1584
Rubi steps
\begin {align*} \int \frac {\left (A+B x^2\right ) \left (b x^2+c x^4\right )^2}{x^3} \, dx &=\int x \left (A+B x^2\right ) \left (b+c x^2\right )^2 \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int (A+B x) (b+c x)^2 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {(-b B+A c) (b+c x)^2}{c}+\frac {B (b+c x)^3}{c}\right ) \, dx,x,x^2\right )\\ &=-\frac {(b B-A c) \left (b+c x^2\right )^3}{6 c^2}+\frac {B \left (b+c x^2\right )^4}{8 c^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{24} x^2 \left (12 A b^2+4 c x^4 (A c+2 b B)+6 b x^2 (2 A c+b B)+3 B c^2 x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 57, normalized size = 1.36 \begin {gather*} \frac {1}{24} x^2 \left (12 A b^2+12 A b c x^2+4 A c^2 x^4+6 b^2 B x^2+8 b B c x^4+3 B c^2 x^6\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.37, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{6} \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac {1}{2} \, A b^{2} x^{2} + \frac {1}{4} \, {\left (B b^{2} + 2 \, A b c\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 53, normalized size = 1.26 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{3} \, B b c x^{6} + \frac {1}{6} \, A c^{2} x^{6} + \frac {1}{4} \, B b^{2} x^{4} + \frac {1}{2} \, A b c x^{4} + \frac {1}{2} \, A b^{2} x^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 52, normalized size = 1.24 \begin {gather*} \frac {B \,c^{2} x^{8}}{8}+\frac {\left (A \,c^{2}+2 b B c \right ) x^{6}}{6}+\frac {A \,b^{2} x^{2}}{2}+\frac {\left (2 A b c +B \,b^{2}\right ) x^{4}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 51, normalized size = 1.21 \begin {gather*} \frac {1}{8} \, B c^{2} x^{8} + \frac {1}{6} \, {\left (2 \, B b c + A c^{2}\right )} x^{6} + \frac {1}{2} \, A b^{2} x^{2} + \frac {1}{4} \, {\left (B b^{2} + 2 \, A b c\right )} x^{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 51, normalized size = 1.21 \begin {gather*} x^4\,\left (\frac {B\,b^2}{4}+\frac {A\,c\,b}{2}\right )+x^6\,\left (\frac {A\,c^2}{6}+\frac {B\,b\,c}{3}\right )+\frac {A\,b^2\,x^2}{2}+\frac {B\,c^2\,x^8}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.08, size = 53, normalized size = 1.26 \begin {gather*} \frac {A b^{2} x^{2}}{2} + \frac {B c^{2} x^{8}}{8} + x^{6} \left (\frac {A c^{2}}{6} + \frac {B b c}{3}\right ) + x^{4} \left (\frac {A b c}{2} + \frac {B b^{2}}{4}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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