3.5.98 \(\int \frac {18+54 x+75 x^2+144 x^3+104 x^4+96 x^5+48 x^6+e^5 (40 x+80 x^2+40 x^3)}{9+18 x+33 x^2+48 x^3+40 x^4+32 x^5+16 x^6} \, dx\)

Optimal. Leaf size=28 \[ 3+2 x+\frac {x^2}{1+x}-\frac {5 e^5}{3+4 x^2} \]

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Rubi [A]  time = 0.14, antiderivative size = 23, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 2, integrand size = 81, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {2074, 261} \begin {gather*} -\frac {5 e^5}{4 x^2+3}+3 x+\frac {1}{x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(18 + 54*x + 75*x^2 + 144*x^3 + 104*x^4 + 96*x^5 + 48*x^6 + E^5*(40*x + 80*x^2 + 40*x^3))/(9 + 18*x + 33*x
^2 + 48*x^3 + 40*x^4 + 32*x^5 + 16*x^6),x]

[Out]

3*x + (1 + x)^(-1) - (5*E^5)/(3 + 4*x^2)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

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 (3-\frac {1}{(1+x)^2}+\frac {40 e^5 x}{\left (3+4 x^2\right )^2}\right ) \, dx\\ &=3 x+\frac {1}{1+x}+\left (40 e^5\right ) \int \frac {x}{\left (3+4 x^2\right )^2} \, dx\\ &=3 x+\frac {1}{1+x}-\frac {5 e^5}{3+4 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 37, normalized size = 1.32 \begin {gather*} 3 x+\frac {3+4 x^2-5 e^5 (1+x)}{3+3 x+4 x^2+4 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18 + 54*x + 75*x^2 + 144*x^3 + 104*x^4 + 96*x^5 + 48*x^6 + E^5*(40*x + 80*x^2 + 40*x^3))/(9 + 18*x
+ 33*x^2 + 48*x^3 + 40*x^4 + 32*x^5 + 16*x^6),x]

[Out]

3*x + (3 + 4*x^2 - 5*E^5*(1 + x))/(3 + 3*x + 4*x^2 + 4*x^3)

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fricas [A]  time = 0.56, size = 45, normalized size = 1.61 \begin {gather*} \frac {12 \, x^{4} + 12 \, x^{3} + 13 \, x^{2} - 5 \, {\left (x + 1\right )} e^{5} + 9 \, x + 3}{4 \, x^{3} + 4 \, x^{2} + 3 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x^3+80*x^2+40*x)*exp(5)+48*x^6+96*x^5+104*x^4+144*x^3+75*x^2+54*x+18)/(16*x^6+32*x^5+40*x^4+48*
x^3+33*x^2+18*x+9),x, algorithm="fricas")

[Out]

(12*x^4 + 12*x^3 + 13*x^2 - 5*(x + 1)*e^5 + 9*x + 3)/(4*x^3 + 4*x^2 + 3*x + 3)

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giac [A]  time = 0.26, size = 38, normalized size = 1.36 \begin {gather*} 3 \, x + \frac {4 \, x^{2} - 5 \, x e^{5} - 5 \, e^{5} + 3}{4 \, x^{3} + 4 \, x^{2} + 3 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x^3+80*x^2+40*x)*exp(5)+48*x^6+96*x^5+104*x^4+144*x^3+75*x^2+54*x+18)/(16*x^6+32*x^5+40*x^4+48*
x^3+33*x^2+18*x+9),x, algorithm="giac")

[Out]

3*x + (4*x^2 - 5*x*e^5 - 5*e^5 + 3)/(4*x^3 + 4*x^2 + 3*x + 3)

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maple [A]  time = 0.12, size = 23, normalized size = 0.82

method	result	size
default	\(3 x -\frac {5 \,{\mathrm e}^{5}}{4 x^{2}+3}+\frac {1}{x +1}\)	\(23\)
risch	\(3 x +\frac {x^{2}-\frac {5 x \,{\mathrm e}^{5}}{4}+\frac {3}{4}-\frac {5 \,{\mathrm e}^{5}}{4}}{x^{3}+x^{2}+\frac {3}{4} x +\frac {3}{4}}\)	\(33\)
norman	\(\frac {12 x^{4}-6+x^{2}-5 x \,{\mathrm e}^{5}-5 \,{\mathrm e}^{5}}{4 x^{3}+4 x^{2}+3 x +3}\)	\(38\)
gosper	\(-\frac {-12 x^{4}+5 x \,{\mathrm e}^{5}-x^{2}+5 \,{\mathrm e}^{5}+6}{4 x^{3}+4 x^{2}+3 x +3}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((40*x^3+80*x^2+40*x)*exp(5)+48*x^6+96*x^5+104*x^4+144*x^3+75*x^2+54*x+18)/(16*x^6+32*x^5+40*x^4+48*x^3+33
*x^2+18*x+9),x,method=_RETURNVERBOSE)

[Out]

3*x-5/(4*x^2+3)*exp(5)+1/(x+1)

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maxima [A]  time = 0.79, size = 38, normalized size = 1.36 \begin {gather*} 3 \, x + \frac {4 \, x^{2} - 5 \, x e^{5} - 5 \, e^{5} + 3}{4 \, x^{3} + 4 \, x^{2} + 3 \, x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x^3+80*x^2+40*x)*exp(5)+48*x^6+96*x^5+104*x^4+144*x^3+75*x^2+54*x+18)/(16*x^6+32*x^5+40*x^4+48*
x^3+33*x^2+18*x+9),x, algorithm="maxima")

[Out]

3*x + (4*x^2 - 5*x*e^5 - 5*e^5 + 3)/(4*x^3 + 4*x^2 + 3*x + 3)

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mupad [B]  time = 0.48, size = 44, normalized size = 1.57 \begin {gather*} \frac {9\,x-5\,{\mathrm {e}}^5-5\,x\,{\mathrm {e}}^5+13\,x^2+12\,x^3+12\,x^4+3}{\left (4\,x^2+3\right )\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((54*x + exp(5)*(40*x + 80*x^2 + 40*x^3) + 75*x^2 + 144*x^3 + 104*x^4 + 96*x^5 + 48*x^6 + 18)/(18*x + 33*x^
2 + 48*x^3 + 40*x^4 + 32*x^5 + 16*x^6 + 9),x)

[Out]

(9*x - 5*exp(5) - 5*x*exp(5) + 13*x^2 + 12*x^3 + 12*x^4 + 3)/((4*x^2 + 3)*(x + 1))

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sympy [A]  time = 0.67, size = 36, normalized size = 1.29 \begin {gather*} 3 x + \frac {4 x^{2} - 5 x e^{5} - 5 e^{5} + 3}{4 x^{3} + 4 x^{2} + 3 x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((40*x**3+80*x**2+40*x)*exp(5)+48*x**6+96*x**5+104*x**4+144*x**3+75*x**2+54*x+18)/(16*x**6+32*x**5+4
0*x**4+48*x**3+33*x**2+18*x+9),x)

[Out]

3*x + (4*x**2 - 5*x*exp(5) - 5*exp(5) + 3)/(4*x**3 + 4*x**2 + 3*x + 3)

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