\(\int \frac {(a+b x)^2}{x^2} \, dx\) [57]

Optimal result

Integrand size = 11, antiderivative size = 20 \[ \int \frac {(a+b x)^2}{x^2} \, dx=-\frac {a^2}{x}+b^2 x+2 a b \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^2}{x^2} \, dx=-\frac {a^2}{x}+2 a b \log (x)+b^2 x \]

[In]

Int[(a + b*x)^2/x^2,x]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (b^2+\frac {a^2}{x^2}+\frac {2 a b}{x}\right ) \, dx \\ & = -\frac {a^2}{x}+b^2 x+2 a b \log (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x^2} \, dx=-\frac {a^2}{x}+b^2 x+2 a b \log (x) \]

[In]

Integrate[(a + b*x)^2/x^2,x]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05

method	result	size
default	\(-\frac {a^{2}}{x}+b^{2} x +2 a b \ln \left (x \right )\)	\(21\)
risch	\(-\frac {a^{2}}{x}+b^{2} x +2 a b \ln \left (x \right )\)	\(21\)
norman	\(\frac {b^{2} x^{2}-a^{2}}{x}+2 a b \ln \left (x \right )\)	\(25\)
parallelrisch	\(\frac {2 a b \ln \left (x \right ) x +b^{2} x^{2}-a^{2}}{x}\)	\(25\)

[In]

int((b*x+a)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {(a+b x)^2}{x^2} \, dx=\frac {b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}}{x} \]

[In]

integrate((b*x+a)^2/x^2,x, algorithm="fricas")

[Out]

(b^2*x^2 + 2*a*b*x*log(x) - a^2)/x

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^2}{x^2} \, dx=- \frac {a^{2}}{x} + 2 a b \log {\left (x \right )} + b^{2} x \]

[In]

integrate((b*x+a)**2/x**2,x)

[Out]

-a**2/x + 2*a*b*log(x) + b**2*x

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x^2} \, dx=b^{2} x + 2 \, a b \log \left (x\right ) - \frac {a^{2}}{x} \]

[In]

integrate((b*x+a)^2/x^2,x, algorithm="maxima")

[Out]

b^2*x + 2*a*b*log(x) - a^2/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {(a+b x)^2}{x^2} \, dx=b^{2} x + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac {a^{2}}{x} \]

[In]

integrate((b*x+a)^2/x^2,x, algorithm="giac")

[Out]

b^2*x + 2*a*b*log(abs(x)) - a^2/x

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^2}{x^2} \, dx=b^2\,x-\frac {a^2}{x}+2\,a\,b\,\ln \left (x\right ) \]

[In]

int((a + b*x)^2/x^2,x)

[Out]

b^2*x - a^2/x + 2*a*b*log(x)