Optimal. Leaf size=247 \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^3 \left (15 a^2 b e-27 a^3 f-7 a b^2 d+3 b^3 c\right )}{12 a b^5}+\frac{a x \left (11 a^2 b e-15 a^3 f-7 a b^2 d+3 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac{x \left (13 a^2 b e-21 a^3 f-7 a b^2 d+3 b^3 c\right )}{2 b^6}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (63 a^2 b e-99 a^3 f-35 a b^2 d+15 b^3 c\right )}{8 b^{13/2}}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^7}{7 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.411223, antiderivative size = 247, 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} \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{4 a \left (a+b x^2\right )^2}-\frac{x^3 \left (15 a^2 b e-27 a^3 f-7 a b^2 d+3 b^3 c\right )}{12 a b^5}+\frac{a x \left (11 a^2 b e-15 a^3 f-7 a b^2 d+3 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac{x \left (13 a^2 b e-21 a^3 f-7 a b^2 d+3 b^3 c\right )}{2 b^6}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (63 a^2 b e-99 a^3 f-35 a b^2 d+15 b^3 c\right )}{8 b^{13/2}}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^7}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1804
Rule 1585
Rule 1257
Rule 1810
Rule 205
Rubi steps
\begin{align*} \int \frac{x^6 \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^7}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^5 \left (\left (3 b c-7 a d+\frac{7 a^2 e}{b}-\frac{7 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^7}{4 a \left (a+b x^2\right )^2}-\frac{\int \frac{x^6 \left (3 b c-7 a d+\frac{7 a^2 e}{b}-\frac{7 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^7}{4 a \left (a+b x^2\right )^2}+\frac{a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}+\frac{\int \frac{-a^2 \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right )+2 a b \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x^2-2 b^2 \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x^4+8 a b^3 (b e-2 a f) x^6+8 a b^4 f x^8}{a+b x^2} \, dx}{8 a b^6}\\ &=\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac{a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}+\frac{\int \left (4 a \left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right )-2 b \left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^2+8 a b^2 (b e-3 a f) x^4+8 a b^3 f x^6+\frac{-15 a^2 b^3 c+35 a^3 b^2 d-63 a^4 b e+99 a^5 f}{a+b x^2}\right ) \, dx}{8 a b^6}\\ &=\frac{\left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right ) x}{2 b^6}-\frac{\left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^3}{12 a b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^7}{7 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac{a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}-\frac{\left (a \left (15 b^3 c-35 a b^2 d+63 a^2 b e-99 a^3 f\right )\right ) \int \frac{1}{a+b x^2} \, dx}{8 b^6}\\ &=\frac{\left (3 b^3 c-7 a b^2 d+13 a^2 b e-21 a^3 f\right ) x}{2 b^6}-\frac{\left (3 b^3 c-7 a b^2 d+15 a^2 b e-27 a^3 f\right ) x^3}{12 a b^5}+\frac{(b e-3 a f) x^5}{5 b^4}+\frac{f x^7}{7 b^3}+\frac{\left (c-\frac{a \left (b^2 d-a b e+a^2 f\right )}{b^3}\right ) x^7}{4 a \left (a+b x^2\right )^2}+\frac{a \left (3 b^3 c-7 a b^2 d+11 a^2 b e-15 a^3 f\right ) x}{8 b^6 \left (a+b x^2\right )}-\frac{\sqrt{a} \left (15 b^3 c-35 a b^2 d+63 a^2 b e-99 a^3 f\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{8 b^{13/2}}\\ \end{align*}
Mathematica [A] time = 0.136549, size = 232, normalized size = 0.94 \[ \frac{a x \left (17 a^2 b e-21 a^3 f-13 a b^2 d+9 b^3 c\right )}{8 b^6 \left (a+b x^2\right )}+\frac{a^2 x \left (-a^2 b e+a^3 f+a b^2 d-b^3 c\right )}{4 b^6 \left (a+b x^2\right )^2}+\frac{x \left (6 a^2 b e-10 a^3 f-3 a b^2 d+b^3 c\right )}{b^6}+\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-63 a^2 b e+99 a^3 f+35 a b^2 d-15 b^3 c\right )}{8 b^{13/2}}+\frac{x^3 \left (6 a^2 f-3 a b e+b^2 d\right )}{3 b^5}+\frac{x^5 (b e-3 a f)}{5 b^4}+\frac{f x^7}{7 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.013, size = 343, normalized size = 1.4 \begin{align*}{\frac{f{x}^{7}}{7\,{b}^{3}}}-{\frac{3\,{x}^{5}af}{5\,{b}^{4}}}+{\frac{{x}^{5}e}{5\,{b}^{3}}}+2\,{\frac{{x}^{3}{a}^{2}f}{{b}^{5}}}-{\frac{a{x}^{3}e}{{b}^{4}}}+{\frac{{x}^{3}d}{3\,{b}^{3}}}-10\,{\frac{{a}^{3}fx}{{b}^{6}}}+6\,{\frac{{a}^{2}ex}{{b}^{5}}}-3\,{\frac{adx}{{b}^{4}}}+{\frac{cx}{{b}^{3}}}-{\frac{21\,{a}^{4}{x}^{3}f}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{17\,{a}^{3}{x}^{3}e}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{13\,{x}^{3}{a}^{2}d}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{9\,a{x}^{3}c}{8\,{b}^{2} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{19\,{a}^{5}fx}{8\,{b}^{6} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{15\,{a}^{4}ex}{8\,{b}^{5} \left ( b{x}^{2}+a \right ) ^{2}}}-{\frac{11\,{a}^{3}dx}{8\,{b}^{4} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{7\,{a}^{2}cx}{8\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{99\,f{a}^{4}}{8\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{63\,{a}^{3}e}{8\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{35\,{a}^{2}d}{8\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,ac}{8\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.30561, size = 1517, normalized size = 6.14 \begin{align*} \left [\frac{240 \, b^{5} f x^{11} + 48 \,{\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 16 \,{\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 112 \,{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 350 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f +{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} + 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) + 210 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{1680 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}, \frac{120 \, b^{5} f x^{11} + 24 \,{\left (7 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 8 \,{\left (35 \, b^{5} d - 63 \, a b^{4} e + 99 \, a^{2} b^{3} f\right )} x^{7} + 56 \,{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{5} + 175 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{3} - 105 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f +{\left (15 \, b^{5} c - 35 \, a b^{4} d + 63 \, a^{2} b^{3} e - 99 \, a^{3} b^{2} f\right )} x^{4} + 2 \,{\left (15 \, a b^{4} c - 35 \, a^{2} b^{3} d + 63 \, a^{3} b^{2} e - 99 \, a^{4} b f\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) + 105 \,{\left (15 \, a^{2} b^{3} c - 35 \, a^{3} b^{2} d + 63 \, a^{4} b e - 99 \, a^{5} f\right )} x}{840 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{2} + a^{2} b^{6}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 16.2142, size = 311, normalized size = 1.26 \begin{align*} - \frac{\sqrt{- \frac{a}{b^{13}}} \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log{\left (- b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{16} + \frac{\sqrt{- \frac{a}{b^{13}}} \left (99 a^{3} f - 63 a^{2} b e + 35 a b^{2} d - 15 b^{3} c\right ) \log{\left (b^{6} \sqrt{- \frac{a}{b^{13}}} + x \right )}}{16} - \frac{x^{3} \left (21 a^{4} b f - 17 a^{3} b^{2} e + 13 a^{2} b^{3} d - 9 a b^{4} c\right ) + x \left (19 a^{5} f - 15 a^{4} b e + 11 a^{3} b^{2} d - 7 a^{2} b^{3} c\right )}{8 a^{2} b^{6} + 16 a b^{7} x^{2} + 8 b^{8} x^{4}} + \frac{f x^{7}}{7 b^{3}} - \frac{x^{5} \left (3 a f - b e\right )}{5 b^{4}} + \frac{x^{3} \left (6 a^{2} f - 3 a b e + b^{2} d\right )}{3 b^{5}} - \frac{x \left (10 a^{3} f - 6 a^{2} b e + 3 a b^{2} d - b^{3} c\right )}{b^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.21404, size = 338, normalized size = 1.37 \begin{align*} -\frac{{\left (15 \, a b^{3} c - 35 \, a^{2} b^{2} d - 99 \, a^{4} f + 63 \, a^{3} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} b^{6}} + \frac{9 \, a b^{4} c x^{3} - 13 \, a^{2} b^{3} d x^{3} - 21 \, a^{4} b f x^{3} + 17 \, a^{3} b^{2} x^{3} e + 7 \, a^{2} b^{3} c x - 11 \, a^{3} b^{2} d x - 19 \, a^{5} f x + 15 \, a^{4} b x e}{8 \,{\left (b x^{2} + a\right )}^{2} b^{6}} + \frac{15 \, b^{18} f x^{7} - 63 \, a b^{17} f x^{5} + 21 \, b^{18} x^{5} e + 35 \, b^{18} d x^{3} + 210 \, a^{2} b^{16} f x^{3} - 105 \, a b^{17} x^{3} e + 105 \, b^{18} c x - 315 \, a b^{17} d x - 1050 \, a^{3} b^{15} f x + 630 \, a^{2} b^{16} x e}{105 \, b^{21}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]