3.112 \(\int \frac{x^7 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\)
Optimal. Leaf size=212 \[ -\frac{\left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (-6 a B c-A b c+2 b^2 B\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
[Out]
((2*b^2*B - A*b*c - 6*a*B*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (x^4*(a*(b*B - 2*A*c)
+ (b^2*B - A*b*c - 2*a*B*c)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*
b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)*ArcTanh[(b + 2*c*x^
2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*B - A*c)*Log[a + b*x^
2 + c*x^4])/(4*c^3)
_______________________________________________________________________________________
Rubi [A] time = 0.761874, antiderivative size = 212, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28
\[ -\frac{\left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{x^2 \left (-6 a B c-A b c+2 b^2 B\right )}{2 c^2 \left (b^2-4 a c\right )}-\frac{x^4 \left (x^2 \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{(2 b B-A c) \log \left (a+b x^2+c x^4\right )}{4 c^3} \]
Antiderivative was successfully verified.
[In] Int[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]
[Out]
((2*b^2*B - A*b*c - 6*a*B*c)*x^2)/(2*c^2*(b^2 - 4*a*c)) - (x^4*(a*(b*B - 2*A*c)
+ (b^2*B - A*b*c - 2*a*B*c)*x^2))/(2*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - ((2*
b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)*ArcTanh[(b + 2*c*x^
2)/Sqrt[b^2 - 4*a*c]])/(2*c^3*(b^2 - 4*a*c)^(3/2)) - ((2*b*B - A*c)*Log[a + b*x^
2 + c*x^4])/(4*c^3)
_______________________________________________________________________________________
Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{3 B a x^{2}}{c \left (- 4 a c + b^{2}\right )} + \frac{B b^{2} x^{2}}{c^{2} \left (- 4 a c + b^{2}\right )} - \frac{b \int ^{x^{2}} A\, dx}{2 c \left (- 4 a c + b^{2}\right )} + \frac{x^{4} \left (a \left (2 A c - B b\right ) - x^{2} \left (- A b c - 2 B a c + B b^{2}\right )\right )}{2 c \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} + \frac{\left (A c - 2 B b\right ) \log{\left (a + b x^{2} + c x^{4} \right )}}{4 c^{3}} - \frac{\left (6 A a b c^{2} - A b^{3} c + 12 B a^{2} c^{2} - 12 B a b^{2} c + 2 B b^{4}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 c^{3} \left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**7*(B*x**2+A)/(c*x**4+b*x**2+a)**2,x)
[Out]
-3*B*a*x**2/(c*(-4*a*c + b**2)) + B*b**2*x**2/(c**2*(-4*a*c + b**2)) - b*Integra
l(A, (x, x**2))/(2*c*(-4*a*c + b**2)) + x**4*(a*(2*A*c - B*b) - x**2*(-A*b*c - 2
*B*a*c + B*b**2))/(2*c*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)) + (A*c - 2*B*b)*lo
g(a + b*x**2 + c*x**4)/(4*c**3) - (6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12
*B*a*b**2*c + 2*B*b**4)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*c**3*(-4*a*
c + b**2)**(3/2))
_______________________________________________________________________________________
Mathematica [A] time = 0.52633, size = 208, normalized size = 0.98 \[ \frac{-\frac{2 \left (a^2 c \left (2 c \left (A+B x^2\right )-3 b B\right )+a b \left (-b c \left (A+4 B x^2\right )+3 A c^2 x^2+b^2 B\right )+b^3 x^2 (b B-A c)\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{2 \left (12 a^2 B c^2+6 a A b c^2-12 a b^2 B c-A b^3 c+2 b^4 B\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+(A c-2 b B) \log \left (a+b x^2+c x^4\right )+2 B c x^2}{4 c^3} \]
Antiderivative was successfully verified.
[In] Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^2,x]
[Out]
(2*B*c*x^2 - (2*(b^3*(b*B - A*c)*x^2 + a^2*c*(-3*b*B + 2*c*(A + B*x^2)) + a*b*(b
^2*B + 3*A*c^2*x^2 - b*c*(A + 4*B*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) -
(2*(2*b^4*B - A*b^3*c - 12*a*b^2*B*c + 6*a*A*b*c^2 + 12*a^2*B*c^2)*ArcTan[(b + 2
*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + (-2*b*B + A*c)*Log[a + b*x^2
+ c*x^4])/(4*c^3)
_______________________________________________________________________________________
Maple [B] time = 0.026, size = 1058, normalized size = 5. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^7*(B*x^2+A)/(c*x^4+b*x^2+a)^2,x)
[Out]
1/2*B*x^2/c^2+3/2/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a*A*b-1/2/c^2/(c*x^4+b*x^2+a
)/(4*a*c-b^2)*x^2*A*b^3+1/c/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*a^2*B-2/c^2/(c*x^4+b
*x^2+a)/(4*a*c-b^2)*x^2*a*b^2*B+1/2/c^3/(c*x^4+b*x^2+a)/(4*a*c-b^2)*x^2*b^4*B+1/
c/(c*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*A-1/2/c^2/(c*x^4+b*x^2+a)*a/(4*a*c-b^2)*A*b^2-
3/2/c^2/(c*x^4+b*x^2+a)*a^2/(4*a*c-b^2)*b*B+1/2/c^3/(c*x^4+b*x^2+a)*a/(4*a*c-b^2
)*b^3*B+1/c/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*a*A-1/4/c^2/(4*a*c-b^2)*
ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*A*b^2-2/c^2/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*
x^2+a))*a*b*B+1/2/c^3/(4*a*c-b^2)*ln((4*a*c-b^2)*(c*x^4+b*x^2+a))*b^3*B-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*a*b-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))*a^2*B+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))*B*a*b^2+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*A-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+12*a*b^4*c-b^6)^(1/2))*b^4*B
_______________________________________________________________________________________
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((B*x^2 + A)*x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
_______________________________________________________________________________________
Fricas [A] time = 0.350885, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="fricas")
[Out]
[-1/4*((2*B*a*b^4 + (2*B*b^4*c + 6*(2*B*a^2 + A*a*b)*c^3 - (12*B*a*b^2 + A*b^3)*
c^2)*x^4 + 6*(2*B*a^3 + A*a^2*b)*c^2 + (2*B*b^5 + 6*(2*B*a^2*b + A*a*b^2)*c^2 -
(12*B*a*b^3 + A*b^4)*c)*x^2 - (12*B*a^2*b^2 + A*a*b^3)*c)*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*(B*b^2*c^2 - 4*B*a*c^3)*x^6 - 2*B*a*b^3 - 4*A*a^2*c^
2 + 2*(B*b^3*c - 4*B*a*b*c^2)*x^4 - 2*(B*b^4 + 3*(2*B*a^2 + A*a*b)*c^2 - (5*B*a*
b^2 + A*b^3)*c)*x^2 + 2*(3*B*a^2*b + A*a*b^2)*c - (2*B*a*b^3 + 4*A*a^2*c^2 + (2*
B*b^3*c + 4*A*a*c^3 - (8*B*a*b + A*b^2)*c^2)*x^4 + (2*B*b^4 + 4*A*a*b*c^2 - (8*B
*a*b^2 + A*b^3)*c)*x^2 - (8*B*a^2*b + A*a*b^2)*c)*log(c*x^4 + b*x^2 + a))*sqrt(b
^2 - 4*a*c))/((a*b^2*c^3 - 4*a^2*c^4 + (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*
b*c^4)*x^2)*sqrt(b^2 - 4*a*c)), 1/4*(2*(2*B*a*b^4 + (2*B*b^4*c + 6*(2*B*a^2 + A*
a*b)*c^3 - (12*B*a*b^2 + A*b^3)*c^2)*x^4 + 6*(2*B*a^3 + A*a^2*b)*c^2 + (2*B*b^5
+ 6*(2*B*a^2*b + A*a*b^2)*c^2 - (12*B*a*b^3 + A*b^4)*c)*x^2 - (12*B*a^2*b^2 + A*
a*b^3)*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (2*(B*b^2*c^
2 - 4*B*a*c^3)*x^6 - 2*B*a*b^3 - 4*A*a^2*c^2 + 2*(B*b^3*c - 4*B*a*b*c^2)*x^4 - 2
*(B*b^4 + 3*(2*B*a^2 + A*a*b)*c^2 - (5*B*a*b^2 + A*b^3)*c)*x^2 + 2*(3*B*a^2*b +
A*a*b^2)*c - (2*B*a*b^3 + 4*A*a^2*c^2 + (2*B*b^3*c + 4*A*a*c^3 - (8*B*a*b + A*b^
2)*c^2)*x^4 + (2*B*b^4 + 4*A*a*b*c^2 - (8*B*a*b^2 + A*b^3)*c)*x^2 - (8*B*a^2*b +
A*a*b^2)*c)*log(c*x^4 + b*x^2 + a))*sqrt(-b^2 + 4*a*c))/((a*b^2*c^3 - 4*a^2*c^4
+ (b^2*c^4 - 4*a*c^5)*x^4 + (b^3*c^3 - 4*a*b*c^4)*x^2)*sqrt(-b^2 + 4*a*c))]
_______________________________________________________________________________________
Sympy [A] time = 103.608, size = 1266, normalized size = 5.97 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**7*(B*x**2+A)/(c*x**4+b*x**2+a)**2,x)
[Out]
B*x**2/(2*c**2) + (-sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**
2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 1
2*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3))*log(x**2 + (8*A*a**2*c**2 - A*a*b
**2*c - 10*B*a**2*b*c + 2*B*a*b**3 - 32*a**2*c**4*(-sqrt(-(4*a*c - b**2)**3)*(6*
A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a
**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)) +
16*a*b**2*c**3*(-sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*
c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*
a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)) - 2*b**4*c**2*(-sqrt(-(4*a*c - b**2
)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c
**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4
*c**3)))/(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4))
+ (sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b
**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6
)) - (-A*c + 2*B*b)/(4*c**3))*log(x**2 + (8*A*a**2*c**2 - A*a*b**2*c - 10*B*a**2
*b*c + 2*B*a*b**3 - 32*a**2*c**4*(sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**
3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2
*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)) + 16*a*b**2*c**3*(s
qrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*
c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) -
(-A*c + 2*B*b)/(4*c**3)) - 2*b**4*c**2*(sqrt(-(4*a*c - b**2)**3)*(6*A*a*b*c**2
- A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)/(4*c**3*(64*a**3*c**3 -
48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)) - (-A*c + 2*B*b)/(4*c**3)))/(6*A*a*b*c*
*2 - A*b**3*c + 12*B*a**2*c**2 - 12*B*a*b**2*c + 2*B*b**4)) + (2*A*a**2*c**2 - A
*a*b**2*c - 3*B*a**2*b*c + B*a*b**3 + x**2*(3*A*a*b*c**2 - A*b**3*c + 2*B*a**2*c
**2 - 4*B*a*b**2*c + B*b**4))/(8*a**2*c**4 - 2*a*b**2*c**3 + x**4*(8*a*c**5 - 2*
b**2*c**4) + x**2*(8*a*b*c**4 - 2*b**3*c**3))
_______________________________________________________________________________________
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((B*x^2 + A)*x^7/(c*x^4 + b*x^2 + a)^2,x, algorithm="giac")
[Out]
Exception raised: TypeError