3.185 \(\int \frac{\left (a+b x^2\right )^2}{x \left (c+d x^2\right )^2} \, dx\)

Optimal. Leaf size=67 \[ -\frac{1}{2} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac{a^2 \log (x)}{c^2}+\frac{(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]

[Out]

(b*c - a*d)^2/(2*c*d^2*(c + d*x^2)) + (a^2*Log[x])/c^2 - ((a^2/c^2 - b^2/d^2)*Lo
g[c + d*x^2])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.154413, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{1}{2} \left (\frac{a^2}{c^2}-\frac{b^2}{d^2}\right ) \log \left (c+d x^2\right )+\frac{a^2 \log (x)}{c^2}+\frac{(b c-a d)^2}{2 c d^2 \left (c+d x^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

(b*c - a*d)^2/(2*c*d^2*(c + d*x^2)) + (a^2*Log[x])/c^2 - ((a^2/c^2 - b^2/d^2)*Lo
g[c + d*x^2])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 29.7366, size = 60, normalized size = 0.9 \[ \frac{a^{2} \log{\left (x^{2} \right )}}{2 c^{2}} - \left (\frac{a^{2}}{2 c^{2}} - \frac{b^{2}}{2 d^{2}}\right ) \log{\left (c + d x^{2} \right )} + \frac{\left (a d - b c\right )^{2}}{2 c d^{2} \left (c + d x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**2/x/(d*x**2+c)**2,x)

[Out]

a**2*log(x**2)/(2*c**2) - (a**2/(2*c**2) - b**2/(2*d**2))*log(c + d*x**2) + (a*d
- b*c)**2/(2*c*d**2*(c + d*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0716916, size = 70, normalized size = 1.04 \[ \frac{2 a^2 \log (x)+\frac{(b c-a d) \left (\left (c+d x^2\right ) (a d+b c) \log \left (c+d x^2\right )+c (b c-a d)\right )}{d^2 \left (c+d x^2\right )}}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*Log[x] + ((b*c - a*d)*(c*(b*c - a*d) + (b*c + a*d)*(c + d*x^2)*Log[c + d*
x^2]))/(d^2*(c + d*x^2)))/(2*c^2)

_______________________________________________________________________________________

Maple [A]  time = 0.021, size = 94, normalized size = 1.4 \[{\frac{{a}^{2}\ln \left ( x \right ) }{{c}^{2}}}+{\frac{{a}^{2}}{2\,c \left ( d{x}^{2}+c \right ) }}-{\frac{ab}{d \left ( d{x}^{2}+c \right ) }}+{\frac{{b}^{2}c}{2\,{d}^{2} \left ( d{x}^{2}+c \right ) }}-{\frac{\ln \left ( d{x}^{2}+c \right ){a}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( d{x}^{2}+c \right ){b}^{2}}{2\,{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^2*ln(x)/c^2+1/2/c/(d*x^2+c)*a^2-1/d/(d*x^2+c)*a*b+1/2*c/d^2/(d*x^2+c)*b^2-1/2/
c^2*ln(d*x^2+c)*a^2+1/2/d^2*ln(d*x^2+c)*b^2

_______________________________________________________________________________________

Maxima [A]  time = 1.34436, size = 116, normalized size = 1.73 \[ \frac{a^{2} \log \left (x^{2}\right )}{2 \, c^{2}} + \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{2 \,{\left (c d^{3} x^{2} + c^{2} d^{2}\right )}} + \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )} \log \left (d x^{2} + c\right )}{2 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^2*log(x^2)/c^2 + 1/2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)/(c*d^3*x^2 + c^2*d^2)
+ 1/2*(b^2*c^2 - a^2*d^2)*log(d*x^2 + c)/(c^2*d^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.235894, size = 157, normalized size = 2.34 \[ \frac{b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2} +{\left (b^{2} c^{3} - a^{2} c d^{2} +{\left (b^{2} c^{2} d - a^{2} d^{3}\right )} x^{2}\right )} \log \left (d x^{2} + c\right ) + 2 \,{\left (a^{2} d^{3} x^{2} + a^{2} c d^{2}\right )} \log \left (x\right )}{2 \,{\left (c^{2} d^{3} x^{2} + c^{3} d^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^2*c^3 - 2*a*b*c^2*d + a^2*c*d^2 + (b^2*c^3 - a^2*c*d^2 + (b^2*c^2*d - a^2
*d^3)*x^2)*log(d*x^2 + c) + 2*(a^2*d^3*x^2 + a^2*c*d^2)*log(x))/(c^2*d^3*x^2 + c
^3*d^2)

_______________________________________________________________________________________

Sympy [A]  time = 5.23226, size = 80, normalized size = 1.19 \[ \frac{a^{2} \log{\left (x \right )}}{c^{2}} + \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{2 c^{2} d^{2} + 2 c d^{3} x^{2}} - \frac{\left (a d - b c\right ) \left (a d + b c\right ) \log{\left (\frac{c}{d} + x^{2} \right )}}{2 c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x/(d*x**2+c)**2,x)

[Out]

a**2*log(x)/c**2 + (a**2*d**2 - 2*a*b*c*d + b**2*c**2)/(2*c**2*d**2 + 2*c*d**3*x
**2) - (a*d - b*c)*(a*d + b*c)*log(c/d + x**2)/(2*c**2*d**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227297, size = 134, normalized size = 2. \[ \frac{a^{2}{\rm ln}\left (x^{2}\right )}{2 \, c^{2}} + \frac{{\left (b^{2} c^{2} - a^{2} d^{2}\right )}{\rm ln}\left ({\left | d x^{2} + c \right |}\right )}{2 \, c^{2} d^{2}} - \frac{b^{2} c^{2} x^{2} - a^{2} d^{2} x^{2} + 2 \, a b c^{2} - 2 \, a^{2} c d}{2 \,{\left (d x^{2} + c\right )} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a^2*ln(x^2)/c^2 + 1/2*(b^2*c^2 - a^2*d^2)*ln(abs(d*x^2 + c))/(c^2*d^2) - 1/2
*(b^2*c^2*x^2 - a^2*d^2*x^2 + 2*a*b*c^2 - 2*a^2*c*d)/((d*x^2 + c)*c^2*d)