Optimal. Leaf size=163 \[ \frac{x^3 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{x \left (5 a^2 b e-7 a^3 f-3 a b^2 d+b^3 c\right )}{2 a b^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^2 b e-7 a^3 f-3 a b^2 d+b^3 c\right )}{2 \sqrt{a} b^{9/2}}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^5}{5 b^2} \]
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Rubi [A] time = 0.228794, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {1804, 1585, 1261, 205} \[ \frac{x^3 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{x \left (5 a^2 b e-7 a^3 f-3 a b^2 d+b^3 c\right )}{2 a b^4}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (5 a^2 b e-7 a^3 f-3 a b^2 d+b^3 c\right )}{2 \sqrt{a} b^{9/2}}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^5}{5 b^2} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1585
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}-\frac{\int \frac{x \left (\left (b c-3 a d+\frac{3 a^2 e}{b}-\frac{3 a^3 f}{b^2}\right ) x-2 a \left (e-\frac{a f}{b}\right ) x^3-2 a f x^5\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}-\frac{\int \frac{x^2 \left (b c-3 a d+\frac{3 a^2 e}{b}-\frac{3 a^3 f}{b^2}-2 a \left (e-\frac{a f}{b}\right ) x^2-2 a f x^4\right )}{a+b x^2} \, dx}{2 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}-\frac{\int \left (c-\frac{a \left (3 b^2 d-5 a b e+7 a^2 f\right )}{b^3}-\frac{2 a (b e-2 a f) x^2}{b^2}-\frac{2 a f x^4}{b}+\frac{-a b^3 c+3 a^2 b^2 d-5 a^3 b e+7 a^4 f}{b^3 \left (a+b x^2\right )}\right ) \, dx}{2 a b}\\ &=-\frac{\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) x}{2 a b^4}+\frac{(b e-2 a f) x^3}{3 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}+\frac{\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{2 b^4}\\ &=-\frac{\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) x}{2 a b^4}+\frac{(b e-2 a f) x^3}{3 b^3}+\frac{f x^5}{5 b^2}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{2 a \left (a+b x^2\right )}+\frac{\left (b^3 c-3 a b^2 d+5 a^2 b e-7 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 \sqrt{a} b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0817565, size = 148, normalized size = 0.91 \[ -\frac{x \left (a^2 b e+a^3 (-f)-a b^2 d+b^3 c\right )}{2 b^4 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-5 a^2 b e+7 a^3 f+3 a b^2 d-b^3 c\right )}{2 \sqrt{a} b^{9/2}}+\frac{x \left (3 a^2 f-2 a b e+b^2 d\right )}{b^4}+\frac{x^3 (b e-2 a f)}{3 b^3}+\frac{f x^5}{5 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 212, normalized size = 1.3 \begin{align*}{\frac{f{x}^{5}}{5\,{b}^{2}}}-{\frac{2\,a{x}^{3}f}{3\,{b}^{3}}}+{\frac{{x}^{3}e}{3\,{b}^{2}}}+3\,{\frac{{a}^{2}fx}{{b}^{4}}}-2\,{\frac{aex}{{b}^{3}}}+{\frac{dx}{{b}^{2}}}+{\frac{x{a}^{3}f}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{2}xe}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{axd}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{cx}{2\,b \left ( b{x}^{2}+a \right ) }}-{\frac{7\,{a}^{3}f}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}e}{2\,{b}^{3}}\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}}}}+{\frac{c}{2\,b}\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 [A] time = 1.49616, size = 902, normalized size = 5.53 \begin{align*} \left [\frac{12 \, a b^{4} f x^{7} + 4 \,{\left (5 \, a b^{4} e - 7 \, a^{2} b^{3} f\right )} x^{5} + 20 \,{\left (3 \, a b^{4} d - 5 \, a^{2} b^{3} e + 7 \, a^{3} b^{2} f\right )} x^{3} + 15 \,{\left (a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f +{\left (b^{4} c - 3 \, a b^{3} d + 5 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} x^{2}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) - 30 \,{\left (a b^{4} c - 3 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x}{60 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, \frac{6 \, a b^{4} f x^{7} + 2 \,{\left (5 \, a b^{4} e - 7 \, a^{2} b^{3} f\right )} x^{5} + 10 \,{\left (3 \, a b^{4} d - 5 \, a^{2} b^{3} e + 7 \, a^{3} b^{2} f\right )} x^{3} + 15 \,{\left (a b^{3} c - 3 \, a^{2} b^{2} d + 5 \, a^{3} b e - 7 \, a^{4} f +{\left (b^{4} c - 3 \, a b^{3} d + 5 \, a^{2} b^{2} e - 7 \, a^{3} b f\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - 15 \,{\left (a b^{4} c - 3 \, a^{2} b^{3} d + 5 \, a^{3} b^{2} e - 7 \, a^{4} b f\right )} x}{30 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.43083, size = 216, normalized size = 1.33 \begin{align*} \frac{x \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right )}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{\sqrt{- \frac{1}{a b^{9}}} \left (7 a^{3} f - 5 a^{2} b e + 3 a b^{2} d - b^{3} c\right ) \log{\left (- a b^{4} \sqrt{- \frac{1}{a b^{9}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a b^{9}}} \left (7 a^{3} f - 5 a^{2} b e + 3 a b^{2} d - b^{3} c\right ) \log{\left (a b^{4} \sqrt{- \frac{1}{a b^{9}}} + x \right )}}{4} + \frac{f x^{5}}{5 b^{2}} - \frac{x^{3} \left (2 a f - b e\right )}{3 b^{3}} + \frac{x \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16362, size = 205, normalized size = 1.26 \begin{align*} \frac{{\left (b^{3} c - 3 \, a b^{2} d - 7 \, a^{3} f + 5 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{4}} - \frac{b^{3} c x - a b^{2} d x - a^{3} f x + a^{2} b x e}{2 \,{\left (b x^{2} + a\right )} b^{4}} + \frac{3 \, b^{8} f x^{5} - 10 \, a b^{7} f x^{3} + 5 \, b^{8} x^{3} e + 15 \, b^{8} d x + 45 \, a^{2} b^{6} f x - 30 \, a b^{7} x e}{15 \, b^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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