3.54 \(\int \frac{(a+b x) (A+B x)}{x} \, dx\)

Optimal. Leaf size=24 \[ x (a B+A b)+a A \log (x)+\frac{1}{2} b B x^2 \]

[Out]

(A*b + a*B)*x + (b*B*x^2)/2 + a*A*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0299539, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071 \[ x (a B+A b)+a A \log (x)+\frac{1}{2} b B x^2 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)*(A + B*x))/x,x]

[Out]

(A*b + a*B)*x + (b*B*x^2)/2 + a*A*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ A a \log{\left (x \right )} + B b \int x\, dx + a \int B\, dx + b \int A\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)*(B*x+A)/x,x)

[Out]

A*a*log(x) + B*b*Integral(x, x) + a*Integral(B, x) + b*Integral(A, x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.00790774, size = 24, normalized size = 1. \[ x (a B+A b)+a A \log (x)+\frac{1}{2} b B x^2 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)*(A + B*x))/x,x]

[Out]

(A*b + a*B)*x + (b*B*x^2)/2 + a*A*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 22, normalized size = 0.9 \[{\frac{bB{x}^{2}}{2}}+Abx+Bax+aA\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)*(B*x+A)/x,x)

[Out]

1/2*b*B*x^2+A*b*x+B*a*x+a*A*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.3494, size = 30, normalized size = 1.25 \[ \frac{1}{2} \, B b x^{2} + A a \log \left (x\right ) +{\left (B a + A b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x,x, algorithm="maxima")

[Out]

1/2*B*b*x^2 + A*a*log(x) + (B*a + A*b)*x

_______________________________________________________________________________________

Fricas [A]  time = 0.20254, size = 30, normalized size = 1.25 \[ \frac{1}{2} \, B b x^{2} + A a \log \left (x\right ) +{\left (B a + A b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x,x, algorithm="fricas")

[Out]

1/2*B*b*x^2 + A*a*log(x) + (B*a + A*b)*x

_______________________________________________________________________________________

Sympy [A]  time = 1.11974, size = 22, normalized size = 0.92 \[ A a \log{\left (x \right )} + \frac{B b x^{2}}{2} + x \left (A b + B a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)*(B*x+A)/x,x)

[Out]

A*a*log(x) + B*b*x**2/2 + x*(A*b + B*a)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.288064, size = 30, normalized size = 1.25 \[ \frac{1}{2} \, B b x^{2} + B a x + A b x + A a{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)/x,x, algorithm="giac")

[Out]

1/2*B*b*x^2 + B*a*x + A*b*x + A*a*ln(abs(x))