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

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 {gather*} 5 x+\frac {1}{x+3}+\frac {25}{4 (x+4)}-\frac {25}{4 x}+\log (x) \end {gather*}

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

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

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 {gather*} \frac {5 \, x^{4} + 35 \, x^{3} + 61 \, x^{2} + {\left (x^{3} + 7 \, x^{2} + 12 \, x\right )} \log \relax (x) - 21 \, x - 75}{x^{3} + 7 \, x^{2} + 12 \, x} \end {gather*}

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 {gather*} 5 \, x + \frac {x^{2} - 21 \, x - 75}{{\left (x + 4\right )} {\left (x + 3\right )} x} + \log \left ({\left | x \right |}\right ) \end {gather*}

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	\(5 x +\ln \relax (x )-\frac {25}{4 x}+\frac {25}{4 \left (4+x \right )}+\frac {1}{3+x}\)	\(24\)
risch	\(5 x +\frac {x^{2}-21 x -75}{x \left (x^{2}+7 x +12\right )}+\ln \relax (x )\)	\(29\)
norman	\(\frac {5 x^{4}-184 x^{2}-441 x -75}{x \left (x^{2}+7 x +12\right )}+\ln \relax (x )\)	\(33\)

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 {gather*} 5 \, x + \frac {x^{2} - 21 \, x - 75}{x^{3} + 7 \, x^{2} + 12 \, x} + \log \relax (x) \end {gather*}

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 {gather*} 5\,x+\ln \relax (x)-\frac {-x^2+21\,x+75}{x^3+7\,x^2+12\,x} \end {gather*}

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 {gather*} 5 x + \frac {x^{2} - 21 x - 75}{x^{3} + 7 x^{2} + 12 x} + \log {\relax (x )} \end {gather*}

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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