Optimal. Leaf size=147 \[ \frac{x \left (-5 a^2 b e+9 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{x \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e-15 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac{f x}{b^3} \]
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Rubi [A] time = 0.150206, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1814, 1157, 388, 205} \[ \frac{x \left (-5 a^2 b e+9 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{x \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e-15 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}}+\frac{f x}{b^3} \]
Antiderivative was successfully verified.
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Rule 1814
Rule 1157
Rule 388
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x^2+e x^4+f x^6}{\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}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{-\frac{3 b^3 c+a b^2 d-a^2 b e+a^3 f}{b^3}-\frac{4 a (b e-a f) x^2}{b^2}-\frac{4 a f x^4}{b}}{\left (a+b x^2\right )^2} \, dx}{4 a}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (3 b^3 c+a b^2 d-5 a^2 b e+9 a^3 f\right ) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\int \frac{\frac{3 b^3 c+a b^2 d+3 a^2 b e-7 a^3 f}{b^3}+\frac{8 a^2 f x^2}{b^2}}{a+b x^2} \, dx}{8 a^2}\\ &=\frac{f x}{b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (3 b^3 c+a b^2 d-5 a^2 b e+9 a^3 f\right ) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (3 b^3 c+a b^2 d+3 a^2 b e-15 a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{8 a^2 b^3}\\ &=\frac{f x}{b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x}{4 a \left (a+b x^2\right )^2}+\frac{\left (3 b^3 c+a b^2 d-5 a^2 b e+9 a^3 f\right ) x}{8 a^2 b^3 \left (a+b x^2\right )}+\frac{\left (3 b^3 c+a b^2 d+3 a^2 b e-15 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 a^{5/2} b^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.114484, size = 141, normalized size = 0.96 \[ \frac{x \left (-a^2 b^2 \left (d+5 e x^2-8 f x^4\right )+a^3 b \left (25 f x^2-3 e\right )+15 a^4 f+a b^3 \left (5 c+d x^2\right )+3 b^4 c x^2\right )}{8 a^2 b^3 \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^2 b e-15 a^3 f+a b^2 d+3 b^3 c\right )}{8 a^{5/2} b^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 234, normalized size = 1.6 \begin{align*}{\frac{fx}{{b}^{3}}}+{\frac{9\,a{x}^{3}f}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{x}^{3}e}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{{x}^{3}d}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}+{\frac{3\,b{x}^{3}c}{8\, \left ( b{x}^{2}+a \right ) ^{2}{a}^{2}}}+{\frac{7\,{a}^{2}fx}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,aex}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{dx}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{5\,cx}{8\, \left ( b{x}^{2}+a \right ) ^{2}a}}-{\frac{15\,af}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,e}{8\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{8\,ab}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{3\,c}{8\,{a}^{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 [A] time = 1.50816, size = 1062, normalized size = 7.22 \begin{align*} \left [\frac{16 \, a^{3} b^{3} f x^{5} + 2 \,{\left (3 \, a b^{5} c + a^{2} b^{4} d - 5 \, a^{3} b^{3} e + 25 \, a^{4} b^{2} f\right )} x^{3} +{\left (3 \, a^{2} b^{3} c + a^{3} b^{2} d + 3 \, a^{4} b e - 15 \, a^{5} f +{\left (3 \, b^{5} c + a b^{4} d + 3 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c + a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 15 \, 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 ) + 2 \,{\left (5 \, a^{2} b^{4} c - a^{3} b^{3} d - 3 \, a^{4} b^{2} e + 15 \, a^{5} b f\right )} x}{16 \,{\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}, \frac{8 \, a^{3} b^{3} f x^{5} +{\left (3 \, a b^{5} c + a^{2} b^{4} d - 5 \, a^{3} b^{3} e + 25 \, a^{4} b^{2} f\right )} x^{3} +{\left (3 \, a^{2} b^{3} c + a^{3} b^{2} d + 3 \, a^{4} b e - 15 \, a^{5} f +{\left (3 \, b^{5} c + a b^{4} d + 3 \, a^{2} b^{3} e - 15 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c + a^{2} b^{3} d + 3 \, a^{3} b^{2} e - 15 \, a^{4} b f\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (5 \, a^{2} b^{4} c - a^{3} b^{3} d - 3 \, a^{4} b^{2} e + 15 \, a^{5} b f\right )} x}{8 \,{\left (a^{3} b^{6} x^{4} + 2 \, a^{4} b^{5} x^{2} + a^{5} b^{4}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.28425, size = 243, normalized size = 1.65 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{7}}} \left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \log{\left (- a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a^{5} b^{7}}} \left (15 a^{3} f - 3 a^{2} b e - a b^{2} d - 3 b^{3} c\right ) \log{\left (a^{3} b^{3} \sqrt{- \frac{1}{a^{5} b^{7}}} + x \right )}}{16} + \frac{x^{3} \left (9 a^{3} b f - 5 a^{2} b^{2} e + a b^{3} d + 3 b^{4} c\right ) + x \left (7 a^{4} f - 3 a^{3} b e - a^{2} b^{2} d + 5 a b^{3} c\right )}{8 a^{4} b^{3} + 16 a^{3} b^{4} x^{2} + 8 a^{2} b^{5} x^{4}} + \frac{f x}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16689, size = 201, normalized size = 1.37 \begin{align*} \frac{f x}{b^{3}} + \frac{{\left (3 \, b^{3} c + a b^{2} d - 15 \, a^{3} f + 3 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{2} b^{3}} + \frac{3 \, b^{4} c x^{3} + a b^{3} d x^{3} + 9 \, a^{3} b f x^{3} - 5 \, a^{2} b^{2} x^{3} e + 5 \, a b^{3} c x - a^{2} b^{2} d x + 7 \, a^{4} f x - 3 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} a^{2} b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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