3.24.77 \(\int \frac {337500+722400 x+4804 x^2+8 x^3+e^3 (5400000 x^2+36000 x^3+60 x^4)}{90000+600 x+x^2} \, dx\)

Optimal. Leaf size=29 \[ -\frac {x}{4+\frac {x}{75}}+4 \left (x+x^2 \left (1+5 e^3 x\right )\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 24, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {27, 1850} \begin {gather*} 20 e^3 x^3+4 x^2+4 x+\frac {22500}{x+300} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(337500 + 722400*x + 4804*x^2 + 8*x^3 + E^3*(5400000*x^2 + 36000*x^3 + 60*x^4))/(90000 + 600*x + x^2),x]

[Out]

4*x + 4*x^2 + 20*E^3*x^3 + 22500/(300 + 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 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {337500+722400 x+4804 x^2+8 x^3+e^3 \left (5400000 x^2+36000 x^3+60 x^4\right )}{(300+x)^2} \, dx\\ &=\int \left (4+8 x+60 e^3 x^2-\frac {22500}{(300+x)^2}\right ) \, dx\\ &=4 x+4 x^2+20 e^3 x^3+\frac {22500}{300+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 38, normalized size = 1.31 \begin {gather*} \frac {4 \left (-26904375-89400 x+301 x^2+x^3+5 e^3 (300+x)^2 \left (90000-300 x+x^2\right )\right )}{300+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(337500 + 722400*x + 4804*x^2 + 8*x^3 + E^3*(5400000*x^2 + 36000*x^3 + 60*x^4))/(90000 + 600*x + x^2
),x]

[Out]

(4*(-26904375 - 89400*x + 301*x^2 + x^3 + 5*E^3*(300 + x)^2*(90000 - 300*x + x^2)))/(300 + x)

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fricas [A]  time = 0.47, size = 33, normalized size = 1.14 \begin {gather*} \frac {4 \, {\left (x^{3} + 301 \, x^{2} + 5 \, {\left (x^{4} + 300 \, x^{3}\right )} e^{3} + 300 \, x + 5625\right )}}{x + 300} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^4+36000*x^3+5400000*x^2)*exp(3)+8*x^3+4804*x^2+722400*x+337500)/(x^2+600*x+90000),x, algorith
m="fricas")

[Out]

4*(x^3 + 301*x^2 + 5*(x^4 + 300*x^3)*e^3 + 300*x + 5625)/(x + 300)

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giac [A]  time = 0.21, size = 23, normalized size = 0.79 \begin {gather*} 20 \, x^{3} e^{3} + 4 \, x^{2} + 4 \, x + \frac {22500}{x + 300} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^4+36000*x^3+5400000*x^2)*exp(3)+8*x^3+4804*x^2+722400*x+337500)/(x^2+600*x+90000),x, algorith
m="giac")

[Out]

20*x^3*e^3 + 4*x^2 + 4*x + 22500/(x + 300)

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maple [A]  time = 0.53, size = 24, normalized size = 0.83

method	result	size
default	\(4 x +20 x^{3} {\mathrm e}^{3}+4 x^{2}+\frac {22500}{x +300}\)	\(24\)
risch	\(4 x +20 x^{3} {\mathrm e}^{3}+4 x^{2}+\frac {22500}{x +300}\)	\(24\)
norman	\(\frac {\left (4+6000 \,{\mathrm e}^{3}\right ) x^{3}+1204 x^{2}+20 x^{4} {\mathrm e}^{3}-337500}{x +300}\)	\(31\)
gosper	\(\frac {20 x^{4} {\mathrm e}^{3}+6000 x^{3} {\mathrm e}^{3}+4 x^{3}+1204 x^{2}-337500}{x +300}\)	\(32\)
meijerg	\(-\frac {9617 x}{4 \left (1+\frac {x}{300}\right )}[phone] \,{\mathrm e}^{3} \left (\frac {x \left (\frac {1}{5400000} x^{3}-\frac {1}{9000} x^{2}+\frac {1}{10} x +60\right )}{4500+15 x}-4 \ln \left (1+\frac {x}{300}\right )\right )+300 \left (10800000 \,{\mathrm e}^{3}+2400\right ) \left (-\frac {x \left (-\frac {1}{45000} x^{2}+\frac {1}{50} x +12\right )}{1200 \left (1+\frac {x}{300}\right )}+3 \ln \left (1+\frac {x}{300}\right )\right )+300 \left (5400000 \,{\mathrm e}^{3}+4804\right ) \left (\frac {x \left (\frac {x}{100}+6\right )}{900+3 x}-2 \ln \left (1+\frac {x}{300}\right )\right )+722400 \ln \left (1+\frac {x}{300}\right )\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60*x^4+36000*x^3+5400000*x^2)*exp(3)+8*x^3+4804*x^2+722400*x+337500)/(x^2+600*x+90000),x,method=_RETURNV
ERBOSE)

[Out]

4*x+20*x^3*exp(3)+4*x^2+22500/(x+300)

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maxima [A]  time = 0.56, size = 23, normalized size = 0.79 \begin {gather*} 20 \, x^{3} e^{3} + 4 \, x^{2} + 4 \, x + \frac {22500}{x + 300} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^4+36000*x^3+5400000*x^2)*exp(3)+8*x^3+4804*x^2+722400*x+337500)/(x^2+600*x+90000),x, algorith
m="maxima")

[Out]

20*x^3*e^3 + 4*x^2 + 4*x + 22500/(x + 300)

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mupad [B]  time = 1.37, size = 23, normalized size = 0.79 \begin {gather*} 4\,x+\frac {22500}{x+300}+20\,x^3\,{\mathrm {e}}^3+4\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((722400*x + exp(3)*(5400000*x^2 + 36000*x^3 + 60*x^4) + 4804*x^2 + 8*x^3 + 337500)/(600*x + x^2 + 90000),x
)

[Out]

4*x + 22500/(x + 300) + 20*x^3*exp(3) + 4*x^2

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sympy [A]  time = 0.10, size = 20, normalized size = 0.69 \begin {gather*} 20 x^{3} e^{3} + 4 x^{2} + 4 x + \frac {22500}{x + 300} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x**4+36000*x**3+5400000*x**2)*exp(3)+8*x**3+4804*x**2+722400*x+337500)/(x**2+600*x+90000),x)

[Out]

20*x**3*exp(3) + 4*x**2 + 4*x + 22500/(x + 300)

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