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

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

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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} $$\begin{array}{c}x+\frac{130}{26-5x}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{1326-260x+25x^2}{(-26+5x{)}^2}dx\\ & =\int (1+\frac{650}{(-26+5x{)}^2})dx\\ & =\frac{130}{26-5x}+x\end{array}\end{array}$$

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

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 $$\begin{array}{c}\frac{5x^2-26x-130}{5x-26}\end{array}$$

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 $$\begin{array}{c}x-\frac{130}{5x-26}\end{array}$$

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	$x-\frac{130}{5x-26}$	$12$
norman	$\frac{5x^2-\frac{1326}{5}}{5x-26}$	$16$
gosper	$\frac{25x^2-1326}{25x-130}$	$17$
meijerg	$-\frac{x}{26(1-\frac{5x}{26})}+\frac{x(-\frac{15x}{26}+6)}{3-\frac{15x}{26}}$	$27$

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 $$\begin{array}{c}x-\frac{130}{5x-26}\end{array}$$

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 $$\begin{array}{c}x-\frac{26}{x-\frac{26}{5}}\end{array}$$

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 $$\begin{array}{c}x-\frac{130}{5x-26}\end{array}$$

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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