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

Optimal. Leaf size=73 \[ -\frac{2 c \log (x) (b c-a d)}{a^3}+\frac{2 c (b c-a d) \log (a+b x)}{a^3}-\frac{(b c-a d)^2}{a^2 b (a+b x)}-\frac{c^2}{a^2 x} \]

[Out]

-(c^2/(a^2*x)) - (b*c - a*d)^2/(a^2*b*(a + b*x)) - (2*c*(b*c - a*d)*Log[x])/a^3
+ (2*c*(b*c - a*d)*Log[a + b*x])/a^3

_______________________________________________________________________________________

Rubi [A]  time = 0.127077, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 c \log (x) (b c-a d)}{a^3}+\frac{2 c (b c-a d) \log (a+b x)}{a^3}-\frac{(b c-a d)^2}{a^2 b (a+b x)}-\frac{c^2}{a^2 x} \]

Antiderivative was successfully verified.

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

[Out]

-(c^2/(a^2*x)) - (b*c - a*d)^2/(a^2*b*(a + b*x)) - (2*c*(b*c - a*d)*Log[x])/a^3
+ (2*c*(b*c - a*d)*Log[a + b*x])/a^3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 17.0389, size = 63, normalized size = 0.86 \[ - \frac{c^{2}}{a^{2} x} - \frac{\left (a d - b c\right )^{2}}{a^{2} b \left (a + b x\right )} + \frac{2 c \left (a d - b c\right ) \log{\left (x \right )}}{a^{3}} - \frac{2 c \left (a d - b c\right ) \log{\left (a + b x \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.128434, size = 67, normalized size = 0.92 \[ \frac{-\frac{a (b c-a d)^2}{b (a+b x)}+2 c \log (x) (a d-b c)+2 c (b c-a d) \log (a+b x)-\frac{a c^2}{x}}{a^3} \]

Antiderivative was successfully verified.

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

[Out]

(-((a*c^2)/x) - (a*(b*c - a*d)^2)/(b*(a + b*x)) + 2*c*(-(b*c) + a*d)*Log[x] + 2*
c*(b*c - a*d)*Log[a + b*x])/a^3

_______________________________________________________________________________________

Maple [A]  time = 0.015, size = 106, normalized size = 1.5 \[ -{\frac{{c}^{2}}{{a}^{2}x}}+2\,{\frac{c\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{{c}^{2}\ln \left ( x \right ) b}{{a}^{3}}}-{\frac{{d}^{2}}{b \left ( bx+a \right ) }}+2\,{\frac{cd}{a \left ( bx+a \right ) }}-{\frac{{c}^{2}b}{{a}^{2} \left ( bx+a \right ) }}-2\,{\frac{c\ln \left ( bx+a \right ) d}{{a}^{2}}}+2\,{\frac{{c}^{2}\ln \left ( bx+a \right ) b}{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^2/x^2/(b*x+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.34229, size = 126, normalized size = 1.73 \[ -\frac{a b c^{2} +{\left (2 \, b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} x}{a^{2} b^{2} x^{2} + a^{3} b x} + \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (b x + a\right )}{a^{3}} - \frac{2 \,{\left (b c^{2} - a c d\right )} \log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.21697, size = 201, normalized size = 2.75 \[ -\frac{a^{2} b c^{2} +{\left (2 \, a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} x - 2 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{2} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \log \left (b x + a\right ) + 2 \,{\left ({\left (b^{3} c^{2} - a b^{2} c d\right )} x^{2} +{\left (a b^{2} c^{2} - a^{2} b c d\right )} x\right )} \log \left (x\right )}{a^{3} b^{2} x^{2} + a^{4} b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a^2*b*c^2 + (2*a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*x - 2*((b^3*c^2 - a*b^2*c*d)
*x^2 + (a*b^2*c^2 - a^2*b*c*d)*x)*log(b*x + a) + 2*((b^3*c^2 - a*b^2*c*d)*x^2 +
(a*b^2*c^2 - a^2*b*c*d)*x)*log(x))/(a^3*b^2*x^2 + a^4*b*x)

_______________________________________________________________________________________

Sympy [A]  time = 5.44414, size = 173, normalized size = 2.37 \[ - \frac{a b c^{2} + x \left (a^{2} d^{2} - 2 a b c d + 2 b^{2} c^{2}\right )}{a^{3} b x + a^{2} b^{2} x^{2}} + \frac{2 c \left (a d - b c\right ) \log{\left (x + \frac{2 a^{2} c d - 2 a b c^{2} - 2 a c \left (a d - b c\right )}{4 a b c d - 4 b^{2} c^{2}} \right )}}{a^{3}} - \frac{2 c \left (a d - b c\right ) \log{\left (x + \frac{2 a^{2} c d - 2 a b c^{2} + 2 a c \left (a d - b c\right )}{4 a b c d - 4 b^{2} c^{2}} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a*b*c**2 + x*(a**2*d**2 - 2*a*b*c*d + 2*b**2*c**2))/(a**3*b*x + a**2*b**2*x**2
) + 2*c*(a*d - b*c)*log(x + (2*a**2*c*d - 2*a*b*c**2 - 2*a*c*(a*d - b*c))/(4*a*b
*c*d - 4*b**2*c**2))/a**3 - 2*c*(a*d - b*c)*log(x + (2*a**2*c*d - 2*a*b*c**2 + 2
*a*c*(a*d - b*c))/(4*a*b*c*d - 4*b**2*c**2))/a**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.294563, size = 150, normalized size = 2.05 \[ \frac{b c^{2}}{a^{3}{\left (\frac{a}{b x + a} - 1\right )}} - \frac{2 \,{\left (b^{2} c^{2} - a b c d\right )}{\rm ln}\left ({\left | -\frac{a}{b x + a} + 1 \right |}\right )}{a^{3} b} - \frac{\frac{b^{3} c^{2}}{b x + a} - \frac{2 \, a b^{2} c d}{b x + a} + \frac{a^{2} b d^{2}}{b x + a}}{a^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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