Optimal. Leaf size=61 \[ -\frac{b x^2 (b c-a d)}{2 d^2}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}+\frac{\left (a+b x^2\right )^2}{4 d} \]
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Rubi [A] time = 0.0489539, 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} \[ -\frac{b x^2 (b c-a d)}{2 d^2}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}+\frac{\left (a+b x^2\right )^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 444
Rule 43
Rubi steps
\begin{align*} \int \frac{x \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{c+d x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{b (b c-a d) x^2}{2 d^2}+\frac{\left (a+b x^2\right )^2}{4 d}+\frac{(b c-a d)^2 \log \left (c+d x^2\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 0.0227169, size = 49, normalized size = 0.8 \[ \frac{b d x^2 \left (4 a d-2 b c+b d x^2\right )+2 (b c-a d)^2 \log \left (c+d x^2\right )}{4 d^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 85, normalized size = 1.4 \begin{align*}{\frac{{b}^{2}{x}^{4}}{4\,d}}+{\frac{ab{x}^{2}}{d}}-{\frac{{b}^{2}c{x}^{2}}{2\,{d}^{2}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,d}}-{\frac{\ln \left ( d{x}^{2}+c \right ) cab}{{d}^{2}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}{c}^{2}}{2\,{d}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.992624, size = 88, normalized size = 1.44 \begin{align*} \frac{b^{2} d x^{4} - 2 \,{\left (b^{2} c - 2 \, a b d\right )} x^{2}}{4 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.2664, size = 140, normalized size = 2.3 \begin{align*} \frac{b^{2} d^{2} x^{4} - 2 \,{\left (b^{2} c d - 2 \, 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 (d x^{2} + c\right )}{4 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.481465, size = 51, normalized size = 0.84 \begin{align*} \frac{b^{2} x^{4}}{4 d} + \frac{x^{2} \left (2 a b d - b^{2} c\right )}{2 d^{2}} + \frac{\left (a d - b c\right )^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17182, size = 90, normalized size = 1.48 \begin{align*} \frac{b^{2} d x^{4} - 2 \, b^{2} c x^{2} + 4 \, a b d x^{2}}{4 \, d^{2}} + \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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