3.126 \(\int \frac{x^7 \left (A+B x^2\right )}{\left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=146 \[ \frac{3 a (A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{x^6 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

-(x^6*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)
+ (3*(A*b - 2*a*B)*x^2*(2*a + b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) +
(3*a*(A*b - 2*a*B)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2
)

_______________________________________________________________________________________

Rubi [A]  time = 0.30925, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{3 a (A b-2 a B) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 x^2 \left (2 a+b x^2\right ) (A b-2 a B)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{x^6 \left (-2 a B+x^2 (-(b B-2 A c))+A b\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

-(x^6*(A*b - 2*a*B - (b*B - 2*A*c)*x^2))/(4*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)^2)
+ (3*(A*b - 2*a*B)*x^2*(2*a + b*x^2))/(4*(b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) +
(3*a*(A*b - 2*a*B)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(b^2 - 4*a*c)^(5/2
)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.4934, size = 138, normalized size = 0.95 \[ \frac{3 a \left (A b - 2 B a\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} - \frac{x^{6} \left (A b - 2 B a + x^{2} \left (2 A c - B b\right )\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{3 x^{2} \left (2 a + b x^{2}\right ) \left (\frac{A b}{2} - B a\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} \]

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)**3,x)

[Out]

3*a*(A*b - 2*B*a)*atanh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(-4*a*c + b**2)**(5/
2) - x**6*(A*b - 2*B*a + x**2*(2*A*c - B*b))/(4*(-4*a*c + b**2)*(a + b*x**2 + c*
x**4)**2) + 3*x**2*(2*a + b*x**2)*(A*b/2 - B*a)/(2*(-4*a*c + b**2)**2*(a + b*x**
2 + c*x**4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.474767, size = 261, normalized size = 1.79 \[ \frac{1}{4} \left (\frac{a^2 c \left (2 c \left (A+B x^2\right )-3 b B\right )+a b \left (-b c \left (A+4 B x^2\right )+3 A c^2 x^2+b^2 B\right )+b^3 x^2 (b B-A c)}{c^3 \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}+\frac{-4 a^2 c^3 \left (4 A+5 B x^2\right )+a b^2 c^2 \left (5 A+16 B x^2\right )+2 a b c^2 \left (11 a B-3 A c x^2\right )-8 a b^3 B c-b^4 c \left (A+2 B x^2\right )+b^5 B}{c^3 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{12 a (A b-2 a B) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^7*(A + B*x^2))/(a + b*x^2 + c*x^4)^3,x]

[Out]

((b^5*B - 8*a*b^3*B*c - b^4*c*(A + 2*B*x^2) - 4*a^2*c^3*(4*A + 5*B*x^2) + a*b^2*
c^2*(5*A + 16*B*x^2) + 2*a*b*c^2*(11*a*B - 3*A*c*x^2))/(c^3*(b^2 - 4*a*c)^2*(a +
b*x^2 + c*x^4)) + (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)))/(c^3*(-b^2 + 4*a*c)*(a + b*x^2 + c*x
^4)^2) - (12*a*(A*b - 2*a*B)*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4
*a*c)^(5/2))/4

_______________________________________________________________________________________

Maple [B]  time = 0.022, size = 398, normalized size = 2.7 \[{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 3\,aAb{c}^{2}+10\,{a}^{2}B{c}^{2}-8\,a{b}^{2}Bc+{b}^{4}B \right ){x}^{6}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) c}}-{\frac{ \left ( 16\,A{a}^{2}{c}^{3}+Aa{b}^{2}{c}^{2}+A{b}^{4}c-2\,B{a}^{2}b{c}^{2}-8\,Ba{b}^{3}c+B{b}^{5} \right ){x}^{4}}{ \left ( 32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4} \right ){c}^{2}}}-{\frac{a \left ( 5\,aAb{c}^{2}+A{b}^{3}c+6\,{a}^{2}B{c}^{2}-10\,a{b}^{2}Bc+{b}^{4}B \right ){x}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ){c}^{2}}}-{\frac{{a}^{2} \left ( 8\,aA{c}^{2}+A{b}^{2}c-10\,abBc+{b}^{3}B \right ) }{2\, \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ){c}^{2}}} \right ) }-3\,{\frac{abA}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+6\,{\frac{B{a}^{2}}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

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)^3,x)

[Out]

1/2*(-(3*A*a*b*c^2+10*B*a^2*c^2-8*B*a*b^2*c+B*b^4)/(16*a^2*c^2-8*a*b^2*c+b^4)/c*
x^6-1/2*(16*A*a^2*c^3+A*a*b^2*c^2+A*b^4*c-2*B*a^2*b*c^2-8*B*a*b^3*c+B*b^5)/(16*a
^2*c^2-8*a*b^2*c+b^4)/c^2*x^4-a*(5*A*a*b*c^2+A*b^3*c+6*B*a^2*c^2-10*B*a*b^2*c+B*
b^4)/c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2-1/2*a^2/c^2*(8*A*a*c^2+A*b^2*c-10*B*a*b*
c+B*b^3)/(16*a^2*c^2-8*a*b^2*c+b^4))/(c*x^4+b*x^2+a)^2-3*a/(16*a^2*c^2-8*a*b^2*c
+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*b+6*a^2/(16*a^2*
c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.265707, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

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)^3,x, algorithm="fricas")

[Out]

[-1/4*(6*((2*B*a^2 - A*a*b)*c^4*x^8 + 2*(2*B*a^2*b - A*a*b^2)*c^3*x^6 + 2*(2*B*a
^3*b - A*a^2*b^2)*c^2*x^2 + (2*(2*B*a^3 - A*a^2*b)*c^3 + (2*B*a^2*b^2 - A*a*b^3)
*c^2)*x^4 + (2*B*a^4 - A*a^3*b)*c^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*(B*b^4*c - 8*B*a*b^2*c^2 + (10*B*a^2 + 3*A*a*b)*c^3)*x^6 + B*a^2*b^3 + 8*
A*a^3*c^2 + (B*b^5 + 16*A*a^2*c^3 - (2*B*a^2*b - A*a*b^2)*c^2 - (8*B*a*b^3 - A*b
^4)*c)*x^4 + 2*(B*a*b^4 + (6*B*a^3 + 5*A*a^2*b)*c^2 - (10*B*a^2*b^2 - A*a*b^3)*c
)*x^2 - (10*B*a^3*b - A*a^2*b^2)*c)*sqrt(b^2 - 4*a*c))/(((b^4*c^4 - 8*a*b^2*c^5
+ 16*a^2*c^6)*x^8 + a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + 2*(b^5*c^3 - 8*a*
b^3*c^4 + 16*a^2*b*c^5)*x^6 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^4 + 2*(a*b^
5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x^2)*sqrt(b^2 - 4*a*c)), 1/4*(12*((2*B*a^2
- A*a*b)*c^4*x^8 + 2*(2*B*a^2*b - A*a*b^2)*c^3*x^6 + 2*(2*B*a^3*b - A*a^2*b^2)*
c^2*x^2 + (2*(2*B*a^3 - A*a^2*b)*c^3 + (2*B*a^2*b^2 - A*a*b^3)*c^2)*x^4 + (2*B*a
^4 - A*a^3*b)*c^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (2*
(B*b^4*c - 8*B*a*b^2*c^2 + (10*B*a^2 + 3*A*a*b)*c^3)*x^6 + B*a^2*b^3 + 8*A*a^3*c
^2 + (B*b^5 + 16*A*a^2*c^3 - (2*B*a^2*b - A*a*b^2)*c^2 - (8*B*a*b^3 - A*b^4)*c)*
x^4 + 2*(B*a*b^4 + (6*B*a^3 + 5*A*a^2*b)*c^2 - (10*B*a^2*b^2 - A*a*b^3)*c)*x^2 -
(10*B*a^3*b - A*a^2*b^2)*c)*sqrt(-b^2 + 4*a*c))/(((b^4*c^4 - 8*a*b^2*c^5 + 16*a
^2*c^6)*x^8 + a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + 2*(b^5*c^3 - 8*a*b^3*c^
4 + 16*a^2*b*c^5)*x^6 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^4 + 2*(a*b^5*c^2
- 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x^2)*sqrt(-b^2 + 4*a*c))]

_______________________________________________________________________________________

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*(B*x**2+A)/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 15.9766, size = 429, normalized size = 2.94 \[ \frac{3 \,{\left (2 \, B a^{2} - A a b\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, B b^{4} c x^{6} - 16 \, B a b^{2} c^{2} x^{6} + 20 \, B a^{2} c^{3} x^{6} + 6 \, A a b c^{3} x^{6} + B b^{5} x^{4} - 8 \, B a b^{3} c x^{4} + A b^{4} c x^{4} - 2 \, B a^{2} b c^{2} x^{4} + A a b^{2} c^{2} x^{4} + 16 \, A a^{2} c^{3} x^{4} + 2 \, B a b^{4} x^{2} - 20 \, B a^{2} b^{2} c x^{2} + 2 \, A a b^{3} c x^{2} + 12 \, B a^{3} c^{2} x^{2} + 10 \, A a^{2} b c^{2} x^{2} + B a^{2} b^{3} - 10 \, B a^{3} b c + A a^{2} b^{2} c + 8 \, A a^{3} c^{2}}{4 \,{\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

3*(2*B*a^2 - A*a*b)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((b^4 - 8*a*b^2*c +
16*a^2*c^2)*sqrt(-b^2 + 4*a*c)) - 1/4*(2*B*b^4*c*x^6 - 16*B*a*b^2*c^2*x^6 + 20*
B*a^2*c^3*x^6 + 6*A*a*b*c^3*x^6 + B*b^5*x^4 - 8*B*a*b^3*c*x^4 + A*b^4*c*x^4 - 2*
B*a^2*b*c^2*x^4 + A*a*b^2*c^2*x^4 + 16*A*a^2*c^3*x^4 + 2*B*a*b^4*x^2 - 20*B*a^2*
b^2*c*x^2 + 2*A*a*b^3*c*x^2 + 12*B*a^3*c^2*x^2 + 10*A*a^2*b*c^2*x^2 + B*a^2*b^3
- 10*B*a^3*b*c + A*a^2*b^2*c + 8*A*a^3*c^2)/((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4
)*(c*x^4 + b*x^2 + a)^2)