Optimal. Leaf size=207 \[ \frac{x^5 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x \left (9 a^2 b e-13 a^3 f-5 a b^2 d+b^3 c\right )}{8 b^5 \left (a+b x^2\right )}-\frac{x \left (13 a^2 b e-25 a^3 f-5 a b^2 d+b^3 c\right )}{4 a b^5}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^2 b e-63 a^3 f-15 a b^2 d+3 b^3 c\right )}{8 \sqrt{a} b^{11/2}}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^5}{5 b^3} \]
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Rubi [A] time = 0.333674, antiderivative size = 207, 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, 1810, 205} \[ \frac{x^5 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x \left (9 a^2 b e-13 a^3 f-5 a b^2 d+b^3 c\right )}{8 b^5 \left (a+b x^2\right )}-\frac{x \left (13 a^2 b e-25 a^3 f-5 a b^2 d+b^3 c\right )}{4 a b^5}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^2 b e-63 a^3 f-15 a b^2 d+3 b^3 c\right )}{8 \sqrt{a} b^{11/2}}+\frac{x^3 (b e-3 a f)}{3 b^4}+\frac{f x^5}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 1804
Rule 1585
Rule 1257
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \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^5}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^3 \left (\left (b c-5 a d+\frac{5 a^2 e}{b}-\frac{5 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^5}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^4 \left (b c-5 a d+\frac{5 a^2 e}{b}-\frac{5 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^5}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac{\int \frac{a \left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right )-2 b \left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x^2+8 a b^2 (b e-2 a f) x^4+8 a b^3 f x^6}{a+b x^2} \, dx}{8 a b^5}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac{\int \left (-2 \left (b^3 c-5 a b^2 d+13 a^2 b e-25 a^3 f\right )+8 a b (b e-3 a f) x^2+8 a b^2 f x^4+\frac{3 a b^3 c-15 a^2 b^2 d+35 a^3 b e-63 a^4 f}{a+b x^2}\right ) \, dx}{8 a b^5}\\ &=-\frac{\left (b^3 c-5 a b^2 d+13 a^2 b e-25 a^3 f\right ) x}{4 a b^5}+\frac{(b e-3 a f) x^3}{3 b^4}+\frac{f x^5}{5 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac{\left (3 b^3 c-15 a b^2 d+35 a^2 b e-63 a^3 f\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^5}\\ &=-\frac{\left (b^3 c-5 a b^2 d+13 a^2 b e-25 a^3 f\right ) x}{4 a b^5}+\frac{(b e-3 a f) x^3}{3 b^4}+\frac{f x^5}{5 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^5}{4 a \left (a+b x^2\right )^2}-\frac{\left (b^3 c-5 a b^2 d+9 a^2 b e-13 a^3 f\right ) x}{8 b^5 \left (a+b x^2\right )}+\frac{\left (3 b^3 c-15 a b^2 d+35 a^2 b e-63 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 \sqrt{a} b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.159392, size = 176, normalized size = 0.85 \[ \frac{x \left (a^2 b^2 \left (225 d-875 e x^2+504 f x^4\right )-525 a^3 b \left (e-3 f x^2\right )+945 a^4 f-a b^3 \left (45 c-375 d x^2+280 e x^4+72 f x^6\right )+b^4 x^2 \left (8 \left (15 d x^2+5 e x^4+3 f x^6\right )-75 c\right )\right )}{120 b^5 \left (a+b x^2\right )^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (35 a^2 b e-63 a^3 f-15 a b^2 d+3 b^3 c\right )}{8 \sqrt{a} b^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 294, normalized size = 1.4 \begin{align*}{\frac{f{x}^{5}}{5\,{b}^{3}}}-{\frac{a{x}^{3}f}{{b}^{4}}}+{\frac{{x}^{3}e}{3\,{b}^{3}}}+6\,{\frac{{a}^{2}fx}{{b}^{5}}}-3\,{\frac{aex}{{b}^{4}}}+{\frac{dx}{{b}^{3}}}+{\frac{17\,{x}^{3}{a}^{3}f}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{x}^{3}{a}^{2}e}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}d}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{5\,{x}^{3}c}{8\,b \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,f{a}^{4}x}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,{a}^{3}ex}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}dx}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{3\,acx}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{63\,{a}^{3}f}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}e}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ad}{8\,{b}^{3}}\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 [A] time = 1.3148, size = 1362, normalized size = 6.58 \begin{align*} \left [\frac{48 \, a b^{5} f x^{9} + 16 \,{\left (5 \, a b^{5} e - 9 \, a^{2} b^{4} f\right )} x^{7} + 16 \,{\left (15 \, a b^{5} d - 35 \, a^{2} b^{4} e + 63 \, a^{3} b^{3} f\right )} x^{5} - 50 \,{\left (3 \, a b^{5} c - 15 \, a^{2} b^{4} d + 35 \, a^{3} b^{3} e - 63 \, a^{4} b^{2} f\right )} x^{3} + 15 \,{\left (3 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 35 \, a^{4} b e - 63 \, a^{5} f +{\left (3 \, b^{5} c - 15 \, a b^{4} d + 35 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c - 15 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 63 \, 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 ) - 30 \,{\left (3 \, a^{2} b^{4} c - 15 \, a^{3} b^{3} d + 35 \, a^{4} b^{2} e - 63 \, a^{5} b f\right )} x}{240 \,{\left (a b^{8} x^{4} + 2 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}, \frac{24 \, a b^{5} f x^{9} + 8 \,{\left (5 \, a b^{5} e - 9 \, a^{2} b^{4} f\right )} x^{7} + 8 \,{\left (15 \, a b^{5} d - 35 \, a^{2} b^{4} e + 63 \, a^{3} b^{3} f\right )} x^{5} - 25 \,{\left (3 \, a b^{5} c - 15 \, a^{2} b^{4} d + 35 \, a^{3} b^{3} e - 63 \, a^{4} b^{2} f\right )} x^{3} + 15 \,{\left (3 \, a^{2} b^{3} c - 15 \, a^{3} b^{2} d + 35 \, a^{4} b e - 63 \, a^{5} f +{\left (3 \, b^{5} c - 15 \, a b^{4} d + 35 \, a^{2} b^{3} e - 63 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (3 \, a b^{4} c - 15 \, a^{2} b^{3} d + 35 \, a^{3} b^{2} e - 63 \, a^{4} b f\right )} x^{2}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - 15 \,{\left (3 \, a^{2} b^{4} c - 15 \, a^{3} b^{3} d + 35 \, a^{4} b^{2} e - 63 \, a^{5} b f\right )} x}{120 \,{\left (a b^{8} x^{4} + 2 \, a^{2} b^{7} x^{2} + a^{3} b^{6}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.8714, size = 279, normalized size = 1.35 \begin{align*} \frac{\sqrt{- \frac{1}{a b^{11}}} \left (63 a^{3} f - 35 a^{2} b e + 15 a b^{2} d - 3 b^{3} c\right ) \log{\left (- a b^{5} \sqrt{- \frac{1}{a b^{11}}} + x \right )}}{16} - \frac{\sqrt{- \frac{1}{a b^{11}}} \left (63 a^{3} f - 35 a^{2} b e + 15 a b^{2} d - 3 b^{3} c\right ) \log{\left (a b^{5} \sqrt{- \frac{1}{a b^{11}}} + x \right )}}{16} + \frac{x^{3} \left (17 a^{3} b f - 13 a^{2} b^{2} e + 9 a b^{3} d - 5 b^{4} c\right ) + x \left (15 a^{4} f - 11 a^{3} b e + 7 a^{2} b^{2} d - 3 a b^{3} c\right )}{8 a^{2} b^{5} + 16 a b^{6} x^{2} + 8 b^{7} x^{4}} + \frac{f x^{5}}{5 b^{3}} - \frac{x^{3} \left (3 a f - b e\right )}{3 b^{4}} + \frac{x \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20482, size = 270, normalized size = 1.3 \begin{align*} \frac{{\left (3 \, b^{3} c - 15 \, a b^{2} d - 63 \, a^{3} f + 35 \, a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{5}} - \frac{5 \, b^{4} c x^{3} - 9 \, a b^{3} d x^{3} - 17 \, a^{3} b f x^{3} + 13 \, a^{2} b^{2} x^{3} e + 3 \, a b^{3} c x - 7 \, a^{2} b^{2} d x - 15 \, a^{4} f x + 11 \, a^{3} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} b^{5}} + \frac{3 \, b^{12} f x^{5} - 15 \, a b^{11} f x^{3} + 5 \, b^{12} x^{3} e + 15 \, b^{12} d x + 90 \, a^{2} b^{10} f x - 45 \, a b^{11} x e}{15 \, b^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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