Optimal. Leaf size=67 \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]
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Rubi [A] time = 0.0681163, antiderivative size = 67, 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} \[ \frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3} \]
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}{\left (c+d x^2\right )^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(a+b x)^2}{(c+d x)^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{(-b c+a d)^2}{d^2 (c+d x)^3}-\frac{2 b (b c-a d)}{d^2 (c+d x)^2}+\frac{b^2}{d^2 (c+d x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{(b c-a d)^2}{4 d^3 \left (c+d x^2\right )^2}+\frac{b (b c-a d)}{d^3 \left (c+d x^2\right )}+\frac{b^2 \log \left (c+d x^2\right )}{2 d^3}\\ \end{align*}
Mathematica [A] time = 0.0240263, size = 75, normalized size = 1.12 \[ \frac{-a^2 d^2-2 a b d \left (c+2 d x^2\right )+b^2 c \left (3 c+4 d x^2\right )+2 b^2 \left (c+d x^2\right )^2 \log \left (c+d x^2\right )}{4 d^3 \left (c+d x^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 105, normalized size = 1.6 \begin{align*}{\frac{{b}^{2}\ln \left ( d{x}^{2}+c \right ) }{2\,{d}^{3}}}-{\frac{{a}^{2}}{4\,d \left ( d{x}^{2}+c \right ) ^{2}}}+{\frac{abc}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{{b}^{2}{c}^{2}}{4\,{d}^{3} \left ( d{x}^{2}+c \right ) ^{2}}}-{\frac{ab}{{d}^{2} \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}c}{{d}^{3} \left ( d{x}^{2}+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01857, size = 117, normalized size = 1.75 \begin{align*} \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2}}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} + \frac{b^{2} \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 = 1.45565, size = 216, normalized size = 3.22 \begin{align*} \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2} + 4 \,{\left (b^{2} c d - a b d^{2}\right )} x^{2} + 2 \,{\left (b^{2} d^{2} x^{4} + 2 \, b^{2} c d x^{2} + b^{2} c^{2}\right )} \log \left (d x^{2} + c\right )}{4 \,{\left (d^{5} x^{4} + 2 \, c d^{4} x^{2} + c^{2} d^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.66126, size = 87, normalized size = 1.3 \begin{align*} \frac{b^{2} \log{\left (c + d x^{2} \right )}}{2 d^{3}} - \frac{a^{2} d^{2} + 2 a b c d - 3 b^{2} c^{2} + x^{2} \left (4 a b d^{2} - 4 b^{2} c d\right )}{4 c^{2} d^{3} + 8 c d^{4} x^{2} + 4 d^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18528, size = 103, normalized size = 1.54 \begin{align*} \frac{b^{2} \log \left ({\left | d x^{2} + c \right |}\right )}{2 \, d^{3}} + \frac{4 \,{\left (b^{2} c - a b d\right )} x^{2} + \frac{3 \, b^{2} c^{2} - 2 \, a b c d - a^{2} d^{2}}{d}}{4 \,{\left (d x^{2} + c\right )}^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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