3.89.73 $\int \frac{900+1194x+1272x^2+955x^3+378x^4+71x^5+5x^6}{144x^2+168x^3+73x^4+14x^5+x^6}dx$

Optimal. Leaf size=24 $$1+\frac{1}{3+x}+5(3+x)-\frac{25}{x(4+x)}+\log (x)$$

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Rubi [A]  time = 0.09, antiderivative size = 27, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 1, integrand size = 57, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.018, Rules used = {2074} $$\begin{array}{c}5x+\frac{1}{x+3}+\frac{25}{4(x+4)}-\frac{25}{4x}+\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(900 + 1194*x + 1272*x^2 + 955*x^3 + 378*x^4 + 71*x^5 + 5*x^6)/(144*x^2 + 168*x^3 + 73*x^4 + 14*x^5 + x^6)
,x]

[Out]

-25/(4*x) + 5*x + (3 + x)^(-1) + 25/(4*(4 + x)) + Log[x]

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int (5+\frac{25}{4x^2}+\frac{1}{x}-\frac{1}{(3+x{)}^2}-\frac{25}{4(4+x{)}^2})dx\\ & =-\frac{25}{4x}+5x+\frac{1}{3+x}+\frac{25}{4(4+x)}+\log (x)\end{array}\end{array}$$

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.33 $$\begin{array}{c}-\frac{25}{4x}+5x+\frac{91+29x}{4(12+7x+x^2)}+\log (x)\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(900 + 1194*x + 1272*x^2 + 955*x^3 + 378*x^4 + 71*x^5 + 5*x^6)/(144*x^2 + 168*x^3 + 73*x^4 + 14*x^5
+ x^6),x]

[Out]

-25/(4*x) + 5*x + (91 + 29*x)/(4*(12 + 7*x + x^2)) + Log[x]

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fricas [B]  time = 0.44, size = 50, normalized size = 2.08 $$\begin{array}{c}\frac{5x^4+35x^3+61x^2+(x^3+7x^2+12x)\log (x)-21x-75}{x^3+7x^2+12x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+71*x^5+378*x^4+955*x^3+1272*x^2+1194*x+900)/(x^6+14*x^5+73*x^4+168*x^3+144*x^2),x, algorithm=
"fricas")

[Out]

(5*x^4 + 35*x^3 + 61*x^2 + (x^3 + 7*x^2 + 12*x)*log(x) - 21*x - 75)/(x^3 + 7*x^2 + 12*x)

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giac [A]  time = 0.14, size = 29, normalized size = 1.21 $$\begin{array}{c}5x+\frac{x^2-21x-75}{(x+4)(x+3)x}+\log \left(\left|x\right|\right)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+71*x^5+378*x^4+955*x^3+1272*x^2+1194*x+900)/(x^6+14*x^5+73*x^4+168*x^3+144*x^2),x, algorithm=
"giac")

[Out]

5*x + (x^2 - 21*x - 75)/((x + 4)*(x + 3)*x) + log(abs(x))

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maple [A]  time = 0.04, size = 24, normalized size = 1.00

method	result	size
default	$5x+\ln (x)-\frac{25}{4x}+\frac{25}{4(4+x)}+\frac{1}{3+x}$	$24$
risch	$5x+\frac{x^2-21x-75}{x(x^2+7x+12)}+\ln (x)$	$29$
norman	$\frac{5x^4-184x^2-441x-75}{x(x^2+7x+12)}+\ln (x)$	$33$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^6+71*x^5+378*x^4+955*x^3+1272*x^2+1194*x+900)/(x^6+14*x^5+73*x^4+168*x^3+144*x^2),x,method=_RETURNVER
BOSE)

[Out]

5*x+ln(x)-25/4/x+25/4/(4+x)+1/(3+x)

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maxima [A]  time = 0.37, size = 29, normalized size = 1.21 $$\begin{array}{c}5x+\frac{x^2-21x-75}{x^3+7x^2+12x}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^6+71*x^5+378*x^4+955*x^3+1272*x^2+1194*x+900)/(x^6+14*x^5+73*x^4+168*x^3+144*x^2),x, algorithm=
"maxima")

[Out]

5*x + (x^2 - 21*x - 75)/(x^3 + 7*x^2 + 12*x) + log(x)

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mupad [B]  time = 0.07, size = 32, normalized size = 1.33 $$\begin{array}{c}5x+\ln (x)-\frac{-x^2+21x+75}{x^3+7x^2+12x}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1194*x + 1272*x^2 + 955*x^3 + 378*x^4 + 71*x^5 + 5*x^6 + 900)/(144*x^2 + 168*x^3 + 73*x^4 + 14*x^5 + x^6)
,x)

[Out]

5*x + log(x) - (21*x - x^2 + 75)/(12*x + 7*x^2 + x^3)

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sympy [A]  time = 0.12, size = 26, normalized size = 1.08 $$\begin{array}{c}5x+\frac{x^2-21x-75}{x^3+7x^2+12x}+\log (x)\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**6+71*x**5+378*x**4+955*x**3+1272*x**2+1194*x+900)/(x**6+14*x**5+73*x**4+168*x**3+144*x**2),x)

[Out]

5*x + (x**2 - 21*x - 75)/(x**3 + 7*x**2 + 12*x) + log(x)

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