Optimal. Leaf size=116 \[ -\frac{4 a^2 D-b x (a C+3 A b)}{8 a^2 b^2 \left (a+b x^2\right )}+\frac{(a C+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}-\frac{a \left (B-\frac{a D}{b}\right )-x (A b-a C)}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.0682323, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {1814, 639, 205} \[ -\frac{4 a^2 D-b x (a C+3 A b)}{8 a^2 b^2 \left (a+b x^2\right )}+\frac{(a C+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}-\frac{a \left (B-\frac{a D}{b}\right )-x (A b-a C)}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 639
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B x+C x^2+D x^3}{\left (a+b x^2\right )^3} \, dx &=-\frac{a \left (B-\frac{a D}{b}\right )-(A b-a C) x}{4 a b \left (a+b x^2\right )^2}-\frac{\int \frac{-3 A-\frac{a C}{b}-\frac{4 a D x}{b}}{\left (a+b x^2\right )^2} \, dx}{4 a}\\ &=-\frac{a \left (B-\frac{a D}{b}\right )-(A b-a C) x}{4 a b \left (a+b x^2\right )^2}-\frac{4 a^2 D-b (3 A b+a C) x}{8 a^2 b^2 \left (a+b x^2\right )}+\frac{(3 A b+a C) \int \frac{1}{a+b x^2} \, dx}{8 a^2 b}\\ &=-\frac{a \left (B-\frac{a D}{b}\right )-(A b-a C) x}{4 a b \left (a+b x^2\right )^2}-\frac{4 a^2 D-b (3 A b+a C) x}{8 a^2 b^2 \left (a+b x^2\right )}+\frac{(3 A b+a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0990765, size = 104, normalized size = 0.9 \[ \frac{\frac{\sqrt{a} \left (-a^2 b (2 B+x (C+4 D x))-2 a^3 D+a b^2 x \left (5 A+C x^2\right )+3 A b^3 x^3\right )}{\left (a+b x^2\right )^2}+\sqrt{b} (a C+3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 111, normalized size = 1. \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( 3\,Ab+aC \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{D{x}^{2}}{2\,b}}+{\frac{ \left ( 5\,Ab-aC \right ) x}{8\,ab}}-{\frac{Bb+aD}{4\,{b}^{2}}} \right ) }+{\frac{3\,A}{8\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{C}{8\,ab}\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 [A] time = 7.9541, size = 184, normalized size = 1.59 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (3 A b + C a\right ) \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (3 A b + C a\right ) \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{16} + \frac{- 2 B a^{2} b - 2 D a^{3} - 4 D a^{2} b x^{2} + x^{3} \left (3 A b^{3} + C a b^{2}\right ) + x \left (5 A a b^{2} - C a^{2} b\right )}{8 a^{4} b^{2} + 16 a^{3} b^{3} x^{2} + 8 a^{2} b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21808, size = 143, normalized size = 1.23 \begin{align*} \frac{{\left (C a + 3 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b} + \frac{C a b^{2} x^{3} + 3 \, A b^{3} x^{3} - 4 \, D a^{2} b x^{2} - C a^{2} b x + 5 \, A a b^{2} x - 2 \, D a^{3} - 2 \, B a^{2} b}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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