Optimal. Leaf size=61 \[ \frac{d x^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{\left (c+d x^2\right )^2}{4 b} \]
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Rubi [A] time = 0.0489202, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {444, 43} \[ \frac{d x^2 (b c-a d)}{2 b^2}+\frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}+\frac{\left (c+d x^2\right )^2}{4 b} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (c+d x^2\right )^2}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(c+d x)^2}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{d (b c-a d) x^2}{2 b^2}+\frac{\left (c+d x^2\right )^2}{4 b}+\frac{(b c-a d)^2 \log \left (a+b x^2\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.0230791, size = 49, normalized size = 0.8 \[ \frac{b d x^2 \left (-2 a d+4 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (a+b x^2\right )}{4 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 85, normalized size = 1.4 \begin{align*}{\frac{{d}^{2}{x}^{4}}{4\,b}}-{\frac{a{d}^{2}{x}^{2}}{2\,{b}^{2}}}+{\frac{d{x}^{2}c}{b}}+{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{2}{d}^{2}}{2\,{b}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) cad}{{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{2}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.988173, size = 89, normalized size = 1.46 \begin{align*} \frac{b d^{2} x^{4} + 2 \,{\left (2 \, b c d - a d^{2}\right )} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42728, size = 140, normalized size = 2.3 \begin{align*} \frac{b^{2} d^{2} x^{4} + 2 \,{\left (2 \, b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (b x^{2} + a\right )}{4 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.503939, size = 51, normalized size = 0.84 \begin{align*} \frac{d^{2} x^{4}}{4 b} - \frac{x^{2} \left (a d^{2} - 2 b c d\right )}{2 b^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12555, size = 90, normalized size = 1.48 \begin{align*} \frac{b d^{2} x^{4} + 4 \, b c d x^{2} - 2 \, a d^{2} x^{2}}{4 \, b^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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