3.1.56 \(\int \frac {(a+b x)^2}{x} \, dx\) [56]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} a^2 \log (x)+2 a b x+\frac {b^2 x^2}{2} \end {gather*}

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 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*} \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 = 0.00, size = 22, normalized size = 1.00 \begin {gather*} 2 a b x+\frac {b^2 x^2}{2}+a^2 \log (x) \end {gather*}

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]

________________________________________________________________________________________

Mathics [A]
time = 1.74, size = 20, normalized size = 0.91 \begin {gather*} a^2 \text {Log}\left [x\right ]+2 a b x+\frac {b^2 x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^2/x^1,x]')

[Out]

a ^ 2 Log[x] + 2 a b x + b ^ 2 x ^ 2 / 2

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 21, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 20, normalized size = 0.91 \begin {gather*} \frac {1}{2} \, b^{2} x^{2} + 2 \, a b x + a^{2} \log \left (x\right ) \end {gather*}

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)

________________________________________________________________________________________

Sympy [A]
time = 0.05, size = 20, normalized size = 0.91 \begin {gather*} a^{2} \log {\left (x \right )} + 2 a b x + \frac {b^{2} x^{2}}{2} \end {gather*}

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

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 23, normalized size = 1.05 \begin {gather*} \frac {1}{2} x^{2} b^{2}+2 x b a+a^{2} \ln \left |x\right | \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 0.03, size = 20, normalized size = 0.91 \begin {gather*} a^2\,\ln \left (x\right )+\frac {b^2\,x^2}{2}+2\,a\,b\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/x,x)

[Out]

a^2*log(x) + (b^2*x^2)/2 + 2*a*b*x

________________________________________________________________________________________