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

Optimal. Leaf size=151 \[ -\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (7 c d^2-e (5 b d-3 a e)\right )}{2 e^{9/2}}+\frac{x \left (7 c d^2-e (5 b d-3 a e)\right )}{2 e^4}-\frac{x^3 \left (7 c d^2-e (5 b d-3 a e)\right )}{6 d e^3}+\frac{x^5 \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x^5}{5 e^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.430298, antiderivative size = 151, 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 \[ -\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (7 c d^2-e (5 b d-3 a e)\right )}{2 e^{9/2}}+\frac{x \left (7 c d^2-e (5 b d-3 a e)\right )}{2 e^4}-\frac{x^3 \left (7 c d^2-e (5 b d-3 a e)\right )}{6 d e^3}+\frac{x^5 \left (a+\frac{d (c d-b e)}{e^2}\right )}{2 d \left (d+e x^2\right )}+\frac{c x^5}{5 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 64.1523, size = 131, normalized size = 0.87 \[ \frac{a x}{e^{2}} - \frac{2 b d x}{e^{3}} + \frac{3 c d^{2} x}{e^{4}} + \frac{c x^{5}}{5 e^{2}} - \frac{\sqrt{d} \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} x}{\sqrt{d}} \right )}}{2 e^{\frac{9}{2}}} + \frac{d x \left (a e^{2} - b d e + c d^{2}\right )}{2 e^{4} \left (d + e x^{2}\right )} + \frac{x^{3} \left (b e - 2 c d\right )}{3 e^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.131685, size = 133, normalized size = 0.88 \[ -\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (3 a e^2-5 b d e+7 c d^2\right )}{2 e^{9/2}}+\frac{x \left (a e^2-2 b d e+3 c d^2\right )}{e^4}+\frac{x \left (a d e^2-b d^2 e+c d^3\right )}{2 e^4 \left (d+e x^2\right )}+\frac{x^3 (b e-2 c d)}{3 e^3}+\frac{c x^5}{5 e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.014, size = 176, normalized size = 1.2 \[{\frac{c{x}^{5}}{5\,{e}^{2}}}+{\frac{b{x}^{3}}{3\,{e}^{2}}}-{\frac{2\,cd{x}^{3}}{3\,{e}^{3}}}+{\frac{ax}{{e}^{2}}}-2\,{\frac{bxd}{{e}^{3}}}+3\,{\frac{c{d}^{2}x}{{e}^{4}}}+{\frac{adx}{2\,{e}^{2} \left ( e{x}^{2}+d \right ) }}-{\frac{xb{d}^{2}}{2\,{e}^{3} \left ( e{x}^{2}+d \right ) }}+{\frac{{d}^{3}xc}{2\,{e}^{4} \left ( e{x}^{2}+d \right ) }}-{\frac{3\,ad}{2\,{e}^{2}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}+{\frac{5\,b{d}^{2}}{2\,{e}^{3}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}}-{\frac{7\,c{d}^{3}}{2\,{e}^{4}}\arctan \left ({ex{\frac{1}{\sqrt{de}}}} \right ){\frac{1}{\sqrt{de}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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^4/(e*x^2 + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.276914, size = 1, normalized size = 0.01 \[ \left [\frac{12 \, c e^{3} x^{7} - 4 \,{\left (7 \, c d e^{2} - 5 \, b e^{3}\right )} x^{5} + 20 \,{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{3} + 15 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2} +{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{2}\right )} \sqrt{-\frac{d}{e}} \log \left (\frac{e x^{2} - 2 \, e x \sqrt{-\frac{d}{e}} - d}{e x^{2} + d}\right ) + 30 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} x}{60 \,{\left (e^{5} x^{2} + d e^{4}\right )}}, \frac{6 \, c e^{3} x^{7} - 2 \,{\left (7 \, c d e^{2} - 5 \, b e^{3}\right )} x^{5} + 10 \,{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{3} - 15 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2} +{\left (7 \, c d^{2} e - 5 \, b d e^{2} + 3 \, a e^{3}\right )} x^{2}\right )} \sqrt{\frac{d}{e}} \arctan \left (\frac{x}{\sqrt{\frac{d}{e}}}\right ) + 15 \,{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} x}{30 \,{\left (e^{5} x^{2} + d e^{4}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/60*(12*c*e^3*x^7 - 4*(7*c*d*e^2 - 5*b*e^3)*x^5 + 20*(7*c*d^2*e - 5*b*d*e^2 +
3*a*e^3)*x^3 + 15*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2 + (7*c*d^2*e - 5*b*d*e^2 + 3*
a*e^3)*x^2)*sqrt(-d/e)*log((e*x^2 - 2*e*x*sqrt(-d/e) - d)/(e*x^2 + d)) + 30*(7*c
*d^3 - 5*b*d^2*e + 3*a*d*e^2)*x)/(e^5*x^2 + d*e^4), 1/30*(6*c*e^3*x^7 - 2*(7*c*d
*e^2 - 5*b*e^3)*x^5 + 10*(7*c*d^2*e - 5*b*d*e^2 + 3*a*e^3)*x^3 - 15*(7*c*d^3 - 5
*b*d^2*e + 3*a*d*e^2 + (7*c*d^2*e - 5*b*d*e^2 + 3*a*e^3)*x^2)*sqrt(d/e)*arctan(x
/sqrt(d/e)) + 15*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2)*x)/(e^5*x^2 + d*e^4)]

_______________________________________________________________________________________

Sympy [A]  time = 5.12089, size = 184, normalized size = 1.22 \[ \frac{c x^{5}}{5 e^{2}} + \frac{x \left (a d e^{2} - b d^{2} e + c d^{3}\right )}{2 d e^{4} + 2 e^{5} x^{2}} + \frac{\sqrt{- \frac{d}{e^{9}}} \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \log{\left (- e^{4} \sqrt{- \frac{d}{e^{9}}} + x \right )}}{4} - \frac{\sqrt{- \frac{d}{e^{9}}} \left (3 a e^{2} - 5 b d e + 7 c d^{2}\right ) \log{\left (e^{4} \sqrt{- \frac{d}{e^{9}}} + x \right )}}{4} + \frac{x^{3} \left (b e - 2 c d\right )}{3 e^{3}} + \frac{x \left (a e^{2} - 2 b d e + 3 c d^{2}\right )}{e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270241, size = 169, normalized size = 1.12 \[ -\frac{{\left (7 \, c d^{3} - 5 \, b d^{2} e + 3 \, a d e^{2}\right )} \arctan \left (\frac{x e^{\frac{1}{2}}}{\sqrt{d}}\right ) e^{\left (-\frac{9}{2}\right )}}{2 \, \sqrt{d}} + \frac{1}{15} \,{\left (3 \, c x^{5} e^{8} - 10 \, c d x^{3} e^{7} + 5 \, b x^{3} e^{8} + 45 \, c d^{2} x e^{6} - 30 \, b d x e^{7} + 15 \, a x e^{8}\right )} e^{\left (-10\right )} + \frac{{\left (c d^{3} x - b d^{2} x e + a d x e^{2}\right )} e^{\left (-4\right )}}{2 \,{\left (x^{2} e + d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(7*c*d^3 - 5*b*d^2*e + 3*a*d*e^2)*arctan(x*e^(1/2)/sqrt(d))*e^(-9/2)/sqrt(d
) + 1/15*(3*c*x^5*e^8 - 10*c*d*x^3*e^7 + 5*b*x^3*e^8 + 45*c*d^2*x*e^6 - 30*b*d*x
*e^7 + 15*a*x*e^8)*e^(-10) + 1/2*(c*d^3*x - b*d^2*x*e + a*d*x*e^2)*e^(-4)/(x^2*e
+ d)