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

Optimal. Leaf size=22 \[ a^2 \log (x)+2 a b x+\frac{b^2 x^2}{2} \]

[Out]

2*a*b*x + (b^2*x^2)/2 + a^2*Log[x]

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Rubi [A]  time = 0.006339, antiderivative size = 22, 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} \[ a^2 \log (x)+2 a b x+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x,x]

[Out]

2*a*b*x + (b^2*x^2)/2 + a^2*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} \, dx &=\int \left (2 a b+\frac{a^2}{x}+b^2 x\right ) \, dx\\ &=2 a b x+\frac{b^2 x^2}{2}+a^2 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0007976, size = 22, normalized size = 1. \[ a^2 \log (x)+2 a b x+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x,x]

[Out]

2*a*b*x + (b^2*x^2)/2 + a^2*Log[x]

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Maple [A]  time = 0.003, size = 21, normalized size = 1. \begin{align*} 2\,abx+{\frac{{b}^{2}{x}^{2}}{2}}+{a}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07302, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2 + 2*a*b*x + a^2*log(x)

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Fricas [A]  time = 1.48218, size = 49, normalized size = 2.23 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b^2*x^2 + 2*a*b*x + a^2*log(x)

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Sympy [A]  time = 0.328913, size = 20, normalized size = 0.91 \begin{align*} a^{2} \log{\left (x \right )} + 2 a b x + \frac{b^{2} x^{2}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*log(x) + 2*a*b*x + b**2*x**2/2

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Giac [A]  time = 1.13502, size = 28, normalized size = 1.27 \begin{align*} \frac{1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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