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

Optimal. Leaf size=41 \[ \frac{B \log \left (a+b x^2\right )}{2 b^2}-\frac{A b-a B}{2 b^2 \left (a+b x^2\right )} \]

[Out]

-(A*b - a*B)/(2*b^2*(a + b*x^2)) + (B*Log[a + b*x^2])/(2*b^2)

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Rubi [A]  time = 0.0869263, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{B \log \left (a+b x^2\right )}{2 b^2}-\frac{A b-a B}{2 b^2 \left (a+b x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

-(A*b - a*B)/(2*b^2*(a + b*x^2)) + (B*Log[a + b*x^2])/(2*b^2)

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Rubi in Sympy [A]  time = 13.3832, size = 32, normalized size = 0.78 \[ \frac{B \log{\left (a + b x^{2} \right )}}{2 b^{2}} - \frac{A b - B a}{2 b^{2} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*log(a + b*x**2)/(2*b**2) - (A*b - B*a)/(2*b**2*(a + b*x**2))

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Mathematica [A]  time = 0.0201065, size = 41, normalized size = 1. \[ \frac{a B-A b}{2 b^2 \left (a+b x^2\right )}+\frac{B \log \left (a+b x^2\right )}{2 b^2} \]

Antiderivative was successfully verified.

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

[Out]

(-(A*b) + a*B)/(2*b^2*(a + b*x^2)) + (B*Log[a + b*x^2])/(2*b^2)

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Maple [A]  time = 0.013, size = 47, normalized size = 1.2 \[{\frac{B\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{2}}}-{\frac{A}{ \left ( 2\,b{x}^{2}+2\,a \right ) b}}+{\frac{Ba}{ \left ( 2\,b{x}^{2}+2\,a \right ){b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(B*x^2+A)/(b*x^2+a)^2,x)

[Out]

1/2*B*ln(b*x^2+a)/b^2-1/2/(b*x^2+a)/b*A+1/2/(b*x^2+a)/b^2*B*a

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Maxima [A]  time = 1.3521, size = 54, normalized size = 1.32 \[ \frac{B a - A b}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} + \frac{B \log \left (b x^{2} + a\right )}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*a - A*b)/(b^3*x^2 + a*b^2) + 1/2*B*log(b*x^2 + a)/b^2

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Fricas [A]  time = 0.214756, size = 59, normalized size = 1.44 \[ \frac{B a - A b +{\left (B b x^{2} + B a\right )} \log \left (b x^{2} + a\right )}{2 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(B*a - A*b + (B*b*x^2 + B*a)*log(b*x^2 + a))/(b^3*x^2 + a*b^2)

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Sympy [A]  time = 2.05519, size = 36, normalized size = 0.88 \[ \frac{B \log{\left (a + b x^{2} \right )}}{2 b^{2}} + \frac{- A b + B a}{2 a b^{2} + 2 b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*log(a + b*x**2)/(2*b**2) + (-A*b + B*a)/(2*a*b**2 + 2*b**3*x**2)

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GIAC/XCAS [A]  time = 0.231593, size = 88, normalized size = 2.15 \[ -\frac{B{\left (\frac{{\rm ln}\left (\frac{{\left | b x^{2} + a \right |}}{{\left (b x^{2} + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x^{2} + a\right )} b}\right )}}{2 \, b} - \frac{A}{2 \,{\left (b x^{2} + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*B*(ln(abs(b*x^2 + a)/((b*x^2 + a)^2*abs(b)))/b - a/((b*x^2 + a)*b))/b - 1/2
*A/((b*x^2 + a)*b)