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

Optimal. Leaf size=181 \[ -\frac{d x \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^5}+\frac{x^3 \left (9 c d^2-e (7 b d-5 a e)\right )}{6 e^4}-\frac{x^5 \left (9 c d^2-e (7 b d-5 a e)\right )}{10 d e^3}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^{11/2}}+\frac{x^7 \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x^7}{7 e^2} \]

[Out]

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

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Rubi [A]  time = 0.505352, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{d x \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^5}+\frac{x^3 \left (9 c d^2-e (7 b d-5 a e)\right )}{6 e^4}-\frac{x^5 \left (9 c d^2-e (7 b d-5 a e)\right )}{10 d e^3}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (9 c d^2-e (7 b d-5 a e)\right )}{2 e^{11/2}}+\frac{x^7 \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x^7}{7 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.3138, size = 158, normalized size = 0.87 \[ \frac{c x^{7}}{7 e^{2}} + \frac{d^{\frac{3}{2}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 e^{\frac{11}{2}}} - \frac{d^{2} x \left (a e^{2} - b d e + c d^{2}\right )}{2 e^{5} \left (d + e x^{2}\right )} - \frac{d x \left (2 a e^{2} - 3 b d e + 4 c d^{2}\right )}{e^{5}} + \frac{x^{5} \left (b e - 2 c d\right )}{5 e^{3}} + \frac{x^{3} \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{3 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*x**7/(7*e**2) + d**(3/2)*(5*a*e**2 - 7*b*d*e + 9*c*d**2)*atan(sqrt(e)*x/sqrt(d
))/(2*e**(11/2)) - d**2*x*(a*e**2 - b*d*e + c*d**2)/(2*e**5*(d + e*x**2)) - d*x*
(2*a*e**2 - 3*b*d*e + 4*c*d**2)/e**5 + x**5*(b*e - 2*c*d)/(5*e**3) + x**3*(a*e**
2 - 2*b*d*e + 3*c*d**2)/(3*e**4)

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Mathematica [A]  time = 0.264586, size = 165, normalized size = 0.91 \[ -\frac{d x \left (2 a e^2-3 b d e+4 c d^2\right )}{e^5}+\frac{x^3 \left (a e^2-2 b d e+3 c d^2\right )}{3 e^4}+\frac{d^{3/2} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (5 a e^2-7 b d e+9 c d^2\right )}{2 e^{11/2}}-\frac{x \left (a d^2 e^2-b d^3 e+c d^4\right )}{2 e^5 \left (d+e x^2\right )}+\frac{x^5 (b e-2 c d)}{5 e^3}+\frac{c x^7}{7 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 214, normalized size = 1.2 \[{\frac{c{x}^{7}}{7\,{e}^{2}}}+{\frac{{x}^{5}b}{5\,{e}^{2}}}-{\frac{2\,cd{x}^{5}}{5\,{e}^{3}}}+{\frac{{x}^{3}a}{3\,{e}^{2}}}-{\frac{2\,b{x}^{3}d}{3\,{e}^{3}}}+{\frac{{x}^{3}c{d}^{2}}{{e}^{4}}}-2\,{\frac{adx}{{e}^{3}}}+3\,{\frac{xb{d}^{2}}{{e}^{4}}}-4\,{\frac{c{d}^{3}x}{{e}^{5}}}-{\frac{a{d}^{2}x}{2\,{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{{d}^{3}xb}{2\,{e}^{4} \left ( e{x}^{2}+d \right ) }}-{\frac{{d}^{4}xc}{2\,{e}^{5} \left ( e{x}^{2}+d \right ) }}+{\frac{5\,a{d}^{2}}{2\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{7\,b{d}^{3}}{2\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{9\,c{d}^{4}}{2\,{e}^{5}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/7*c*x^7/e^2+1/5/e^2*x^5*b-2/5/e^3*c*d*x^5+1/3/e^2*x^3*a-2/3/e^3*x^3*b*d+1/e^4*
x^3*c*d^2-2/e^3*d*a*x+3/e^4*b*d^2*x-4/e^5*d^3*c*x-1/2*d^2/e^3*x/(e*x^2+d)*a+1/2*
d^3/e^4*x/(e*x^2+d)*b-1/2*d^4/e^5*x/(e*x^2+d)*c+5/2*d^2/e^3/(d*e)^(1/2)*arctan(x
*e/(d*e)^(1/2))*a-7/2*d^3/e^4/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*b+9/2*d^4/e^5/
(d*e)^(1/2)*arctan(x*e/(d*e)^(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((c*x^4 + b*x^2 + a)*x^6/(e*x^2 + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265578, size = 1, normalized size = 0.01 \[ \left [\frac{60 \, c e^{4} x^{9} - 12 \,{\left (9 \, c d e^{3} - 7 \, b e^{4}\right )} x^{7} + 28 \,{\left (9 \, c d^{2} e^{2} - 7 \, b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 140 \,{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3} + 105 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2} +{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{2}\right )} \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} + 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) - 210 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} x}{420 \,{\left (e^{6} x^{2} + d e^{5}\right )}}, \frac{30 \, c e^{4} x^{9} - 6 \,{\left (9 \, c d e^{3} - 7 \, b e^{4}\right )} x^{7} + 14 \,{\left (9 \, c d^{2} e^{2} - 7 \, b d e^{3} + 5 \, a e^{4}\right )} x^{5} - 70 \,{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{3} + 105 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2} +{\left (9 \, c d^{3} e - 7 \, b d^{2} e^{2} + 5 \, a d e^{3}\right )} x^{2}\right )} \sqrt{\frac{d}{e}} \arctan \left (\frac{x}{\sqrt{\frac{d}{e}}}\right ) - 105 \,{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} x}{210 \,{\left (e^{6} x^{2} + d e^{5}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/420*(60*c*e^4*x^9 - 12*(9*c*d*e^3 - 7*b*e^4)*x^7 + 28*(9*c*d^2*e^2 - 7*b*d*e^
3 + 5*a*e^4)*x^5 - 140*(9*c*d^3*e - 7*b*d^2*e^2 + 5*a*d*e^3)*x^3 + 105*(9*c*d^4
- 7*b*d^3*e + 5*a*d^2*e^2 + (9*c*d^3*e - 7*b*d^2*e^2 + 5*a*d*e^3)*x^2)*sqrt(-d/e
)*log((e*x^2 + 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) - 210*(9*c*d^4 - 7*b*d^3*e + 5
*a*d^2*e^2)*x)/(e^6*x^2 + d*e^5), 1/210*(30*c*e^4*x^9 - 6*(9*c*d*e^3 - 7*b*e^4)*
x^7 + 14*(9*c*d^2*e^2 - 7*b*d*e^3 + 5*a*e^4)*x^5 - 70*(9*c*d^3*e - 7*b*d^2*e^2 +
5*a*d*e^3)*x^3 + 105*(9*c*d^4 - 7*b*d^3*e + 5*a*d^2*e^2 + (9*c*d^3*e - 7*b*d^2*
e^2 + 5*a*d*e^3)*x^2)*sqrt(d/e)*arctan(x/sqrt(d/e)) - 105*(9*c*d^4 - 7*b*d^3*e +
5*a*d^2*e^2)*x)/(e^6*x^2 + d*e^5)]

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Sympy [A]  time = 5.5182, size = 313, normalized size = 1.73 \[ \frac{c x^{7}}{7 e^{2}} - \frac{x \left (a d^{2} e^{2} - b d^{3} e + c d^{4}\right )}{2 d e^{5} + 2 e^{6} x^{2}} - \frac{\sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log{\left (- \frac{e^{5} \sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} + \frac{\sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right ) \log{\left (\frac{e^{5} \sqrt{- \frac{d^{3}}{e^{11}}} \left (5 a e^{2} - 7 b d e + 9 c d^{2}\right )}{5 a d e^{2} - 7 b d^{2} e + 9 c d^{3}} + x \right )}}{4} + \frac{x^{5} \left (b e - 2 c d\right )}{5 e^{3}} + \frac{x^{3} \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{3 e^{4}} - \frac{x \left (2 a d e^{2} - 3 b d^{2} e + 4 c d^{3}\right )}{e^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.270438, size = 216, normalized size = 1.19 \[ \frac{{\left (9 \, c d^{4} - 7 \, b d^{3} e + 5 \, a d^{2} e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{11}{2}\right )}}{2 \, \sqrt{d}} + \frac{1}{105} \,{\left (15 \, c x^{7} e^{12} - 42 \, c d x^{5} e^{11} + 21 \, b x^{5} e^{12} + 105 \, c d^{2} x^{3} e^{10} - 70 \, b d x^{3} e^{11} - 420 \, c d^{3} x e^{9} + 35 \, a x^{3} e^{12} + 315 \, b d^{2} x e^{10} - 210 \, a d x e^{11}\right )} e^{\left (-14\right )} - \frac{{\left (c d^{4} x - b d^{3} x e + a d^{2} x e^{2}\right )} e^{\left (-5\right )}}{2 \,{\left (x^{2} e + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(9*c*d^4 - 7*b*d^3*e + 5*a*d^2*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-11/2)/sqrt
(d) + 1/105*(15*c*x^7*e^12 - 42*c*d*x^5*e^11 + 21*b*x^5*e^12 + 105*c*d^2*x^3*e^1
0 - 70*b*d*x^3*e^11 - 420*c*d^3*x*e^9 + 35*a*x^3*e^12 + 315*b*d^2*x*e^10 - 210*a
*d*x*e^11)*e^(-14) - 1/2*(c*d^4*x - b*d^3*x*e + a*d^2*x*e^2)*e^(-5)/(x^2*e + d)