3.1792 \(\int \frac{(a+b x)^2}{a c+(b c+a d) x+b d x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0605462, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069 \[ \frac{b x}{d}-\frac{(b c-a d) \log (c+d x)}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0133209, size = 25, normalized size = 0.96 \[ \frac{(a d-b c) \log (c+d x)}{d^2}+\frac{b x}{d} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 1.2 \[{\frac{bx}{d}}+{\frac{\ln \left ( dx+c \right ) a}{d}}-{\frac{\ln \left ( dx+c \right ) bc}{{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.748402, size = 35, normalized size = 1.35 \[ \frac{b x}{d} - \frac{{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x/d - (b*c - a*d)*log(d*x + c)/d^2

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Fricas [A]  time = 0.203786, size = 34, normalized size = 1.31 \[ \frac{b d x -{\left (b c - a d\right )} \log \left (d x + c\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.25226, size = 20, normalized size = 0.77 \[ \frac{b x}{d} + \frac{\left (a d - b c\right ) \log{\left (c + d x \right )}}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x/d + (a*d - b*c)*log(c + d*x)/d**2

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GIAC/XCAS [A]  time = 0.212336, size = 36, normalized size = 1.38 \[ \frac{b x}{d} - \frac{{\left (b c - a d\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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