3.873 \(\int \frac{x^{11}}{\left (a+b x^2+c x^4\right )^3} \, dx\)
Optimal. Leaf size=209 \[ \frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{b x^2 \left (b^2-7 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^4 \left (b x^2 \left (b^2-10 a c\right )+a \left (b^2-16 a c\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^3} \]
[Out]
-(b*(b^2 - 7*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) + (x^8*(2*a + b*x^2))/(4*(b^2 - 4
*a*c)*(a + b*x^2 + c*x^4)^2) + (x^4*(a*(b^2 - 16*a*c) + b*(b^2 - 10*a*c)*x^2))/(
4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*Ar
cTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + Log[a + b*
x^2 + c*x^4]/(4*c^3)
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Rubi [A] time = 0.840838, antiderivative size = 209, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444
\[ \frac{b \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{5/2}}-\frac{b x^2 \left (b^2-7 a c\right )}{2 c^2 \left (b^2-4 a c\right )^2}+\frac{x^4 \left (b x^2 \left (b^2-10 a c\right )+a \left (b^2-16 a c\right )\right )}{4 c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{x^8 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{\log \left (a+b x^2+c x^4\right )}{4 c^3} \]
Antiderivative was successfully verified.
[In] Int[x^11/(a + b*x^2 + c*x^4)^3,x]
[Out]
-(b*(b^2 - 7*a*c)*x^2)/(2*c^2*(b^2 - 4*a*c)^2) + (x^8*(2*a + b*x^2))/(4*(b^2 - 4
*a*c)*(a + b*x^2 + c*x^4)^2) + (x^4*(a*(b^2 - 16*a*c) + b*(b^2 - 10*a*c)*x^2))/(
4*c*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (b*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*Ar
cTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(5/2)) + Log[a + b*
x^2 + c*x^4]/(4*c^3)
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{b \left (30 a^{2} c^{2} - 10 a b^{2} c + b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{3} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{x^{8} \left (2 a + b x^{2}\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{x^{4} \left (a \left (- 16 a c + b^{2}\right ) + b x^{2} \left (- 10 a c + b^{2}\right )\right )}{4 c \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} - \frac{\left (- 7 a c + b^{2}\right ) \int ^{x^{2}} b\, dx}{2 c^{2} \left (- 4 a c + b^{2}\right )^{2}} + \frac{\log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**11/(c*x**4+b*x**2+a)**3,x)
[Out]
b*(30*a**2*c**2 - 10*a*b**2*c + b**4)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/
(2*c**3*(-4*a*c + b**2)**(5/2)) + x**8*(2*a + b*x**2)/(4*(-4*a*c + b**2)*(a + b*
x**2 + c*x**4)**2) + x**4*(a*(-16*a*c + b**2) + b*x**2*(-10*a*c + b**2))/(4*c*(-
4*a*c + b**2)**2*(a + b*x**2 + c*x**4)) - (-7*a*c + b**2)*Integral(b, (x, x**2))
/(2*c**2*(-4*a*c + b**2)**2) + log(a + b*x**2 + c*x**4)/(4*c**3)
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Mathematica [A] time = 0.584226, size = 244, normalized size = 1.17 \[ \frac{-\frac{2 b c \left (30 a^2 c^2-10 a b^2 c+b^4\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{2 a^3 c^2+a^2 b c \left (5 c x^2-4 b\right )+a b^3 \left (b-5 c x^2\right )+b^5 x^2}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{32 a^3 c^3-39 a^2 b^2 c^2+50 a^2 b c^3 x^2+11 a b^4 c-30 a b^3 c^2 x^2-b^6+4 b^5 c x^2}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+c \log \left (a+b x^2+c x^4\right )}{4 c^4} \]
Antiderivative was successfully verified.
[In] Integrate[x^11/(a + b*x^2 + c*x^4)^3,x]
[Out]
((-b^6 + 11*a*b^4*c - 39*a^2*b^2*c^2 + 32*a^3*c^3 + 4*b^5*c*x^2 - 30*a*b^3*c^2*x
^2 + 50*a^2*b*c^3*x^2)/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) + (2*a^3*c^2 + b^5*
x^2 + a*b^3*(b - 5*c*x^2) + a^2*b*c*(-4*b + 5*c*x^2))/((b^2 - 4*a*c)*(a + b*x^2
+ c*x^4)^2) - (2*b*c*(b^4 - 10*a*b^2*c + 30*a^2*c^2)*ArcTan[(b + 2*c*x^2)/Sqrt[-
b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(5/2) + c*Log[a + b*x^2 + c*x^4])/(4*c^4)
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Maple [B] time = 0.033, size = 819, normalized size = 3.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^11/(c*x^4+b*x^2+a)^3,x)
[Out]
1/2*(1/c^2*b*(25*a^2*c^2-15*a*b^2*c+2*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+1/2*(3
2*a^3*c^3+11*a^2*b^2*c^2-19*a*b^4*c+3*b^6)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4+a*
b*(31*a^2*c^2-22*a*b^2*c+3*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c^3*x^2+3/2*a^2*(8*a^
2*c^2-7*a*b^2*c+b^4)/c^3/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2+1/4/c^3*l
n(c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))-15/(1024*a^5*c^9-1280*a^4*b^2*
c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a
^2*c^2-8*a*b^2*c+b^4)*c^3*x^2+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-12
80*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*a^2
*b*c+5/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c
^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b^2*c+b^4)*c^3*x^2+c^2*(16*a^2*c^2-
8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6
+20*a*b^8*c^5-b^10*c^4)^(1/2))*a*b^3-1/2/(1024*a^5*c^9-1280*a^4*b^2*c^8+640*a^3*
b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2)*arctan((2*(16*a^2*c^2-8*a*b
^2*c+b^4)*c^3*x^2+c^2*(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^9-1280*a^4*b^2*c
^8+640*a^3*b^4*c^7-160*a^2*b^6*c^6+20*a*b^8*c^5-b^10*c^4)^(1/2))*b^5/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(x^11/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.298331, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")
[Out]
[1/4*(((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^2*b^5 - 10*a^3*b^3*c + 30
*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*x^6 + (b^7 - 8*a*b^5*c +
10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^4 + 2*(a*b^6 - 10*a^2*b^4*c + 30*a^3*b^2*c^2)*x
^2)*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)) + (2*(2*b^5*c - 15*a*b^3*c^2 +
25*a^2*b*c^3)*x^6 + 3*a^2*b^4 - 21*a^3*b^2*c + 24*a^4*c^2 + (3*b^6 - 19*a*b^4*c
+ 11*a^2*b^2*c^2 + 32*a^3*c^3)*x^4 + 2*(3*a*b^5 - 22*a^2*b^3*c + 31*a^3*b*c^2)*x
^2 + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2
*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3
)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt
(b^2 - 4*a*c))/((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c
^6 + 16*a^2*c^7)*x^8 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^6 + (b^6*c^3 -
6*a*b^4*c^4 + 32*a^3*c^6)*x^4 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x^
2)*sqrt(b^2 - 4*a*c)), -1/4*(2*((b^5*c^2 - 10*a*b^3*c^3 + 30*a^2*b*c^4)*x^8 + a^
2*b^5 - 10*a^3*b^3*c + 30*a^4*b*c^2 + 2*(b^6*c - 10*a*b^4*c^2 + 30*a^2*b^2*c^3)*
x^6 + (b^7 - 8*a*b^5*c + 10*a^2*b^3*c^2 + 60*a^3*b*c^3)*x^4 + 2*(a*b^6 - 10*a^2*
b^4*c + 30*a^3*b^2*c^2)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a
*c)) - (2*(2*b^5*c - 15*a*b^3*c^2 + 25*a^2*b*c^3)*x^6 + 3*a^2*b^4 - 21*a^3*b^2*c
+ 24*a^4*c^2 + (3*b^6 - 19*a*b^4*c + 11*a^2*b^2*c^2 + 32*a^3*c^3)*x^4 + 2*(3*a*
b^5 - 22*a^2*b^3*c + 31*a^3*b*c^2)*x^2 + ((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*x
^8 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*x^6 + a^2*b^4 - 8*a^3*b^2*c + 16*a^4
*c^2 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*x^4 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^
2)*x^2)*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4*a*c))/((a^2*b^4*c^3 - 8*a^3*b^2*c^
4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^8 + 2*(b^5*c^4 - 8*a*b^3
*c^5 + 16*a^2*b*c^6)*x^6 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^4 + 2*(a*b^5*c
^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x^2)*sqrt(-b^2 + 4*a*c))]
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Sympy [A] time = 54.5537, size = 1520, normalized size = 7.27 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**11/(c*x**4+b*x**2+a)**3,x)
[Out]
(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024*a
**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*
b**8*c - b**10)) + 1/(4*c**3))*log(x**2 + (-128*a**3*c**5*(-b*sqrt(-(4*a*c - b**
2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024*a**5*c**5 - 1280*a**4*b
**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 1/(
4*c**3)) + 32*a**3*c**2 + 96*a**2*b**2*c**4*(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**
2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640
*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 1/(4*c**3)) - 9*a
**2*b**2*c - 24*a*b**4*c**3*(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b*
*2*c + b**4)/(4*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3
- 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 1/(4*c**3)) + a*b**4 + 2*b**6*c**
2*(-b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024
*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*
a*b**8*c - b**10)) + 1/(4*c**3)))/(30*a**2*b*c**2 - 10*a*b**3*c + b**5)) + (b*sq
rt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024*a**5*c*
*5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c
- b**10)) + 1/(4*c**3))*log(x**2 + (-128*a**3*c**5*(b*sqrt(-(4*a*c - b**2)**5)*
(30*a**2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**
4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 1/(4*c**3)
) + 32*a**3*c**2 + 96*a**2*b**2*c**4*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 -
10*a*b**2*c + b**4)/(4*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b*
*4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c - b**10)) + 1/(4*c**3)) - 9*a**2*b**2
*c - 24*a*b**4*c**3*(b*sqrt(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b*
*4)/(4*c**3*(1024*a**5*c**5 - 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**
2*b**6*c**2 + 20*a*b**8*c - b**10)) + 1/(4*c**3)) + a*b**4 + 2*b**6*c**2*(b*sqrt
(-(4*a*c - b**2)**5)*(30*a**2*c**2 - 10*a*b**2*c + b**4)/(4*c**3*(1024*a**5*c**5
- 1280*a**4*b**2*c**4 + 640*a**3*b**4*c**3 - 160*a**2*b**6*c**2 + 20*a*b**8*c -
b**10)) + 1/(4*c**3)))/(30*a**2*b*c**2 - 10*a*b**3*c + b**5)) + (24*a**4*c**2 -
21*a**3*b**2*c + 3*a**2*b**4 + x**6*(50*a**2*b*c**3 - 30*a*b**3*c**2 + 4*b**5*c
) + x**4*(32*a**3*c**3 + 11*a**2*b**2*c**2 - 19*a*b**4*c + 3*b**6) + x**2*(62*a*
*3*b*c**2 - 44*a**2*b**3*c + 6*a*b**5))/(64*a**4*c**5 - 32*a**3*b**2*c**4 + 4*a*
*2*b**4*c**3 + x**8*(64*a**2*c**7 - 32*a*b**2*c**6 + 4*b**4*c**5) + x**6*(128*a*
*2*b*c**6 - 64*a*b**3*c**5 + 8*b**5*c**4) + x**4*(128*a**3*c**6 - 24*a*b**4*c**4
+ 4*b**6*c**3) + x**2*(128*a**3*b*c**5 - 64*a**2*b**3*c**4 + 8*a*b**5*c**3))
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GIAC/XCAS [A] time = 15.6605, size = 413, normalized size = 1.98 \[ -\frac{{\left (b^{5} - 10 \, a b^{3} c + 30 \, a^{2} b c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{3 \, b^{4} c^{2} x^{8} - 24 \, a b^{2} c^{3} x^{8} + 48 \, a^{2} c^{4} x^{8} - 2 \, b^{5} c x^{6} + 12 \, a b^{3} c^{2} x^{6} - 4 \, a^{2} b c^{3} x^{6} - 3 \, b^{6} x^{4} + 20 \, a b^{4} c x^{4} - 22 \, a^{2} b^{2} c^{2} x^{4} + 32 \, a^{3} c^{3} x^{4} - 6 \, a b^{5} x^{2} + 40 \, a^{2} b^{3} c x^{2} - 28 \, a^{3} b c^{2} x^{2} - 3 \, a^{2} b^{4} + 18 \, a^{3} b^{2} c}{8 \,{\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} + \frac{{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^11/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")
[Out]
-1/2*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/
((b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*sqrt(-b^2 + 4*a*c)) - 1/8*(3*b^4*c^2*x^8 -
24*a*b^2*c^3*x^8 + 48*a^2*c^4*x^8 - 2*b^5*c*x^6 + 12*a*b^3*c^2*x^6 - 4*a^2*b*c^
3*x^6 - 3*b^6*x^4 + 20*a*b^4*c*x^4 - 22*a^2*b^2*c^2*x^4 + 32*a^3*c^3*x^4 - 6*a*b
^5*x^2 + 40*a^2*b^3*c*x^2 - 28*a^3*b*c^2*x^2 - 3*a^2*b^4 + 18*a^3*b^2*c)/((b^4*c
^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*(c*x^4 + b*x^2 + a)^2) + 1/4*ln(c*x^4 + b*x^2 + a
)/c^3