Optimal. Leaf size=97 \[ -\frac{\left (-2 a B c-A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{B x^2}{2 c} \]
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Rubi [A] time = 0.116055, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1251, 773, 634, 618, 206, 628} \[ -\frac{\left (-2 a B c-A b c+b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac{B x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 773
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (A+B x^2\right )}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x (A+B x)}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{B x^2}{2 c}+\frac{\operatorname{Subst}\left (\int \frac{-a B+(-b B+A c) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c}\\ &=\frac{B x^2}{2 c}-\frac{(b B-A c) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}+\frac{\left (b^2 B-A b c-2 a B c\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac{B x^2}{2 c}-\frac{(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}-\frac{\left (b^2 B-A b c-2 a B c\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2}\\ &=\frac{B x^2}{2 c}-\frac{\left (b^2 B-A b c-2 a B c\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c}}-\frac{(b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end{align*}
Mathematica [A] time = 0.0680335, size = 93, normalized size = 0.96 \[ \frac{\frac{2 \left (-2 a B c-A b c+b^2 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+(A c-b B) \log \left (a+b x^2+c x^4\right )+2 B c x^2}{4 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 175, normalized size = 1.8 \begin{align*}{\frac{B{x}^{2}}{2\,c}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) A}{4\,c}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bB}{4\,{c}^{2}}}-{\frac{aB}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{Ab}{2\,c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}B}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62066, size = 670, normalized size = 6.91 \begin{align*} \left [\frac{2 \,{\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c +{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (B b^{3} + 4 \, A a c^{2} -{\left (4 \, B a b + A b^{2}\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac{2 \,{\left (B b^{2} c - 4 \, B a c^{2}\right )} x^{2} - 2 \,{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) -{\left (B b^{3} + 4 \, A a c^{2} -{\left (4 \, B a b + A b^{2}\right )} c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.97837, size = 434, normalized size = 4.47 \begin{align*} \frac{B x^{2}}{2 c} + \left (- \frac{- A c + B b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - B a b - 8 a c^{2} \left (- \frac{- A c + B b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac{- A c + B b}{4 c^{2}} - \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{A b c + 2 B a c - B b^{2}} \right )} + \left (- \frac{- A c + B b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{2} + \frac{2 A a c - B a b - 8 a c^{2} \left (- \frac{- A c + B b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right ) + 2 b^{2} c \left (- \frac{- A c + B b}{4 c^{2}} + \frac{\sqrt{- 4 a c + b^{2}} \left (A b c + 2 B a c - B b^{2}\right )}{4 c^{2} \left (4 a c - b^{2}\right )}\right )}{A b c + 2 B a c - B b^{2}} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17829, size = 123, normalized size = 1.27 \begin{align*} \frac{B x^{2}}{2 \, c} - \frac{{\left (B b - A c\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac{{\left (B b^{2} - 2 \, B a c - A b c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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