3.69.19 \(\int \frac {1326-260 x+25 x^2}{676-260 x+25 x^2} \, dx\)

Optimal. Leaf size=14 \[ x+\frac {5 x}{\frac {26}{5}-x} \]

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Rubi [A]  time = 0.01, antiderivative size = 11, normalized size of antiderivative = 0.79, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {27, 683} \begin {gather*} x+\frac {130}{26-5 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1326 - 260*x + 25*x^2)/(676 - 260*x + 25*x^2),x]

[Out]

130/(26 - 5*x) + x

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 683

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] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1326-260 x+25 x^2}{(-26+5 x)^2} \, dx\\ &=\int \left (1+\frac {650}{(-26+5 x)^2}\right ) \, dx\\ &=\frac {130}{26-5 x}+x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.79 \begin {gather*} \frac {130}{26-5 x}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1326 - 260*x + 25*x^2)/(676 - 260*x + 25*x^2),x]

[Out]

130/(26 - 5*x) + x

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fricas [A]  time = 0.50, size = 18, normalized size = 1.29 \begin {gather*} \frac {5 \, x^{2} - 26 \, x - 130}{5 \, x - 26} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^2-260*x+1326)/(25*x^2-260*x+676),x, algorithm="fricas")

[Out]

(5*x^2 - 26*x - 130)/(5*x - 26)

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giac [A]  time = 0.13, size = 11, normalized size = 0.79 \begin {gather*} x - \frac {130}{5 \, x - 26} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^2-260*x+1326)/(25*x^2-260*x+676),x, algorithm="giac")

[Out]

x - 130/(5*x - 26)

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maple [A]  time = 0.10, size = 10, normalized size = 0.71

method	result	size
risch	\(x -\frac {26}{x -\frac {26}{5}}\)	\(10\)
default	\(x -\frac {130}{5 x -26}\)	\(12\)
norman	\(\frac {5 x^{2}-\frac {1326}{5}}{5 x -26}\)	\(16\)
gosper	\(\frac {25 x^{2}-1326}{25 x -130}\)	\(17\)
meijerg	\(-\frac {x}{26 \left (1-\frac {5 x}{26}\right )}+\frac {x \left (-\frac {15 x}{26}+6\right )}{3-\frac {15 x}{26}}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*x^2-260*x+1326)/(25*x^2-260*x+676),x,method=_RETURNVERBOSE)

[Out]

x-26/(x-26/5)

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maxima [A]  time = 0.38, size = 11, normalized size = 0.79 \begin {gather*} x - \frac {130}{5 \, x - 26} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x^2-260*x+1326)/(25*x^2-260*x+676),x, algorithm="maxima")

[Out]

x - 130/(5*x - 26)

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mupad [B]  time = 4.05, size = 9, normalized size = 0.64 \begin {gather*} x-\frac {26}{x-\frac {26}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((25*x^2 - 260*x + 1326)/(25*x^2 - 260*x + 676),x)

[Out]

x - 26/(x - 26/5)

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sympy [A]  time = 0.07, size = 7, normalized size = 0.50 \begin {gather*} x - \frac {130}{5 x - 26} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((25*x**2-260*x+1326)/(25*x**2-260*x+676),x)

[Out]

x - 130/(5*x - 26)

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