Optimal. Leaf size=101 \[ -\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-3 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}+\frac{D x}{b^2} \]
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Rubi [A] time = 0.117818, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1804, 1810, 635, 205, 260} \[ -\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-3 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}+\frac{D x}{b^2} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1810
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^2} \, dx &=-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{\int \frac{-a \left (B-\frac{a D}{b}\right )-2 a C x-2 a D x^2}{a+b x^2} \, dx}{2 a b}\\ &=-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}-\frac{\int \left (-\frac{2 a D}{b}-\frac{a (b B-3 a D)+2 a b C x}{b \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=\frac{D x}{b^2}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac{\int \frac{a (b B-3 a D)+2 a b C x}{a+b x^2} \, dx}{2 a b^2}\\ &=\frac{D x}{b^2}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac{C \int \frac{x}{a+b x^2} \, dx}{b}+\frac{(b B-3 a D) \int \frac{1}{a+b x^2} \, dx}{2 b^2}\\ &=\frac{D x}{b^2}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{2 a b \left (a+b x^2\right )}+\frac{(b B-3 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}\\ \end{align*}
Mathematica [A] time = 0.0485397, size = 92, normalized size = 0.91 \[ \frac{a C+a D x-A b-b B x}{2 b^2 \left (a+b x^2\right )}-\frac{(3 a D-b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{5/2}}+\frac{C \log \left (a+b x^2\right )}{2 b^2}+\frac{D x}{b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 127, normalized size = 1.3 \begin{align*}{\frac{Dx}{{b}^{2}}}-{\frac{Bx}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{aDx}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{A}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{aC}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{C\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}+{\frac{B}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,aD}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.65598, size = 212, normalized size = 2.1 \begin{align*} \frac{D x}{b^{2}} + \left (\frac{C}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log{\left (x + \frac{2 C a - 4 a b^{2} \left (\frac{C}{2 b^{2}} - \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \left (\frac{C}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right ) \log{\left (x + \frac{2 C a - 4 a b^{2} \left (\frac{C}{2 b^{2}} + \frac{\sqrt{- a b^{5}} \left (- B b + 3 D a\right )}{4 a b^{5}}\right )}{- B b + 3 D a} \right )} + \frac{- A b + C a + x \left (- B b + D a\right )}{2 a b^{2} + 2 b^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14644, size = 109, normalized size = 1.08 \begin{align*} \frac{D x}{b^{2}} + \frac{C \log \left (b x^{2} + a\right )}{2 \, b^{2}} - \frac{{\left (3 \, D a - B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{2}} + \frac{C a - A b +{\left (D a - B b\right )} x}{2 \,{\left (b x^{2} + a\right )} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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