3.61 \(\int \frac{x^7 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=320 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (12 a^2 c^3 e-b^3 c (c d-20 a f)-12 a b^2 c^2 e+6 a b c^2 (c d-5 a f)-3 b^5 f+2 b^4 c e\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{x^4 \left (-2 c (4 a f+b e)+3 b^2 f+4 c^2 d\right )}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (x^2 \left (-\left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )}{4 c^4}+\frac{x^2 \left (-b c (c d-11 a f)-6 a c^2 e-3 b^3 f+2 b^2 c e\right )}{2 c^3 \left (b^2-4 a c\right )} \]

[Out]

((2*b^2*c*e - 6*a*c^2*e - 3*b^3*f - b*c*(c*d - 11*a*f))*x^2)/(2*c^3*(b^2 - 4*a*c
)) + ((4*c^2*d + 3*b^2*f - 2*c*(b*e + 4*a*f))*x^4)/(4*c^2*(b^2 - 4*a*c)) + (x^6*
(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x^2))/(2*c*(b^2 -
4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*c*e - 12*a*b^2*c^2*e + 12*a^2*c^3*e - 3*b
^5*f - b^3*c*(c*d - 20*a*f) + 6*a*b*c^2*(c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqr
t[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(3/2)) + ((c^2*d + 3*b^2*f - 2*c*(b*e + a*
f))*Log[a + b*x^2 + c*x^4])/(4*c^4)

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Rubi [A]  time = 2.46279, antiderivative size = 320, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.233 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (12 a^2 c^3 e-b^3 c (c d-20 a f)-12 a b^2 c^2 e+6 a b c^2 (c d-5 a f)-3 b^5 f+2 b^4 c e\right )}{2 c^4 \left (b^2-4 a c\right )^{3/2}}+\frac{x^4 \left (-2 c (4 a f+b e)+3 b^2 f+4 c^2 d\right )}{4 c^2 \left (b^2-4 a c\right )}+\frac{x^6 \left (x^2 \left (-\left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )-b (a f+c d)+2 a c e\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )}{4 c^4}+\frac{x^2 \left (-b c (c d-11 a f)-6 a c^2 e-3 b^3 f+2 b^2 c e\right )}{2 c^3 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

((2*b^2*c*e - 6*a*c^2*e - 3*b^3*f - b*c*(c*d - 11*a*f))*x^2)/(2*c^3*(b^2 - 4*a*c
)) + ((4*c^2*d + 3*b^2*f - 2*c*(b*e + 4*a*f))*x^4)/(4*c^2*(b^2 - 4*a*c)) + (x^6*
(2*a*c*e - b*(c*d + a*f) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x^2))/(2*c*(b^2 -
4*a*c)*(a + b*x^2 + c*x^4)) - ((2*b^4*c*e - 12*a*b^2*c^2*e + 12*a^2*c^3*e - 3*b
^5*f - b^3*c*(c*d - 20*a*f) + 6*a*b*c^2*(c*d - 5*a*f))*ArcTanh[(b + 2*c*x^2)/Sqr
t[b^2 - 4*a*c]])/(2*c^4*(b^2 - 4*a*c)^(3/2)) + ((c^2*d + 3*b^2*f - 2*c*(b*e + a*
f))*Log[a + b*x^2 + c*x^4])/(4*c^4)

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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**7*(f*x**4+e*x**2+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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Mathematica [A]  time = 1.0414, size = 309, normalized size = 0.97 \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-12 a^2 c^3 e+b^3 c (c d-20 a f)+12 a b^2 c^2 e+6 a b c^2 (5 a f-c d)+3 b^5 f-2 b^4 c e\right )}{\left (4 a c-b^2\right )^{3/2}}+\frac{2 \left (2 a^3 c^2 f+a^2 c \left (-4 b^2 f+b c \left (3 e+5 f x^2\right )-2 c^2 \left (d+e x^2\right )\right )+a b \left (b^3 f-b^2 c \left (e+5 f x^2\right )+b c^2 \left (d+4 e x^2\right )-3 c^3 d x^2\right )+b^3 x^2 \left (b^2 f-b c e+c^2 d\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\log \left (a+b x^2+c x^4\right ) \left (-2 c (a f+b e)+3 b^2 f+c^2 d\right )+2 c x^2 (c e-2 b f)+c^2 f x^4}{4 c^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(d + e*x^2 + f*x^4))/(a + b*x^2 + c*x^4)^2,x]

[Out]

(2*c*(c*e - 2*b*f)*x^2 + c^2*f*x^4 + (2*(2*a^3*c^2*f + b^3*(c^2*d - b*c*e + b^2*
f)*x^2 + a*b*(b^3*f - 3*c^3*d*x^2 + b*c^2*(d + 4*e*x^2) - b^2*c*(e + 5*f*x^2)) +
a^2*c*(-4*b^2*f - 2*c^2*(d + e*x^2) + b*c*(3*e + 5*f*x^2))))/((b^2 - 4*a*c)*(a
+ b*x^2 + c*x^4)) + (2*(-2*b^4*c*e + 12*a*b^2*c^2*e - 12*a^2*c^3*e + 3*b^5*f + b
^3*c*(c*d - 20*a*f) + 6*a*b*c^2*(-(c*d) + 5*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2
+ 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (c^2*d + 3*b^2*f - 2*c*(b*e + a*f))*Log[a + b
*x^2 + c*x^4])/(4*c^4)

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Maple [B]  time = 0.026, size = 1764, normalized size = 5.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(f*x^4+e*x^2+d)/(c*x^4+b*x^2+a)^2,x)

[Out]

1/4/c^2*x^4*f+1/2/c^2*x^2*e-1/c^2/(c*x^4+b*x^2+a)*a^3/(4*a*c-b^2)*f+1/c/(c*x^4+b
*x^2+a)*a^2/(4*a*c-b^2)*d-1/2/c^4/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b^5*f-1/2/c^2/
(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b^3*d-3/2/c^2/(c*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*b*
e+1/2/c^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b^4*e-1/c^3*b*f*x^2+1/c/(4*a*c-b^2)*ln
((4*a*c-b^2)*(c*x^4+b*x^2+a))*a*d-1/c^3/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^
6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+1
2*a*b^4*c-b^6)^(1/2))*b^4*e-3/4/c^4/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*
b^4*f+1/2/c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^3*e-1/4/c^2/(4*a*c-b
^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^2*d-3/c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^
4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2
*c^2+12*a*b^4*c-b^6)^(1/2))*a*b*d-1/2/c^4/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*b^4*f-6/
c/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(
4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*e*a^2+3/2/c^4/(6
4*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*
c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^5*f+1/2/c^2/(64*a^
3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4*a*c-b^
2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*b^3*d-2/c^2/(4*a*c-b^2)*
ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*a^2*f+7/2/c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+
b*x^2+a))*a*b^2*f-2/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*a*b*e-10/c^3
/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(4
*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^3*f+2/c^3/(c*
x^4+b*x^2+a)*a^2/(4*a*c-b^2)*b^2*f+1/2/c^3/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*b^3*e+1
5/c^2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x
^2+(4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a^2*b*f+6/c^
2/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2)*arctan((2*(4*a*c-b^2)*c*x^2+(
4*a*c-b^2)*b)/(64*a^3*c^3-48*a^2*b^2*c^2+12*a*b^4*c-b^6)^(1/2))*a*b^2*e-1/2/c^2/
(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*b^2*d+1/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a^2*e+5/
2/c^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*b^3*f-2/c^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*
x^2*a*b^2*e+3/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*b*d-5/2/c^2/(c*x^4+b*x^2+a)/
(4*a*c-b^2)*x^2*a^2*b*f

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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^4 + e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.657548, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")

[Out]

[-1/4*((((b^3*c^3 - 6*a*b*c^4)*d - 2*(b^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e + (3*
b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*f)*x^4 + ((b^4*c^2 - 6*a*b^2*c^3)*d - 2*(b^
5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e + (3*b^6 - 20*a*b^4*c + 30*a^2*b^2*c^2)*f)*x^
2 + (a*b^3*c^2 - 6*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c^2 + 6*a^3*c^3)*e + (3
*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*f)*log(-(b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^
2)*x^2 - (2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2
+ a)) - ((b^2*c^3 - 4*a*c^4)*f*x^8 + (2*(b^2*c^3 - 4*a*c^4)*e - 3*(b^3*c^2 - 4*
a*b*c^3)*f)*x^6 + (2*(b^3*c^2 - 4*a*b*c^3)*e - (4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c
^3)*f)*x^4 + 2*((b^3*c^2 - 3*a*b*c^3)*d - (b^4*c - 5*a*b^2*c^2 + 6*a^2*c^3)*e +
(b^5 - 7*a*b^3*c + 13*a^2*b*c^2)*f)*x^2 + 2*(a*b^2*c^2 - 2*a^2*c^3)*d - 2*(a*b^3
*c - 3*a^2*b*c^2)*e + 2*(a*b^4 - 4*a^2*b^2*c + 2*a^3*c^2)*f + (((b^2*c^3 - 4*a*c
^4)*d - 2*(b^3*c^2 - 4*a*b*c^3)*e + (3*b^4*c - 14*a*b^2*c^2 + 8*a^2*c^3)*f)*x^4
+ ((b^3*c^2 - 4*a*b*c^3)*d - 2*(b^4*c - 4*a*b^2*c^2)*e + (3*b^5 - 14*a*b^3*c + 8
*a^2*b*c^2)*f)*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d - 2*(a*b^3*c - 4*a^2*b*c^2)*e + (
3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*f)*log(c*x^4 + b*x^2 + a))*sqrt(b^2 - 4*a*c)
)/((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)*x^4 + (b^3*c^4 - 4*a*b*c^5)*x^2)
*sqrt(b^2 - 4*a*c)), -1/4*(2*(((b^3*c^3 - 6*a*b*c^4)*d - 2*(b^4*c^2 - 6*a*b^2*c^
3 + 6*a^2*c^4)*e + (3*b^5*c - 20*a*b^3*c^2 + 30*a^2*b*c^3)*f)*x^4 + ((b^4*c^2 -
6*a*b^2*c^3)*d - 2*(b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e + (3*b^6 - 20*a*b^4*c +
30*a^2*b^2*c^2)*f)*x^2 + (a*b^3*c^2 - 6*a^2*b*c^3)*d - 2*(a*b^4*c - 6*a^2*b^2*c
^2 + 6*a^3*c^3)*e + (3*a*b^5 - 20*a^2*b^3*c + 30*a^3*b*c^2)*f)*arctan(-(2*c*x^2
+ b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - ((b^2*c^3 - 4*a*c^4)*f*x^8 + (2*(b^2*c^
3 - 4*a*c^4)*e - 3*(b^3*c^2 - 4*a*b*c^3)*f)*x^6 + (2*(b^3*c^2 - 4*a*b*c^3)*e - (
4*b^4*c - 17*a*b^2*c^2 + 4*a^2*c^3)*f)*x^4 + 2*((b^3*c^2 - 3*a*b*c^3)*d - (b^4*c
- 5*a*b^2*c^2 + 6*a^2*c^3)*e + (b^5 - 7*a*b^3*c + 13*a^2*b*c^2)*f)*x^2 + 2*(a*b
^2*c^2 - 2*a^2*c^3)*d - 2*(a*b^3*c - 3*a^2*b*c^2)*e + 2*(a*b^4 - 4*a^2*b^2*c + 2
*a^3*c^2)*f + (((b^2*c^3 - 4*a*c^4)*d - 2*(b^3*c^2 - 4*a*b*c^3)*e + (3*b^4*c - 1
4*a*b^2*c^2 + 8*a^2*c^3)*f)*x^4 + ((b^3*c^2 - 4*a*b*c^3)*d - 2*(b^4*c - 4*a*b^2*
c^2)*e + (3*b^5 - 14*a*b^3*c + 8*a^2*b*c^2)*f)*x^2 + (a*b^2*c^2 - 4*a^2*c^3)*d -
2*(a*b^3*c - 4*a^2*b*c^2)*e + (3*a*b^4 - 14*a^2*b^2*c + 8*a^3*c^2)*f)*log(c*x^4
+ b*x^2 + a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c^4 - 4*a^2*c^5 + (b^2*c^5 - 4*a*c^6)
*x^4 + (b^3*c^4 - 4*a*b*c^5)*x^2)*sqrt(-b^2 + 4*a*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(f*x**4+e*x**2+d)/(c*x**4+b*x**2+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((f*x^4 + e*x^2 + d)*x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")

[Out]

Exception raised: TypeError