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

Optimal. Leaf size=94 \[ -\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a B+x^2 (-(b B-2 A c))+A b}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

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

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Rubi [A]  time = 0.183677, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{(b B-2 A c) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{-2 a B+x^2 (-(b B-2 A c))+A b}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 22.5939, size = 83, normalized size = 0.88 \[ \frac{\left (2 A c - B b\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{A b - 2 B a + x^{2} \left (2 A c - B b\right )}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.143549, size = 101, normalized size = 1.07 \[ \frac{\frac{2 (b B-2 A c) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{B \left (2 a+b x^2\right )-A \left (b+2 c x^2\right )}{a+b x^2+c x^4}}{2 \left (b^2-4 a c\right )} \]

Antiderivative was successfully verified.

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

[Out]

((B*(2*a + b*x^2) - A*(b + 2*c*x^2))/(a + b*x^2 + c*x^4) + (2*(b*B - 2*A*c)*ArcT
an[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c])/(2*(b^2 - 4*a*c))

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Maple [A]  time = 0.008, size = 127, normalized size = 1.4 \[{\frac{ \left ( 2\,Ac-bB \right ){x}^{2}+Ab-2\,Ba}{ \left ( 8\,ac-2\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+2\,{\frac{Ac}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{bB\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*((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/(4*a*c-b^2)^(3/2)*
arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*c-1/(4*a*c-b^2)^(3/2)*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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.290959, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{4} + B a b - 2 \, A a c +{\left (B b^{2} - 2 \, A b c\right )} x^{2}\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left ({\left (B b - 2 \, A c\right )} x^{2} + 2 \, B a - A b\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left ({\left (B b c - 2 \, A c^{2}\right )} x^{4} + B a b - 2 \, A a c +{\left (B b^{2} - 2 \, A b c\right )} x^{2}\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (B b - 2 \, A c\right )} x^{2} + 2 \, B a - A b\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(((B*b*c - 2*A*c^2)*x^4 + B*a*b - 2*A*a*c + (B*b^2 - 2*A*b*c)*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)*sq
rt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + ((B*b - 2*A*c)*x^2 + 2*B*a - A*b)*sqrt(b
^2 - 4*a*c))/(((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a^2*c + (b^3 - 4*a*b*c)*x^2)*sq
rt(b^2 - 4*a*c)), 1/2*(2*((B*b*c - 2*A*c^2)*x^4 + B*a*b - 2*A*a*c + (B*b^2 - 2*A
*b*c)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) + ((B*b - 2*A
*c)*x^2 + 2*B*a - A*b)*sqrt(-b^2 + 4*a*c))/(((b^2*c - 4*a*c^2)*x^4 + a*b^2 - 4*a
^2*c + (b^3 - 4*a*b*c)*x^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [A]  time = 12.2112, size = 374, normalized size = 3.98 \[ \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right ) \log{\left (x^{2} + \frac{- 2 A b c + B b^{2} - 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right ) + 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right ) - b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{2} - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right ) \log{\left (x^{2} + \frac{- 2 A b c + B b^{2} + 16 a^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right ) - 8 a b^{2} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right ) + b^{4} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \left (- 2 A c + B b\right )}{- 4 A c^{2} + 2 B b c} \right )}}{2} - \frac{- A b + 2 B a + x^{2} \left (- 2 A c + B b\right )}{8 a^{2} c - 2 a b^{2} + x^{4} \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \left (8 a b c - 2 b^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1/(4*a*c - b**2)**3)*(-2*A*c + B*b)*log(x**2 + (-2*A*b*c + B*b**2 - 16*a**
2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(-2*A*c + B*b) + 8*a*b**2*c*sqrt(-1/(4*a*c - b
**2)**3)*(-2*A*c + B*b) - b**4*sqrt(-1/(4*a*c - b**2)**3)*(-2*A*c + B*b))/(-4*A*
c**2 + 2*B*b*c))/2 - sqrt(-1/(4*a*c - b**2)**3)*(-2*A*c + B*b)*log(x**2 + (-2*A*
b*c + B*b**2 + 16*a**2*c**2*sqrt(-1/(4*a*c - b**2)**3)*(-2*A*c + B*b) - 8*a*b**2
*c*sqrt(-1/(4*a*c - b**2)**3)*(-2*A*c + B*b) + b**4*sqrt(-1/(4*a*c - b**2)**3)*(
-2*A*c + B*b))/(-4*A*c**2 + 2*B*b*c))/2 - (-A*b + 2*B*a + x**2*(-2*A*c + B*b))/(
8*a**2*c - 2*a*b**2 + x**4*(8*a*c**2 - 2*b**2*c) + x**2*(8*a*b*c - 2*b**3))

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

[Out]

Exception raised: TypeError