3.78.36 $\int \frac{-120+11x-8x^2}{125+5x+8x^2}dx$

Optimal. Leaf size=19 $$-x+\log (25+x+4x\log \left(e^{2x/5}\right))$$

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Rubi [A]  time = 0.02, antiderivative size = 15, 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 = {1657, 628} $$\begin{array}{c}\log (8x^2+5x+125)-x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-120 + 11*x - 8*x^2)/(125 + 5*x + 8*x^2),x]

[Out]

-x + Log[125 + 5*x + 8*x^2]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (-1+\frac{5+16x}{125+5x+8x^2})dx\\ & =-x+\int \frac{5+16x}{125+5x+8x^2}dx\\ & =-x+\log (125+5x+8x^2)\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 15, normalized size = 0.79 $$\begin{array}{c}-x+\log (125+5x+8x^2)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-120 + 11*x - 8*x^2)/(125 + 5*x + 8*x^2),x]

[Out]

-x + Log[125 + 5*x + 8*x^2]

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fricas [A]  time = 0.54, size = 15, normalized size = 0.79 $$\begin{array}{c}-x+\log (8x^2+5x+125)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^2+11*x-120)/(8*x^2+5*x+125),x, algorithm="fricas")

[Out]

-x + log(8*x^2 + 5*x + 125)

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giac [A]  time = 0.16, size = 15, normalized size = 0.79 $$\begin{array}{c}-x+\log (8x^2+5x+125)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^2+11*x-120)/(8*x^2+5*x+125),x, algorithm="giac")

[Out]

-x + log(8*x^2 + 5*x + 125)

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maple [A]  time = 0.51, size = 16, normalized size = 0.84

method	result	size
default	$-x+\ln (8x^2+5x+125)$	$16$
norman	$-x+\ln (8x^2+5x+125)$	$16$
risch	$-x+\ln (8x^2+5x+125)$	$16$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x^2+11*x-120)/(8*x^2+5*x+125),x,method=_RETURNVERBOSE)

[Out]

-x+ln(8*x^2+5*x+125)

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maxima [A]  time = 0.37, size = 15, normalized size = 0.79 $$\begin{array}{c}-x+\log (8x^2+5x+125)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x^2+11*x-120)/(8*x^2+5*x+125),x, algorithm="maxima")

[Out]

-x + log(8*x^2 + 5*x + 125)

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mupad [B]  time = 0.06, size = 15, normalized size = 0.79 $$\begin{array}{c}\ln (8x^2+5x+125)-x\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(8*x^2 - 11*x + 120)/(5*x + 8*x^2 + 125),x)

[Out]

log(5*x + 8*x^2 + 125) - x

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sympy [A]  time = 0.10, size = 12, normalized size = 0.63 $$\begin{array}{c}-x+\log (8x^2+5x+125)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x**2+11*x-120)/(8*x**2+5*x+125),x)

[Out]

-x + log(8*x**2 + 5*x + 125)

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