Optimal. Leaf size=88 \[ \frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}-\frac{(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{d^3 x^4}{4 b^2} \]
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Rubi [A] time = 0.0933621, antiderivative size = 88, 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} \[ \frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}-\frac{(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}+\frac{d^3 x^4}{4 b^2} \]
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 )^3}{\left (a+b x^2\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(c+d x)^3}{(a+b x)^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d^2 (3 b c-2 a d)}{b^3}+\frac{d^3 x}{b^2}+\frac{(b c-a d)^3}{b^3 (a+b x)^2}+\frac{3 d (b c-a d)^2}{b^3 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=\frac{d^2 (3 b c-2 a d) x^2}{2 b^3}+\frac{d^3 x^4}{4 b^2}-\frac{(b c-a d)^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 d (b c-a d)^2 \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0440927, size = 127, normalized size = 1.44 \[ \frac{-3 a^2 b c d^2+a^3 d^3+3 a b^2 c^2 d-b^3 c^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 \left (a^2 d^3-2 a b c d^2+b^2 c^2 d\right ) \log \left (a+b x^2\right )}{2 b^4}+\frac{d^2 x^2 (3 b c-2 a d)}{2 b^3}+\frac{d^3 x^4}{4 b^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 168, normalized size = 1.9 \begin{align*}{\frac{{d}^{3}{x}^{4}}{4\,{b}^{2}}}-{\frac{a{d}^{3}{x}^{2}}{{b}^{3}}}+{\frac{3\,{d}^{2}{x}^{2}c}{2\,{b}^{2}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){d}^{3}{a}^{2}}{2\,{b}^{4}}}-3\,{\frac{\ln \left ( b{x}^{2}+a \right ){d}^{2}ca}{{b}^{3}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ) d{c}^{2}}{2\,{b}^{2}}}+{\frac{{a}^{3}{d}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,{a}^{2}c{d}^{2}}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,ad{c}^{2}}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{{c}^{3}}{2\,b \left ( b{x}^{2}+a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0022, size = 167, normalized size = 1.9 \begin{align*} -\frac{b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}}{2 \,{\left (b^{5} x^{2} + a b^{4}\right )}} + \frac{b d^{3} x^{4} + 2 \,{\left (3 \, b c d^{2} - 2 \, a d^{3}\right )} x^{2}}{4 \, b^{3}} + \frac{3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (b x^{2} + a\right )}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.4785, size = 365, normalized size = 4.15 \begin{align*} \frac{b^{3} d^{3} x^{6} - 2 \, b^{3} c^{3} + 6 \, a b^{2} c^{2} d - 6 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} + 3 \,{\left (2 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 2 \,{\left (3 \, a b^{2} c d^{2} - 2 \, a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3} +{\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.37117, size = 112, normalized size = 1.27 \begin{align*} \frac{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{d^{3} x^{4}}{4 b^{2}} - \frac{x^{2} \left (2 a d^{3} - 3 b c d^{2}\right )}{2 b^{3}} + \frac{3 d \left (a d - b c\right )^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15526, size = 247, normalized size = 2.81 \begin{align*} \frac{{\left (d^{3} + \frac{6 \,{\left (b^{2} c d^{2} - a b d^{3}\right )}}{{\left (b x^{2} + a\right )} b}\right )}{\left (b x^{2} + a\right )}^{2}}{4 \, b^{4}} - \frac{3 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} \log \left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{2 \, b^{4}} - \frac{\frac{b^{5} c^{3}}{b x^{2} + a} - \frac{3 \, a b^{4} c^{2} d}{b x^{2} + a} + \frac{3 \, a^{2} b^{3} c d^{2}}{b x^{2} + a} - \frac{a^{3} b^{2} d^{3}}{b x^{2} + a}}{2 \, b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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