Optimal. Leaf size=87 \[ \frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{\left (c+d x^2\right )^3}{6 b} \]
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Rubi [A] time = 0.0811352, antiderivative size = 87, 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} \[ \frac{d x^2 (b c-a d)^2}{2 b^3}+\frac{\left (c+d x^2\right )^2 (b c-a d)}{4 b^2}+\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}+\frac{\left (c+d x^2\right )^3}{6 b} \]
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 )^3}{a+b x^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(c+d x)^3}{a+b x} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx,x,x^2\right )\\ &=\frac{d (b c-a d)^2 x^2}{2 b^3}+\frac{(b c-a d) \left (c+d x^2\right )^2}{4 b^2}+\frac{\left (c+d x^2\right )^3}{6 b}+\frac{(b c-a d)^3 \log \left (a+b x^2\right )}{2 b^4}\\ \end{align*}
Mathematica [A] time = 0.0303811, size = 82, normalized size = 0.94 \[ \frac{b d x^2 \left (6 a^2 d^2-3 a b d \left (6 c+d x^2\right )+b^2 \left (18 c^2+9 c d x^2+2 d^2 x^4\right )\right )+6 (b c-a d)^3 \log \left (a+b x^2\right )}{12 b^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 149, normalized size = 1.7 \begin{align*}{\frac{{d}^{3}{x}^{6}}{6\,b}}-{\frac{{d}^{3}{x}^{4}a}{4\,{b}^{2}}}+{\frac{3\,{d}^{2}{x}^{4}c}{4\,b}}+{\frac{{d}^{3}{x}^{2}{a}^{2}}{2\,{b}^{3}}}-{\frac{3\,{d}^{2}{x}^{2}ac}{2\,{b}^{2}}}+{\frac{3\,d{x}^{2}{c}^{2}}{2\,b}}-{\frac{\ln \left ( b{x}^{2}+a \right ){a}^{3}{d}^{3}}{2\,{b}^{4}}}+{\frac{3\,\ln \left ( b{x}^{2}+a \right ){a}^{2}c{d}^{2}}{2\,{b}^{3}}}-{\frac{3\,\ln \left ( b{x}^{2}+a \right ) a{c}^{2}d}{2\,{b}^{2}}}+{\frac{\ln \left ( b{x}^{2}+a \right ){c}^{3}}{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9848, size = 161, normalized size = 1.85 \begin{align*} \frac{2 \, b^{2} d^{3} x^{6} + 3 \,{\left (3 \, b^{2} c d^{2} - a b d^{3}\right )} x^{4} + 6 \,{\left (3 \, b^{2} c^{2} d - 3 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}}{12 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} 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 [A] time = 1.44448, size = 244, normalized size = 2.8 \begin{align*} \frac{2 \, b^{3} d^{3} x^{6} + 3 \,{\left (3 \, b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{4} + 6 \,{\left (3 \, b^{3} c^{2} d - 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{2} + 6 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left (b x^{2} + a\right )}{12 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.625289, size = 88, normalized size = 1.01 \begin{align*} \frac{d^{3} x^{6}}{6 b} - \frac{x^{4} \left (a d^{3} - 3 b c d^{2}\right )}{4 b^{2}} + \frac{x^{2} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{2 b^{3}} - \frac{\left (a d - b c\right )^{3} \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 [A] time = 1.16644, size = 167, normalized size = 1.92 \begin{align*} \frac{2 \, b^{2} d^{3} x^{6} + 9 \, b^{2} c d^{2} x^{4} - 3 \, a b d^{3} x^{4} + 18 \, b^{2} c^{2} d x^{2} - 18 \, a b c d^{2} x^{2} + 6 \, a^{2} d^{3} x^{2}}{12 \, b^{3}} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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