3.124 \(\int \frac{x^6 \left (c+d x^2+e x^4+f x^6\right )}{\left (a+b x^2\right )^2} \, dx\)

Optimal. Leaf size=240 \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{a x \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{2 b^6}+\frac{x^3 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{6 b^5}-\frac{x^5 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{10 a b^4}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{2 b^{13/2}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2} \]

[Out]

-(a*(5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x)/(2*b^6) + ((5*b^3*c - 7*a*b^
2*d + 9*a^2*b*e - 11*a^3*f)*x^3)/(6*b^5) - ((5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 1
1*a^3*f)*x^5)/(10*a*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^9)/(9*b^2) + ((c -
(a*(b^2*d - a*b*e + a^2*f))/b^3)*x^7)/(2*a*(a + b*x^2)) + (a^(3/2)*(5*b^3*c - 7
*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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Rubi [A]  time = 0.648624, antiderivative size = 240, 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 \[ \frac{x^7 \left (c-\frac{a \left (a^2 f-a b e+b^2 d\right )}{b^3}\right )}{2 a \left (a+b x^2\right )}-\frac{a x \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{2 b^6}+\frac{x^3 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{6 b^5}-\frac{x^5 \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{10 a b^4}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (-11 a^3 f+9 a^2 b e-7 a b^2 d+5 b^3 c\right )}{2 b^{13/2}}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]

[Out]

-(a*(5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*x)/(2*b^6) + ((5*b^3*c - 7*a*b^
2*d + 9*a^2*b*e - 11*a^3*f)*x^3)/(6*b^5) - ((5*b^3*c - 7*a*b^2*d + 9*a^2*b*e - 1
1*a^3*f)*x^5)/(10*a*b^4) + ((b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^9)/(9*b^2) + ((c -
(a*(b^2*d - a*b*e + a^2*f))/b^3)*x^7)/(2*a*(a + b*x^2)) + (a^(3/2)*(5*b^3*c - 7
*a*b^2*d + 9*a^2*b*e - 11*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 0.237628, size = 227, normalized size = 0.95 \[ \frac{x^5 \left (3 a^2 f-2 a b e+b^2 d\right )}{5 b^4}+\frac{a x \left (5 a^3 f-4 a^2 b e+3 a b^2 d-2 b^3 c\right )}{b^6}+\frac{x^3 \left (-4 a^3 f+3 a^2 b e-2 a b^2 d+b^3 c\right )}{3 b^5}-\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (11 a^3 f-9 a^2 b e+7 a b^2 d-5 b^3 c\right )}{2 b^{13/2}}-\frac{x \left (a^5 (-f)+a^4 b e-a^3 b^2 d+a^2 b^3 c\right )}{2 b^6 \left (a+b x^2\right )}+\frac{x^7 (b e-2 a f)}{7 b^3}+\frac{f x^9}{9 b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^6*(c + d*x^2 + e*x^4 + f*x^6))/(a + b*x^2)^2,x]

[Out]

(a*(-2*b^3*c + 3*a*b^2*d - 4*a^2*b*e + 5*a^3*f)*x)/b^6 + ((b^3*c - 2*a*b^2*d + 3
*a^2*b*e - 4*a^3*f)*x^3)/(3*b^5) + ((b^2*d - 2*a*b*e + 3*a^2*f)*x^5)/(5*b^4) + (
(b*e - 2*a*f)*x^7)/(7*b^3) + (f*x^9)/(9*b^2) - ((a^2*b^3*c - a^3*b^2*d + a^4*b*e
- a^5*f)*x)/(2*b^6*(a + b*x^2)) - (a^(3/2)*(-5*b^3*c + 7*a*b^2*d - 9*a^2*b*e +
11*a^3*f)*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(2*b^(13/2))

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Maple [A]  time = 0.017, size = 309, normalized size = 1.3 \[{\frac{f{x}^{9}}{9\,{b}^{2}}}-{\frac{2\,{x}^{7}af}{7\,{b}^{3}}}+{\frac{{x}^{7}e}{7\,{b}^{2}}}+{\frac{3\,{x}^{5}{a}^{2}f}{5\,{b}^{4}}}-{\frac{2\,{x}^{5}ae}{5\,{b}^{3}}}+{\frac{{x}^{5}d}{5\,{b}^{2}}}-{\frac{4\,{x}^{3}{a}^{3}f}{3\,{b}^{5}}}+{\frac{{x}^{3}{a}^{2}e}{{b}^{4}}}-{\frac{2\,a{x}^{3}d}{3\,{b}^{3}}}+{\frac{{x}^{3}c}{3\,{b}^{2}}}+5\,{\frac{{a}^{4}fx}{{b}^{6}}}-4\,{\frac{{a}^{3}ex}{{b}^{5}}}+3\,{\frac{{a}^{2}dx}{{b}^{4}}}-2\,{\frac{acx}{{b}^{3}}}+{\frac{{a}^{5}xf}{2\,{b}^{6} \left ( b{x}^{2}+a \right ) }}-{\frac{{a}^{4}xe}{2\,{b}^{5} \left ( b{x}^{2}+a \right ) }}+{\frac{{a}^{3}xd}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}-{\frac{x{a}^{2}c}{2\,{b}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{11\,{a}^{5}f}{2\,{b}^{6}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{9\,{a}^{4}e}{2\,{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{7\,{a}^{3}d}{2\,{b}^{4}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{a}^{2}c}{2\,{b}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(f*x^6+e*x^4+d*x^2+c)/(b*x^2+a)^2,x)

[Out]

1/9*f*x^9/b^2-2/7/b^3*x^7*a*f+1/7/b^2*x^7*e+3/5/b^4*x^5*a^2*f-2/5/b^3*x^5*a*e+1/
5/b^2*x^5*d-4/3/b^5*x^3*a^3*f+1/b^4*x^3*a^2*e-2/3/b^3*x^3*a*d+1/3/b^2*x^3*c+5/b^
6*a^4*f*x-4/b^5*a^3*e*x+3/b^4*a^2*d*x-2/b^3*a*c*x+1/2*a^5/b^6*x/(b*x^2+a)*f-1/2*
a^4/b^5*x/(b*x^2+a)*e+1/2*a^3/b^4*x/(b*x^2+a)*d-1/2*a^2/b^3*x/(b*x^2+a)*c-11/2*a
^5/b^6/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*f+9/2*a^4/b^5/(a*b)^(1/2)*arctan(x*b/
(a*b)^(1/2))*e-7/2*a^3/b^4/(a*b)^(1/2)*arctan(x*b/(a*b)^(1/2))*d+5/2*a^2/b^3/(a*
b)^(1/2)*arctan(x*b/(a*b)^(1/2))*c

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^6/(b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236943, size = 1, normalized size = 0. \[ \left [\frac{140 \, b^{5} f x^{11} + 20 \,{\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 36 \,{\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 84 \,{\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 420 \,{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} - 315 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f +{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 630 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{1260 \,{\left (b^{7} x^{2} + a b^{6}\right )}}, \frac{70 \, b^{5} f x^{11} + 10 \,{\left (9 \, b^{5} e - 11 \, a b^{4} f\right )} x^{9} + 18 \,{\left (7 \, b^{5} d - 9 \, a b^{4} e + 11 \, a^{2} b^{3} f\right )} x^{7} + 42 \,{\left (5 \, b^{5} c - 7 \, a b^{4} d + 9 \, a^{2} b^{3} e - 11 \, a^{3} b^{2} f\right )} x^{5} - 210 \,{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{3} + 315 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f +{\left (5 \, a b^{4} c - 7 \, a^{2} b^{3} d + 9 \, a^{3} b^{2} e - 11 \, a^{4} b f\right )} x^{2}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{x}{\sqrt{\frac{a}{b}}}\right ) - 315 \,{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d + 9 \, a^{4} b e - 11 \, a^{5} f\right )} x}{630 \,{\left (b^{7} x^{2} + a b^{6}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^6/(b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[1/1260*(140*b^5*f*x^11 + 20*(9*b^5*e - 11*a*b^4*f)*x^9 + 36*(7*b^5*d - 9*a*b^4*
e + 11*a^2*b^3*f)*x^7 + 84*(5*b^5*c - 7*a*b^4*d + 9*a^2*b^3*e - 11*a^3*b^2*f)*x^
5 - 420*(5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*e - 11*a^4*b*f)*x^3 - 315*(5*a^2*b^
3*c - 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f + (5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*
e - 11*a^4*b*f)*x^2)*sqrt(-a/b)*log((b*x^2 - 2*b*x*sqrt(-a/b) - a)/(b*x^2 + a))
- 630*(5*a^2*b^3*c - 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f)*x)/(b^7*x^2 + a*b^6), 1
/630*(70*b^5*f*x^11 + 10*(9*b^5*e - 11*a*b^4*f)*x^9 + 18*(7*b^5*d - 9*a*b^4*e +
11*a^2*b^3*f)*x^7 + 42*(5*b^5*c - 7*a*b^4*d + 9*a^2*b^3*e - 11*a^3*b^2*f)*x^5 -
210*(5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*e - 11*a^4*b*f)*x^3 + 315*(5*a^2*b^3*c
- 7*a^3*b^2*d + 9*a^4*b*e - 11*a^5*f + (5*a*b^4*c - 7*a^2*b^3*d + 9*a^3*b^2*e -
11*a^4*b*f)*x^2)*sqrt(a/b)*arctan(x/sqrt(a/b)) - 315*(5*a^2*b^3*c - 7*a^3*b^2*d
+ 9*a^4*b*e - 11*a^5*f)*x)/(b^7*x^2 + a*b^6)]

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Sympy [A]  time = 4.61709, size = 430, normalized size = 1.79 \[ \frac{x \left (a^{5} f - a^{4} b e + a^{3} b^{2} d - a^{2} b^{3} c\right )}{2 a b^{6} + 2 b^{7} x^{2}} + \frac{\sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log{\left (- \frac{b^{6} \sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} - \frac{\sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right ) \log{\left (\frac{b^{6} \sqrt{- \frac{a^{3}}{b^{13}}} \left (11 a^{3} f - 9 a^{2} b e + 7 a b^{2} d - 5 b^{3} c\right )}{11 a^{4} f - 9 a^{3} b e + 7 a^{2} b^{2} d - 5 a b^{3} c} + x \right )}}{4} + \frac{f x^{9}}{9 b^{2}} - \frac{x^{7} \left (2 a f - b e\right )}{7 b^{3}} + \frac{x^{5} \left (3 a^{2} f - 2 a b e + b^{2} d\right )}{5 b^{4}} - \frac{x^{3} \left (4 a^{3} f - 3 a^{2} b e + 2 a b^{2} d - b^{3} c\right )}{3 b^{5}} + \frac{x \left (5 a^{4} f - 4 a^{3} b e + 3 a^{2} b^{2} d - 2 a b^{3} c\right )}{b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(f*x**6+e*x**4+d*x**2+c)/(b*x**2+a)**2,x)

[Out]

x*(a**5*f - a**4*b*e + a**3*b**2*d - a**2*b**3*c)/(2*a*b**6 + 2*b**7*x**2) + sqr
t(-a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b**3*c)*log(-b**6*sqrt(-
a**3/b**13)*(11*a**3*f - 9*a**2*b*e + 7*a*b**2*d - 5*b**3*c)/(11*a**4*f - 9*a**3
*b*e + 7*a**2*b**2*d - 5*a*b**3*c) + x)/4 - sqrt(-a**3/b**13)*(11*a**3*f - 9*a**
2*b*e + 7*a*b**2*d - 5*b**3*c)*log(b**6*sqrt(-a**3/b**13)*(11*a**3*f - 9*a**2*b*
e + 7*a*b**2*d - 5*b**3*c)/(11*a**4*f - 9*a**3*b*e + 7*a**2*b**2*d - 5*a*b**3*c)
+ x)/4 + f*x**9/(9*b**2) - x**7*(2*a*f - b*e)/(7*b**3) + x**5*(3*a**2*f - 2*a*b
*e + b**2*d)/(5*b**4) - x**3*(4*a**3*f - 3*a**2*b*e + 2*a*b**2*d - b**3*c)/(3*b*
*5) + x*(5*a**4*f - 4*a**3*b*e + 3*a**2*b**2*d - 2*a*b**3*c)/b**6

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GIAC/XCAS [A]  time = 0.216489, size = 340, normalized size = 1.42 \[ \frac{{\left (5 \, a^{2} b^{3} c - 7 \, a^{3} b^{2} d - 11 \, a^{5} f + 9 \, a^{4} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} b^{6}} - \frac{a^{2} b^{3} c x - a^{3} b^{2} d x - a^{5} f x + a^{4} b x e}{2 \,{\left (b x^{2} + a\right )} b^{6}} + \frac{35 \, b^{16} f x^{9} - 90 \, a b^{15} f x^{7} + 45 \, b^{16} x^{7} e + 63 \, b^{16} d x^{5} + 189 \, a^{2} b^{14} f x^{5} - 126 \, a b^{15} x^{5} e + 105 \, b^{16} c x^{3} - 210 \, a b^{15} d x^{3} - 420 \, a^{3} b^{13} f x^{3} + 315 \, a^{2} b^{14} x^{3} e - 630 \, a b^{15} c x + 945 \, a^{2} b^{14} d x + 1575 \, a^{4} b^{12} f x - 1260 \, a^{3} b^{13} x e}{315 \, b^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^6 + e*x^4 + d*x^2 + c)*x^6/(b*x^2 + a)^2,x, algorithm="giac")

[Out]

1/2*(5*a^2*b^3*c - 7*a^3*b^2*d - 11*a^5*f + 9*a^4*b*e)*arctan(b*x/sqrt(a*b))/(sq
rt(a*b)*b^6) - 1/2*(a^2*b^3*c*x - a^3*b^2*d*x - a^5*f*x + a^4*b*x*e)/((b*x^2 + a
)*b^6) + 1/315*(35*b^16*f*x^9 - 90*a*b^15*f*x^7 + 45*b^16*x^7*e + 63*b^16*d*x^5
+ 189*a^2*b^14*f*x^5 - 126*a*b^15*x^5*e + 105*b^16*c*x^3 - 210*a*b^15*d*x^3 - 42
0*a^3*b^13*f*x^3 + 315*a^2*b^14*x^3*e - 630*a*b^15*c*x + 945*a^2*b^14*d*x + 1575
*a^4*b^12*f*x - 1260*a^3*b^13*x*e)/b^18