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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0116563, antiderivative size = 36, normalized size of antiderivative = 1., 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 = 0.0006213, size = 36, normalized size = 1. $-\frac{1}{2 x^2}+\frac{8}{3 x^3}-\frac{11}{2 x^4}+\frac{24}{5 x^5}-\frac{3}{2 x^6}$

Antiderivative was successfully veriﬁed.

[In]

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)

________________________________________________________________________________________

Maple [A]  time = 0.041, size = 27, normalized size = 0.8 \begin{align*} -{\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*}

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 = 1.00922, size = 34, normalized size = 0.94 \begin{align*} -\frac{15 \, x^{4} - 80 \, x^{3} + 165 \, x^{2} - 144 \, x + 45}{30 \, x^{6}} \end{align*}

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

________________________________________________________________________________________

Fricas [A]  time = 1.90917, size = 70, normalized size = 1.94 \begin{align*} -\frac{15 \, x^{4} - 80 \, x^{3} + 165 \, x^{2} - 144 \, x + 45}{30 \, x^{6}} \end{align*}

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

________________________________________________________________________________________

Sympy [A]  time = 0.116787, size = 26, normalized size = 0.72 \begin{align*} - \frac{15 x^{4} - 80 x^{3} + 165 x^{2} - 144 x + 45}{30 x^{6}} \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.10473, size = 34, normalized size = 0.94 \begin{align*} -\frac{15 \, x^{4} - 80 \, x^{3} + 165 \, x^{2} - 144 \, x + 45}{30 \, x^{6}} \end{align*}

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