Optimal. Leaf size=119 \[ \frac{(3 a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left (a+b x^2\right )}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2} \]
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Rubi [A] time = 0.114108, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1804, 1814, 12, 205} \[ \frac{(3 a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{2 (a C+A b)-x (b B-5 a D)}{8 a b^2 \left (a+b x^2\right )}-\frac{x \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1814
Rule 12
Rule 205
Rubi steps
\begin{align*} \int \frac{x \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{\int \frac{-a \left (B-\frac{a D}{b}\right )-2 (A b+a C) x-4 a D x^2}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{2 (A b+a C)-(b B-5 a D) x}{8 a b^2 \left (a+b x^2\right )}+\frac{\int \frac{a \left (B+\frac{3 a D}{b}\right )}{a+b x^2} \, dx}{8 a^2 b}\\ &=-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{2 (A b+a C)-(b B-5 a D) x}{8 a b^2 \left (a+b x^2\right )}+\frac{(b B+3 a D) \int \frac{1}{a+b x^2} \, dx}{8 a b^2}\\ &=-\frac{x \left (a \left (B-\frac{a D}{b}\right )-(A b-a C) x\right )}{4 a b \left (a+b x^2\right )^2}-\frac{2 (A b+a C)-(b B-5 a D) x}{8 a b^2 \left (a+b x^2\right )}+\frac{(b B+3 a D) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.111549, size = 99, normalized size = 0.83 \[ \frac{\frac{\sqrt{b} \left (-a^2 (2 C+3 D x)-a b \left (2 A+x \left (B+4 C x+5 D x^2\right )\right )+b^2 B x^3\right )}{a \left (a+b x^2\right )^2}+\frac{(3 a D+b B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}}{8 b^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 110, normalized size = 0.9 \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Bb-5\,aD \right ){x}^{3}}{8\,ab}}-{\frac{C{x}^{2}}{2\,b}}-{\frac{ \left ( Bb+3\,aD \right ) x}{8\,{b}^{2}}}-{\frac{Ab+aC}{4\,{b}^{2}}} \right ) }+{\frac{B}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,D}{8\,{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 [A] time = 13.4165, size = 177, normalized size = 1.49 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (B b + 3 D a\right ) \log{\left (- a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (B b + 3 D a\right ) \log{\left (a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} + x \right )}}{16} - \frac{2 A a b + 2 C a^{2} + 4 C a b x^{2} + x^{3} \left (- B b^{2} + 5 D a b\right ) + x \left (B a b + 3 D a^{2}\right )}{8 a^{3} b^{2} + 16 a^{2} b^{3} x^{2} + 8 a b^{4} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19612, size = 131, normalized size = 1.1 \begin{align*} \frac{{\left (3 \, D a + B b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{2}} - \frac{5 \, D a b x^{3} - B b^{2} x^{3} + 4 \, C a b x^{2} + 3 \, D a^{2} x + B a b x + 2 \, C a^{2} + 2 \, A a b}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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