3.1102 \(\int \frac{a+b x+c x^2}{b d+2 c d x} \, dx\)

Optimal. Leaf size=48 \[ -\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d} \]

[Out]

(b*x)/(4*c*d) + x^2/(4*d) - ((b^2 - 4*a*c)*Log[b + 2*c*x])/(8*c^2*d)

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Rubi [A]  time = 0.0895044, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d}+\frac{b x}{4 c d}+\frac{x^2}{4 d} \]

Antiderivative was successfully verified.

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

[Out]

(b*x)/(4*c*d) + x^2/(4*d) - ((b^2 - 4*a*c)*Log[b + 2*c*x])/(8*c^2*d)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \int \frac{1}{4}\, dx}{c d} + \frac{\int x\, dx}{2 d} - \frac{\left (- a c + \frac{b^{2}}{4}\right ) \log{\left (b + 2 c x \right )}}{2 c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*Integral(1/4, x)/(c*d) + Integral(x, x)/(2*d) - (-a*c + b**2/4)*log(b + 2*c*x)
/(2*c**2*d)

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Mathematica [A]  time = 0.0246102, size = 37, normalized size = 0.77 \[ \frac{2 c x (b+c x)-\left (b^2-4 a c\right ) \log (b+2 c x)}{8 c^2 d} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*x*(b + c*x) - (b^2 - 4*a*c)*Log[b + 2*c*x])/(8*c^2*d)

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Maple [A]  time = 0.005, size = 54, normalized size = 1.1 \[{\frac{{x}^{2}}{4\,d}}+{\frac{bx}{4\,cd}}+{\frac{\ln \left ( 2\,cx+b \right ) a}{2\,cd}}-{\frac{\ln \left ( 2\,cx+b \right ){b}^{2}}{8\,{c}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*x^2/d+1/4*b*x/c/d+1/2/d/c*ln(2*c*x+b)*a-1/8/d/c^2*ln(2*c*x+b)*b^2

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Maxima [A]  time = 0.675951, size = 55, normalized size = 1.15 \[ \frac{c x^{2} + b x}{4 \, c d} - \frac{{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*(c*x^2 + b*x)/(c*d) - 1/8*(b^2 - 4*a*c)*log(2*c*x + b)/(c^2*d)

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Fricas [A]  time = 0.2077, size = 53, normalized size = 1.1 \[ \frac{2 \, c^{2} x^{2} + 2 \, b c x -{\left (b^{2} - 4 \, a c\right )} \log \left (2 \, c x + b\right )}{8 \, c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(2*c^2*x^2 + 2*b*c*x - (b^2 - 4*a*c)*log(2*c*x + b))/(c^2*d)

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Sympy [A]  time = 1.34084, size = 37, normalized size = 0.77 \[ \frac{b x}{4 c d} + \frac{x^{2}}{4 d} + \frac{\left (4 a c - b^{2}\right ) \log{\left (b + 2 c x \right )}}{8 c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x/(4*c*d) + x**2/(4*d) + (4*a*c - b**2)*log(b + 2*c*x)/(8*c**2*d)

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GIAC/XCAS [A]  time = 0.213689, size = 63, normalized size = 1.31 \[ -\frac{{\left (b^{2} - 4 \, a c\right )}{\rm ln}\left ({\left | 2 \, c x + b \right |}\right )}{8 \, c^{2} d} + \frac{c^{2} d x^{2} + b c d x}{4 \, c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(b^2 - 4*a*c)*ln(abs(2*c*x + b))/(c^2*d) + 1/4*(c^2*d*x^2 + b*c*d*x)/(c^2*d
^2)