3.1.58 \(\int \frac {(a+b x)^2}{x^3} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 24, 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 = {43} \begin {gather*} -\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2/x^3,x]

[Out]

-a^2/(2*x^2) - (2*a*b)/x + b^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^3} \, dx &=\int \left (\frac {a^2}{x^3}+\frac {2 a b}{x^2}+\frac {b^2}{x}\right ) \, dx\\ &=-\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 24, normalized size = 1.00 \begin {gather*} -\frac {a^2}{2 x^2}-\frac {2 a b}{x}+b^2 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2/x^3,x]

[Out]

-1/2*a^2/x^2 - (2*a*b)/x + b^2*Log[x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b x)^2}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x)^2/x^3,x]

[Out]

IntegrateAlgebraic[(a + b*x)^2/x^3, x]

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fricas [A]  time = 0.91, size = 26, normalized size = 1.08 \begin {gather*} \frac {2 \, b^{2} x^{2} \log \relax (x) - 4 \, a b x - a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^3,x, algorithm="fricas")

[Out]

1/2*(2*b^2*x^2*log(x) - 4*a*b*x - a^2)/x^2

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giac [A]  time = 1.18, size = 22, normalized size = 0.92 \begin {gather*} b^{2} \log \left ({\left | x \right |}\right ) - \frac {4 \, a b x + a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^3,x, algorithm="giac")

[Out]

b^2*log(abs(x)) - 1/2*(4*a*b*x + a^2)/x^2

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maple [A]  time = 0.01, size = 23, normalized size = 0.96 \begin {gather*} b^{2} \ln \relax (x )-\frac {2 a b}{x}-\frac {a^{2}}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2/x^3,x)

[Out]

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

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maxima [A]  time = 1.18, size = 21, normalized size = 0.88 \begin {gather*} b^{2} \log \relax (x) - \frac {4 \, a b x + a^{2}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2/x^3,x, algorithm="maxima")

[Out]

b^2*log(x) - 1/2*(4*a*b*x + a^2)/x^2

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mupad [B]  time = 0.04, size = 23, normalized size = 0.96 \begin {gather*} b^2\,\ln \relax (x)-\frac {\frac {a^2}{2}+2\,b\,x\,a}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^2/x^3,x)

[Out]

b^2*log(x) - (a^2/2 + 2*a*b*x)/x^2

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sympy [A]  time = 0.17, size = 22, normalized size = 0.92 \begin {gather*} b^{2} \log {\relax (x )} + \frac {- a^{2} - 4 a b x}{2 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2/x**3,x)

[Out]

b**2*log(x) + (-a**2 - 4*a*b*x)/(2*x**2)

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