Optimal. Leaf size=42 \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]
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Rubi [A] time = 0.0592418, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {444, 43} \[ \frac{\left (a+b x^2\right )^3 (b c-a d)}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int x \left (a+b x^2\right )^2 \left (c+d x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b x)^2 (c+d x) \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(b c-a d) (a+b x)^2}{b}+\frac{d (a+b x)^3}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{(b c-a d) \left (a+b x^2\right )^3}{6 b^2}+\frac{d \left (a+b x^2\right )^4}{8 b^2}\\ \end{align*}
Mathematica [A] time = 0.0123594, size = 51, normalized size = 1.21 \[ \frac{1}{24} x^2 \left (12 a^2 c+4 b x^4 (2 a d+b c)+6 a x^2 (a d+2 b c)+3 b^2 d x^6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 52, normalized size = 1.2 \begin{align*}{\frac{{b}^{2}d{x}^{8}}{8}}+{\frac{ \left ( 2\,abd+{b}^{2}c \right ){x}^{6}}{6}}+{\frac{ \left ({a}^{2}d+2\,abc \right ){x}^{4}}{4}}+{\frac{{a}^{2}c{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00039, size = 69, normalized size = 1.64 \begin{align*} \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \,{\left (b^{2} c + 2 \, a b d\right )} x^{6} + \frac{1}{2} \, a^{2} c x^{2} + \frac{1}{4} \,{\left (2 \, a b c + a^{2} d\right )} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03948, size = 128, normalized size = 3.05 \begin{align*} \frac{1}{8} x^{8} d b^{2} + \frac{1}{6} x^{6} c b^{2} + \frac{1}{3} x^{6} d b a + \frac{1}{2} x^{4} c b a + \frac{1}{4} x^{4} d a^{2} + \frac{1}{2} x^{2} c a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.066984, size = 53, normalized size = 1.26 \begin{align*} \frac{a^{2} c x^{2}}{2} + \frac{b^{2} d x^{8}}{8} + x^{6} \left (\frac{a b d}{3} + \frac{b^{2} c}{6}\right ) + x^{4} \left (\frac{a^{2} d}{4} + \frac{a b c}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12865, size = 72, normalized size = 1.71 \begin{align*} \frac{1}{8} \, b^{2} d x^{8} + \frac{1}{6} \, b^{2} c x^{6} + \frac{1}{3} \, a b d x^{6} + \frac{1}{2} \, a b c x^{4} + \frac{1}{4} \, a^{2} d x^{4} + \frac{1}{2} \, a^{2} c x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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