3.103.70 \(\int \frac {-25+10 x+4 x^2+375 x^4-150 x^5+15 x^6}{25 x^2-10 x^3+x^4} \, dx\)

Optimal. Leaf size=24 \[ 1-e^4+\frac {1}{x}+\frac {x}{5-x}+5 x^3 \]

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Rubi [A]  time = 0.06, antiderivative size = 18, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1594, 27, 1620} \begin {gather*} 5 x^3+\frac {5}{5-x}+\frac {1}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-25 + 10*x + 4*x^2 + 375*x^4 - 150*x^5 + 15*x^6)/(25*x^2 - 10*x^3 + x^4),x]

[Out]

5/(5 - x) + x^(-1) + 5*x^3

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-25+10 x+4 x^2+375 x^4-150 x^5+15 x^6}{x^2 \left (25-10 x+x^2\right )} \, dx\\ &=\int \frac {-25+10 x+4 x^2+375 x^4-150 x^5+15 x^6}{(-5+x)^2 x^2} \, dx\\ &=\int \left (\frac {5}{(-5+x)^2}-\frac {1}{x^2}+15 x^2\right ) \, dx\\ &=\frac {5}{5-x}+\frac {1}{x}+5 x^3\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 0.67 \begin {gather*} -\frac {5}{-5+x}+\frac {1}{x}+5 x^3 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-25 + 10*x + 4*x^2 + 375*x^4 - 150*x^5 + 15*x^6)/(25*x^2 - 10*x^3 + x^4),x]

[Out]

-5/(-5 + x) + x^(-1) + 5*x^3

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fricas [A]  time = 0.84, size = 25, normalized size = 1.04 \begin {gather*} \frac {5 \, x^{5} - 25 \, x^{4} - 4 \, x - 5}{x^{2} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x^6-150*x^5+375*x^4+4*x^2+10*x-25)/(x^4-10*x^3+25*x^2),x, algorithm="fricas")

[Out]

(5*x^5 - 25*x^4 - 4*x - 5)/(x^2 - 5*x)

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giac [A]  time = 0.24, size = 22, normalized size = 0.92 \begin {gather*} 5 \, x^{3} - \frac {4 \, x + 5}{x^{2} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x^6-150*x^5+375*x^4+4*x^2+10*x-25)/(x^4-10*x^3+25*x^2),x, algorithm="giac")

[Out]

5*x^3 - (4*x + 5)/(x^2 - 5*x)

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maple [A]  time = 0.02, size = 17, normalized size = 0.71

method	result	size
default	\(5 x^{3}-\frac {5}{x -5}+\frac {1}{x}\)	\(17\)
risch	\(5 x^{3}+\frac {-4 x -5}{\left (x -5\right ) x}\)	\(21\)
gosper	\(\frac {5 x^{5}-25 x^{4}-4 x -5}{x \left (x -5\right )}\)	\(25\)
norman	\(\frac {5 x^{5}-25 x^{4}-4 x -5}{x \left (x -5\right )}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((15*x^6-150*x^5+375*x^4+4*x^2+10*x-25)/(x^4-10*x^3+25*x^2),x,method=_RETURNVERBOSE)

[Out]

5*x^3-5/(x-5)+1/x

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maxima [A]  time = 0.36, size = 22, normalized size = 0.92 \begin {gather*} 5 \, x^{3} - \frac {4 \, x + 5}{x^{2} - 5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x^6-150*x^5+375*x^4+4*x^2+10*x-25)/(x^4-10*x^3+25*x^2),x, algorithm="maxima")

[Out]

5*x^3 - (4*x + 5)/(x^2 - 5*x)

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mupad [B]  time = 6.85, size = 21, normalized size = 0.88 \begin {gather*} 5\,x^3-\frac {4\,x+5}{x\,\left (x-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x + 4*x^2 + 375*x^4 - 150*x^5 + 15*x^6 - 25)/(25*x^2 - 10*x^3 + x^4),x)

[Out]

5*x^3 - (4*x + 5)/(x*(x - 5))

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sympy [A]  time = 0.13, size = 17, normalized size = 0.71 \begin {gather*} 5 x^{3} + \frac {- 4 x - 5}{x^{2} - 5 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((15*x**6-150*x**5+375*x**4+4*x**2+10*x-25)/(x**4-10*x**3+25*x**2),x)

[Out]

5*x**3 + (-4*x - 5)/(x**2 - 5*x)

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