3.57 \(\int \frac{(a+b x)^2}{x^2} \, dx\)

Optimal. Leaf size=20 \[ -\frac{a^2}{x}+2 a b \log (x)+b^2 x \]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

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Rubi [A]  time = 0.0081317, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{a^2}{x}+2 a b \log (x)+b^2 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^2,x]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

Rule 43

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*} \int \frac{(a+b x)^2}{x^2} \, dx &=\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]  time = 0.0010358, size = 20, normalized size = 1. \[ -\frac{a^2}{x}+2 a b \log (x)+b^2 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^2,x]

[Out]

-(a^2/x) + b^2*x + 2*a*b*Log[x]

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Maple [A]  time = 0.006, size = 21, normalized size = 1.1 \begin{align*} -{\frac{{a}^{2}}{x}}+{b}^{2}x+2\,ab\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02448, size = 27, normalized size = 1.35 \begin{align*} b^{2} x + 2 \, a b \log \left (x\right ) - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*x + 2*a*b*log(x) - a^2/x

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Fricas [A]  time = 1.59555, size = 49, normalized size = 2.45 \begin{align*} \frac{b^{2} x^{2} + 2 \, a b x \log \left (x\right ) - a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 0.30444, size = 17, normalized size = 0.85 \begin{align*} - \frac{a^{2}}{x} + 2 a b \log{\left (x \right )} + b^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a**2/x + 2*a*b*log(x) + b**2*x

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Giac [A]  time = 1.15834, size = 28, normalized size = 1.4 \begin{align*} b^{2} x + 2 \, a b \log \left ({\left | x \right |}\right ) - \frac{a^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*x + 2*a*b*log(abs(x)) - a^2/x