Optimal. Leaf size=167 \[ \frac{x^3 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x \left (-7 a^2 b e+11 a^3 f+3 a b^2 d+b^3 c\right )}{8 a b^4 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^2 b e+35 a^3 f+3 a b^2 d+b^3 c\right )}{8 a^{3/2} b^{9/2}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^3}{3 b^3} \]
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Rubi [A] time = 0.262115, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1804, 1585, 1257, 1153, 205} \[ \frac{x^3 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x \left (-7 a^2 b e+11 a^3 f+3 a b^2 d+b^3 c\right )}{8 a b^4 \left (a+b x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^2 b e+35 a^3 f+3 a b^2 d+b^3 c\right )}{8 a^{3/2} b^{9/2}}+\frac{x (b e-3 a f)}{b^4}+\frac{f x^3}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1585
Rule 1257
Rule 1153
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 )^3} \, dx &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{4 a \left (a+b x^2\right )^2}-\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-4 a \left (e-\frac{a f}{b}\right ) x^3-4 a f x^5\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^2 \left (-b c-3 a d+\frac{3 a^2 e}{b}-\frac{3 a^3 f}{b^2}-4 a \left (e-\frac{a f}{b}\right ) x^2-4 a f x^4\right )}{\left (a+b x^2\right )^2} \, dx}{4 a b}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c+3 a b^2 d-7 a^2 b e+11 a^3 f\right ) x}{8 a b^4 \left (a+b x^2\right )}+\frac{\int \frac{b^3 c+3 a b^2 d-7 a^2 b e+11 a^3 f+8 a b (b e-2 a f) x^2+8 a b^2 f x^4}{a+b x^2} \, dx}{8 a b^4}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c+3 a b^2 d-7 a^2 b e+11 a^3 f\right ) x}{8 a b^4 \left (a+b x^2\right )}+\frac{\int \left (8 a (b e-3 a f)+8 a b f x^2+\frac{b^3 c+3 a b^2 d-15 a^2 b e+35 a^3 f}{a+b x^2}\right ) \, dx}{8 a b^4}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^3}{3 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c+3 a b^2 d-7 a^2 b e+11 a^3 f\right ) x}{8 a b^4 \left (a+b x^2\right )}+\frac{\left (b^3 c+3 a b^2 d-15 a^2 b e+35 a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{8 a b^4}\\ &=\frac{(b e-3 a f) x}{b^4}+\frac{f x^3}{3 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^3}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c+3 a b^2 d-7 a^2 b e+11 a^3 f\right ) x}{8 a b^4 \left (a+b x^2\right )}+\frac{\left (b^3 c+3 a b^2 d-15 a^2 b e+35 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{3/2} b^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.120743, size = 156, normalized size = 0.93 \[ \frac{x \left (a^2 b^2 \left (-9 d+75 e x^2-56 f x^4\right )+5 a^3 b \left (9 e-35 f x^2\right )-105 a^4 f+a b^3 \left (-3 c-15 d x^2+24 e x^4+8 f x^6\right )+3 b^4 c x^2\right )}{24 a b^4 \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-15 a^2 b e+35 a^3 f+3 a b^2 d+b^3 c\right )}{8 a^{3/2} b^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 259, normalized size = 1.6 \begin{align*}{\frac{f{x}^{3}}{3\,{b}^{3}}}-3\,{\frac{afx}{{b}^{4}}}+{\frac{ex}{{b}^{3}}}-{\frac{13\,{x}^{3}{a}^{2}f}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}e}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{x}^{3}d}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{x}^{3}c}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{11\,{a}^{3}fx}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}ex}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,adx}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{cx}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{35\,{a}^{2}f}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ae}{8\,{b}^{3}}\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}}}}+{\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 [A] time = 1.5005, size = 1195, normalized size = 7.16 \begin{align*} \left [\frac{16 \, a^{2} b^{4} f x^{7} + 16 \,{\left (3 \, a^{2} b^{4} e - 7 \, a^{3} b^{3} f\right )} x^{5} + 2 \,{\left (3 \, a b^{5} c - 15 \, a^{2} b^{4} d + 75 \, a^{3} b^{3} e - 175 \, a^{4} b^{2} f\right )} x^{3} - 3 \,{\left (a^{2} b^{3} c + 3 \, a^{3} b^{2} d - 15 \, a^{4} b e + 35 \, a^{5} f +{\left (b^{5} c + 3 \, a b^{4} d - 15 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (a b^{4} c + 3 \, a^{2} b^{3} d - 15 \, a^{3} b^{2} e + 35 \, a^{4} 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 ) - 6 \,{\left (a^{2} b^{4} c + 3 \, a^{3} b^{3} d - 15 \, a^{4} b^{2} e + 35 \, a^{5} b f\right )} x}{48 \,{\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}, \frac{8 \, a^{2} b^{4} f x^{7} + 8 \,{\left (3 \, a^{2} b^{4} e - 7 \, a^{3} b^{3} f\right )} x^{5} +{\left (3 \, a b^{5} c - 15 \, a^{2} b^{4} d + 75 \, a^{3} b^{3} e - 175 \, a^{4} b^{2} f\right )} x^{3} + 3 \,{\left (a^{2} b^{3} c + 3 \, a^{3} b^{2} d - 15 \, a^{4} b e + 35 \, a^{5} f +{\left (b^{5} c + 3 \, a b^{4} d - 15 \, a^{2} b^{3} e + 35 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (a b^{4} c + 3 \, a^{2} b^{3} d - 15 \, a^{3} b^{2} e + 35 \, a^{4} b f\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - 3 \,{\left (a^{2} b^{4} c + 3 \, a^{3} b^{3} d - 15 \, a^{4} b^{2} e + 35 \, a^{5} b f\right )} x}{24 \,{\left (a^{2} b^{7} x^{4} + 2 \, a^{3} b^{6} x^{2} + a^{4} b^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.9172, size = 258, normalized size = 1.54 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (35 a^{3} f - 15 a^{2} b e + 3 a b^{2} d + b^{3} c\right ) \log{\left (- a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{16} + \frac{\sqrt{- \frac{1}{a^{3} b^{9}}} \left (35 a^{3} f - 15 a^{2} b e + 3 a b^{2} d + b^{3} c\right ) \log{\left (a^{2} b^{4} \sqrt{- \frac{1}{a^{3} b^{9}}} + x \right )}}{16} - \frac{x^{3} \left (13 a^{3} b f - 9 a^{2} b^{2} e + 5 a b^{3} d - b^{4} c\right ) + x \left (11 a^{4} f - 7 a^{3} b e + 3 a^{2} b^{2} d + a b^{3} c\right )}{8 a^{3} b^{4} + 16 a^{2} b^{5} x^{2} + 8 a b^{6} x^{4}} + \frac{f x^{3}}{3 b^{3}} - \frac{x \left (3 a f - b e\right )}{b^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22776, size = 234, normalized size = 1.4 \begin{align*} \frac{{\left (b^{3} c + 3 \, a b^{2} d + 35 \, a^{3} f - 15 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a b^{4}} + \frac{b^{4} c x^{3} - 5 \, a b^{3} d x^{3} - 13 \, a^{3} b f x^{3} + 9 \, a^{2} b^{2} x^{3} e - a b^{3} c x - 3 \, a^{2} b^{2} d x - 11 \, a^{4} f x + 7 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a b^{4}} + \frac{b^{6} f x^{3} - 9 \, a b^{5} f x + 3 \, b^{6} x e}{3 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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