3.2174 \(\int \frac {(3-4 x+x^2)^2}{x^7} \, dx\)

Optimal. Leaf size=36 \[ -\frac {3}{2 x^6}+\frac {24}{5 x^5}-\frac {11}{2 x^4}+\frac {8}{3 x^3}-\frac {1}{2 x^2} \]

[Out]

-3/2/x^6+24/5/x^5-11/2/x^4+8/3/x^3-1/2/x^2

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {698} \[ -\frac {1}{2 x^2}+\frac {8}{3 x^3}-\frac {11}{2 x^4}+\frac {24}{5 x^5}-\frac {3}{2 x^6} \]

Antiderivative was successfully verified.

[In]

Int[(3 - 4*x + x^2)^2/x^7,x]

[Out]

-3/(2*x^6) + 24/(5*x^5) - 11/(2*x^4) + 8/(3*x^3) - 1/(2*x^2)

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {\left (3-4 x+x^2\right )^2}{x^7} \, dx &=\int \left (\frac {9}{x^7}-\frac {24}{x^6}+\frac {22}{x^5}-\frac {8}{x^4}+\frac {1}{x^3}\right ) \, dx\\ &=-\frac {3}{2 x^6}+\frac {24}{5 x^5}-\frac {11}{2 x^4}+\frac {8}{3 x^3}-\frac {1}{2 x^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 36, normalized size = 1.00 \[ -\frac {3}{2 x^6}+\frac {24}{5 x^5}-\frac {11}{2 x^4}+\frac {8}{3 x^3}-\frac {1}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 - 4*x + x^2)^2/x^7,x]

[Out]

-3/(2*x^6) + 24/(5*x^5) - 11/(2*x^4) + 8/(3*x^3) - 1/(2*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 25, normalized size = 0.69 \[ -\frac {15 \, x^{4} - 80 \, x^{3} + 165 \, x^{2} - 144 \, x + 45}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^7,x, algorithm="fricas")

[Out]

-1/30*(15*x^4 - 80*x^3 + 165*x^2 - 144*x + 45)/x^6

________________________________________________________________________________________

giac [A]  time = 0.16, size = 25, normalized size = 0.69 \[ -\frac {15 \, x^{4} - 80 \, x^{3} + 165 \, x^{2} - 144 \, x + 45}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^7,x, algorithm="giac")

[Out]

-1/30*(15*x^4 - 80*x^3 + 165*x^2 - 144*x + 45)/x^6

________________________________________________________________________________________

maple [A]  time = 0.05, size = 27, normalized size = 0.75 \[ -\frac {1}{2 x^{2}}+\frac {8}{3 x^{3}}-\frac {11}{2 x^{4}}+\frac {24}{5 x^{5}}-\frac {3}{2 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2-4*x+3)^2/x^7,x)

[Out]

-3/2/x^6+24/5/x^5-11/2/x^4+8/3/x^3-1/2/x^2

________________________________________________________________________________________

maxima [A]  time = 1.07, size = 25, normalized size = 0.69 \[ -\frac {15 \, x^{4} - 80 \, x^{3} + 165 \, x^{2} - 144 \, x + 45}{30 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2-4*x+3)^2/x^7,x, algorithm="maxima")

[Out]

-1/30*(15*x^4 - 80*x^3 + 165*x^2 - 144*x + 45)/x^6

________________________________________________________________________________________

mupad [B]  time = 0.02, size = 25, normalized size = 0.69 \[ -\frac {15\,x^4-80\,x^3+165\,x^2-144\,x+45}{30\,x^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 - 4*x + 3)^2/x^7,x)

[Out]

-(165*x^2 - 144*x - 80*x^3 + 15*x^4 + 45)/(30*x^6)

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 24, normalized size = 0.67 \[ \frac {- 15 x^{4} + 80 x^{3} - 165 x^{2} + 144 x - 45}{30 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2-4*x+3)**2/x**7,x)

[Out]

(-15*x**4 + 80*x**3 - 165*x**2 + 144*x - 45)/(30*x**6)

________________________________________________________________________________________