Optimal. Leaf size=136 \[ \frac{(3 a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{x (-2 x (b B-3 a D)+3 a C+A b)}{8 a b^2 \left (a+b x^2\right )}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac{D \log \left (a+b x^2\right )}{2 b^3} \]
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Rubi [A] time = 0.158462, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {1804, 635, 205, 260} \[ \frac{(3 a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}-\frac{x (-2 x (b B-3 a D)+3 a C+A b)}{8 a b^2 \left (a+b x^2\right )}-\frac{x^2 \left (a \left (B-\frac{a D}{b}\right )-x (A b-a C)\right )}{4 a b \left (a+b x^2\right )^2}+\frac{D \log \left (a+b x^2\right )}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^2 \left (A+B x+C x^2+D x^3\right )}{\left (a+b x^2\right )^3} \, dx &=-\frac{x^2 \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{x \left (-2 a \left (B-\frac{a D}{b}\right )-(A b+3 a C) x-4 a D x^2\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=-\frac{x^2 \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{x (A b+3 a C-2 (b B-3 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac{\int \frac{a (A b+3 a C)+8 a^2 D x}{a+b x^2} \, dx}{8 a^2 b^2}\\ &=-\frac{x^2 \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{x (A b+3 a C-2 (b B-3 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac{(A b+3 a C) \int \frac{1}{a+b x^2} \, dx}{8 a b^2}+\frac{D \int \frac{x}{a+b x^2} \, dx}{b^2}\\ &=-\frac{x^2 \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{x (A b+3 a C-2 (b B-3 a D) x)}{8 a b^2 \left (a+b x^2\right )}+\frac{(A b+3 a C) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{5/2}}+\frac{D \log \left (a+b x^2\right )}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.0932766, size = 122, normalized size = 0.9 \[ \frac{\frac{-2 a^2 D+2 a b (B+C x)-2 A b^2 x}{\left (a+b x^2\right )^2}+\frac{8 a^2 D-a b (4 B+5 C x)+A b^2 x}{a \left (a+b x^2\right )}+\frac{\sqrt{b} (3 a C+A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2}}+4 D \log \left (a+b x^2\right )}{8 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 133, normalized size = 1. \begin{align*}{\frac{1}{ \left ( b{x}^{2}+a \right ) ^{2}} \left ({\frac{ \left ( Ab-5\,aC \right ){x}^{3}}{8\,ab}}-{\frac{ \left ( Bb-2\,aD \right ){x}^{2}}{2\,{b}^{2}}}-{\frac{ \left ( Ab+3\,aC \right ) x}{8\,{b}^{2}}}-{\frac{a \left ( Bb-3\,aD \right ) }{4\,{b}^{3}}} \right ) }+{\frac{D\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}}+{\frac{A}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,C}{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 [B] time = 18.1106, size = 303, normalized size = 2.23 \begin{align*} \left (\frac{D}{2 b^{3}} - \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right ) \log{\left (x + \frac{- 8 D a^{2} + 16 a^{2} b^{3} \left (\frac{D}{2 b^{3}} - \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right )}{A b^{2} + 3 C a b} \right )} + \left (\frac{D}{2 b^{3}} + \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right ) \log{\left (x + \frac{- 8 D a^{2} + 16 a^{2} b^{3} \left (\frac{D}{2 b^{3}} + \frac{\sqrt{- a^{3} b^{7}} \left (A b + 3 C a\right )}{16 a^{3} b^{6}}\right )}{A b^{2} + 3 C a b} \right )} - \frac{2 B a^{2} b - 6 D a^{3} + x^{3} \left (- A b^{3} + 5 C a b^{2}\right ) + x^{2} \left (4 B a b^{2} - 8 D a^{2} b\right ) + x \left (A a b^{2} + 3 C a^{2} b\right )}{8 a^{3} b^{3} + 16 a^{2} b^{4} x^{2} + 8 a b^{5} x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19087, size = 173, normalized size = 1.27 \begin{align*} \frac{D \log \left (b x^{2} + a\right )}{2 \, b^{3}} + \frac{{\left (3 \, C a + A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{2}} - \frac{{\left (5 \, C a b - A b^{2}\right )} x^{3} - 4 \,{\left (2 \, D a^{2} - B a b\right )} x^{2} +{\left (3 \, C a^{2} + A a b\right )} x - \frac{2 \,{\left (3 \, D a^{3} - B a^{2} b\right )}}{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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