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

Optimal. Leaf size=139 \[ \frac{3 c (b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (b+2 c x^2\right ) (b B-2 A c)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{-2 a B+x^2 (-(b B-2 A c))+A b}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.243689, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{3 c (b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{3 \left (b+2 c x^2\right ) (b B-2 A c)}{4 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{-2 a B+x^2 (-(b B-2 A c))+A b}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 30.0505, size = 131, normalized size = 0.94 \[ - \frac{3 c \left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} + \frac{3 \left (b + 2 c x^{2}\right ) \left (A c - \frac{B b}{2}\right )}{2 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} - \frac{A b - 2 B a + x^{2} \left (2 A c - B b\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.235279, size = 142, normalized size = 1.02 \[ \frac{-\frac{12 c (b B-2 A c) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{\left (b^2-4 a c\right ) \left (B \left (2 a+b x^2\right )-A \left (b+2 c x^2\right )\right )}{\left (a+b x^2+c x^4\right )^2}-\frac{3 \left (b+2 c x^2\right ) (b B-2 A c)}{a+b x^2+c x^4}}{4 \left (b^2-4 a c\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 262, normalized size = 1.9 \[{\frac{ \left ( 2\,Ac-bB \right ){x}^{2}+Ab-2\,Ba}{ \left ( 16\,ac-4\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}}}+3\,{\frac{{c}^{2}{x}^{2}A}{ \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}-{\frac{3\,bc{x}^{2}B}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+{\frac{3\,Abc}{2\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}-{\frac{3\,{b}^{2}B}{4\, \left ( 4\,ac-{b}^{2} \right ) ^{2} \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+6\,{\frac{A{c}^{2}}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-3\,{\frac{Bbc}{ \left ( 4\,ac-{b}^{2} \right ) ^{5/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((2*A*c-B*b)*x^2+A*b-2*B*a)/(4*a*c-b^2)/(c*x^4+b*x^2+a)^2+3/(4*a*c-b^2)^2/(c
*x^4+b*x^2+a)*x^2*c^2*A-3/2/(4*a*c-b^2)^2/(c*x^4+b*x^2+a)*x^2*c*b*B+3/2/(4*a*c-b
^2)^2/(c*x^4+b*x^2+a)*b*A*c-3/4/(4*a*c-b^2)^2/(c*x^4+b*x^2+a)*b^2*B+6/(4*a*c-b^2
)^(5/2)*c^2*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A-3/(4*a*c-b^2)^(5/2)*c*arctan
((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*B

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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((B*x^2 + A)*x/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.267678, 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/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")

[Out]

[-1/4*(6*((B*b*c^3 - 2*A*c^4)*x^8 + 2*(B*b^2*c^2 - 2*A*b*c^3)*x^6 + B*a^2*b*c -
2*A*a^2*c^2 + (B*b^3*c - 4*A*a*c^3 + 2*(B*a*b - A*b^2)*c^2)*x^4 + 2*(B*a*b^2*c -
2*A*a*b*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)) + (6*(B*b*c^2 -
2*A*c^3)*x^6 + 9*(B*b^2*c - 2*A*b*c^2)*x^4 + B*a*b^2 + A*b^3 + 2*(B*b^3 - 10*A*
a*c^2 + (5*B*a*b - 2*A*b^2)*c)*x^2 + 2*(4*B*a^2 - 5*A*a*b)*c)*sqrt(b^2 - 4*a*c))
/(((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)*sqrt(b^2 - 4*a*c)), -1/4*(12*((
B*b*c^3 - 2*A*c^4)*x^8 + 2*(B*b^2*c^2 - 2*A*b*c^3)*x^6 + B*a^2*b*c - 2*A*a^2*c^2
+ (B*b^3*c - 4*A*a*c^3 + 2*(B*a*b - A*b^2)*c^2)*x^4 + 2*(B*a*b^2*c - 2*A*a*b*c^
2)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + (6*(B*b*c^2 -
2*A*c^3)*x^6 + 9*(B*b^2*c - 2*A*b*c^2)*x^4 + B*a*b^2 + A*b^3 + 2*(B*b^3 - 10*A*a
*c^2 + (5*B*a*b - 2*A*b^2)*c)*x^2 + 2*(4*B*a^2 - 5*A*a*b)*c)*sqrt(-b^2 + 4*a*c))
/(((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)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 50.1615, size = 661, normalized size = 4.76 \[ \frac{3 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) \log{\left (x^{2} + \frac{- 6 A b c^{2} + 3 B b^{2} c - 192 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) + 144 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) - 36 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) + 3 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{2} - \frac{3 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) \log{\left (x^{2} + \frac{- 6 A b c^{2} + 3 B b^{2} c + 192 a^{3} c^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) - 144 a^{2} b^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) + 36 a b^{4} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right ) - 3 b^{6} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (- 2 A c + B b\right )}{- 12 A c^{3} + 6 B b c^{2}} \right )}}{2} - \frac{- 10 A a b c + A b^{3} + 8 B a^{2} c + B a b^{2} + x^{6} \left (- 12 A c^{3} + 6 B b c^{2}\right ) + x^{4} \left (- 18 A b c^{2} + 9 B b^{2} c\right ) + x^{2} \left (- 20 A a c^{2} - 4 A b^{2} c + 10 B a b c + 2 B b^{3}\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*c*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b)*log(x**2 + (-6*A*b*c**2 + 3*B*b**2
*c - 192*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b) + 144*a**2*b**2*c**
3*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b) - 36*a*b**4*c**2*sqrt(-1/(4*a*c - b*
*2)**5)*(-2*A*c + B*b) + 3*b**6*c*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b))/(-1
2*A*c**3 + 6*B*b*c**2))/2 - 3*c*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b)*log(x*
*2 + (-6*A*b*c**2 + 3*B*b**2*c + 192*a**3*c**4*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*
c + B*b) - 144*a**2*b**2*c**3*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b) + 36*a*b
**4*c**2*sqrt(-1/(4*a*c - b**2)**5)*(-2*A*c + B*b) - 3*b**6*c*sqrt(-1/(4*a*c - b
**2)**5)*(-2*A*c + B*b))/(-12*A*c**3 + 6*B*b*c**2))/2 - (-10*A*a*b*c + A*b**3 +
8*B*a**2*c + B*a*b**2 + x**6*(-12*A*c**3 + 6*B*b*c**2) + x**4*(-18*A*b*c**2 + 9*
B*b**2*c) + x**2*(-20*A*a*c**2 - 4*A*b**2*c + 10*B*a*b*c + 2*B*b**3))/(64*a**4*c
**2 - 32*a**3*b**2*c + 4*a**2*b**4 + x**8*(64*a**2*c**4 - 32*a*b**2*c**3 + 4*b**
4*c**2) + x**6*(128*a**2*b*c**3 - 64*a*b**3*c**2 + 8*b**5*c) + x**4*(128*a**3*c*
*3 - 24*a*b**4*c + 4*b**6) + x**2*(128*a**3*b*c**2 - 64*a**2*b**3*c + 8*a*b**5))

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GIAC/XCAS [A]  time = 15.9525, size = 281, normalized size = 2.02 \[ -\frac{3 \,{\left (B b c - 2 \, A c^{2}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{6 \, B b c^{2} x^{6} - 12 \, A c^{3} x^{6} + 9 \, B b^{2} c x^{4} - 18 \, A b c^{2} x^{4} + 2 \, B b^{3} x^{2} + 10 \, B a b c x^{2} - 4 \, A b^{2} c x^{2} - 20 \, A a c^{2} x^{2} + B a b^{2} + A b^{3} + 8 \, B a^{2} c - 10 \, A a b c}{4 \,{\left (c x^{4} + b x^{2} + a\right )}^{2}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*x/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")

[Out]

-3*(B*b*c - 2*A*c^2)*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*(6*B*b*c^2*x^6 - 12*A*c^3*x^6 + 9*B*b^2*
c*x^4 - 18*A*b*c^2*x^4 + 2*B*b^3*x^2 + 10*B*a*b*c*x^2 - 4*A*b^2*c*x^2 - 20*A*a*c
^2*x^2 + B*a*b^2 + A*b^3 + 8*B*a^2*c - 10*A*a*b*c)/((c*x^4 + b*x^2 + a)^2*(b^4 -
8*a*b^2*c + 16*a^2*c^2))