3.1346 \(\int \frac{(a+b x)^2}{(c+d x)^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0849753, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ -\frac{(b c-a d)^2}{d^3 (c+d x)}-\frac{2 b (b c-a d) \log (c+d x)}{d^3}+\frac{b^2 x}{d^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/(c + d*x)^2,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 b \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{3}} + \frac{\int b^{2}\, dx}{d^{2}} - \frac{\left (a d - b c\right )^{2}}{d^{3} \left (c + d x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0628187, size = 47, normalized size = 0.92 \[ \frac{-\frac{(b c-a d)^2}{c+d x}+2 b (a d-b c) \log (c+d x)+b^2 d x}{d^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 86, normalized size = 1.7 \[{\frac{{b}^{2}x}{{d}^{2}}}+2\,{\frac{b\ln \left ( dx+c \right ) a}{{d}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( dx+c \right ) c}{{d}^{3}}}-{\frac{{a}^{2}}{d \left ( dx+c \right ) }}+2\,{\frac{abc}{{d}^{2} \left ( dx+c \right ) }}-{\frac{{b}^{2}{c}^{2}}{{d}^{3} \left ( dx+c \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/(d*x+c)^2,x)

[Out]

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

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Maxima [A]  time = 1.34676, size = 90, normalized size = 1.76 \[ \frac{b^{2} x}{d^{2}} - \frac{b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}}{d^{4} x + c d^{3}} - \frac{2 \,{\left (b^{2} c - a b d\right )} \log \left (d x + c\right )}{d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.204406, size = 124, normalized size = 2.43 \[ \frac{b^{2} d^{2} x^{2} + b^{2} c d x - b^{2} c^{2} + 2 \, a b c d - a^{2} d^{2} - 2 \,{\left (b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x\right )} \log \left (d x + c\right )}{d^{4} x + c d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.963175, size = 60, normalized size = 1.18 \[ \frac{b^{2} x}{d^{2}} + \frac{2 b \left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{3}} - \frac{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}}{c d^{3} + d^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219853, size = 132, normalized size = 2.59 \[ \frac{{\left (d x + c\right )} b^{2}}{d^{3}} + \frac{2 \,{\left (b^{2} c - a b d\right )}{\rm ln}\left (\frac{{\left | d x + c \right |}}{{\left (d x + c\right )}^{2}{\left | d \right |}}\right )}{d^{3}} - \frac{\frac{b^{2} c^{2} d}{d x + c} - \frac{2 \, a b c d^{2}}{d x + c} + \frac{a^{2} d^{3}}{d x + c}}{d^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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