3.1754 \(\int \frac{a c+(b c+a d) x+b d x^2}{(a+b x)^3} \, dx\)

Optimal. Leaf size=32 \[ \frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0623704, antiderivative size = 32, 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 \[ \frac{d \log (a+b x)}{b^2}-\frac{b c-a d}{b^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.1272, size = 26, normalized size = 0.81 \[ \frac{d \log{\left (a + b x \right )}}{b^{2}} + \frac{a d - b c}{b^{2} \left (a + b x\right )} \]

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)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0192003, size = 31, normalized size = 0.97 \[ \frac{a d-b c}{b^2 (a+b x)}+\frac{d \log (a+b x)}{b^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*c + (b*c + a*d)*x + b*d*x^2)/(a + b*x)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 39, normalized size = 1.2 \[{\frac{d\ln \left ( bx+a \right ) }{{b}^{2}}}+{\frac{ad}{ \left ( bx+a \right ){b}^{2}}}-{\frac{c}{b \left ( bx+a \right ) }} \]

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)^3,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.740994, size = 47, normalized size = 1.47 \[ -\frac{b c - a d}{b^{3} x + a b^{2}} + \frac{d \log \left (b x + a\right )}{b^{2}} \]

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)^3,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.206167, size = 53, normalized size = 1.66 \[ -\frac{b c - a d -{\left (b d x + a d\right )} \log \left (b x + a\right )}{b^{3} x + a b^{2}} \]

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)^3,x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 1.43684, size = 27, normalized size = 0.84 \[ \frac{a d - b c}{a b^{2} + b^{3} x} + \frac{d \log{\left (a + b x \right )}}{b^{2}} \]

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)**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210462, size = 45, normalized size = 1.41 \[ \frac{d{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{2}} - \frac{b c - a d}{{\left (b x + a\right )} b^{2}} \]

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)^3,x, algorithm="giac")

[Out]

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