3.1753 \(\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

[Out]

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

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Rubi [A]  time = 0.0592084, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074 \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.013455, size = 25, normalized size = 1. \[ \frac{(b c-a d) \log (a+b x)}{b^2}+\frac{d x}{b} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 1.3 \[{\frac{dx}{b}}-{\frac{\ln \left ( bx+a \right ) ad}{{b}^{2}}}+{\frac{\ln \left ( bx+a \right ) c}{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.727036, size = 34, normalized size = 1.36 \[ \frac{d x}{b} + \frac{{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/b + (b*c - a*d)*log(b*x + a)/b^2

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Fricas [A]  time = 0.20209, size = 32, normalized size = 1.28 \[ \frac{b d x +{\left (b c - a d\right )} \log \left (b x + a\right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.24681, size = 20, normalized size = 0.8 \[ \frac{d x}{b} - \frac{\left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x/b - (a*d - b*c)*log(a + b*x)/b**2

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GIAC/XCAS [A]  time = 0.211378, size = 158, normalized size = 6.32 \[ b d{\left (\frac{2 \, a{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{3}} + \frac{b x + a}{b^{3}} - \frac{a^{2}}{{\left (b x + a\right )} b^{3}}\right )} - \frac{{\left (b c + a d\right )}{\left (\frac{{\rm ln}\left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b} - \frac{a}{{\left (b x + a\right )} b}\right )}}{b} - \frac{a c}{{\left (b x + a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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