3.37.12 \(\int \frac {75+27 e^4+18 x+27 x^2+e^2 (18+54 x)}{25+9 e^4+18 e^2 x+9 x^2} \, dx\)

Optimal. Leaf size=18 \[ 3 (4+x)+\log \left (25+9 \left (e^2+x\right )^2\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 3, number of rules used = 2, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.044, Rules used = {1657, 628} \begin {gather*} \log \left (9 x^2+18 e^2 x+9 e^4+25\right )+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(75 + 27*E^4 + 18*x + 27*x^2 + E^2*(18 + 54*x))/(25 + 9*E^4 + 18*E^2*x + 9*x^2),x]

[Out]

3*x + Log[25 + 9*E^4 + 18*E^2*x + 9*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 {gather*} \begin {aligned} \text {integral} &=\int \left (3+\frac {18 \left (e^2+x\right )}{25+9 e^4+18 e^2 x+9 x^2}\right ) \, dx\\ &=3 x+18 \int \frac {e^2+x}{25+9 e^4+18 e^2 x+9 x^2} \, dx\\ &=3 x+\log \left (25+9 e^4+18 e^2 x+9 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.50 \begin {gather*} 3 \left (x+\frac {1}{3} \log \left (25+9 e^4+18 e^2 x+9 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(75 + 27*E^4 + 18*x + 27*x^2 + E^2*(18 + 54*x))/(25 + 9*E^4 + 18*E^2*x + 9*x^2),x]

[Out]

3*(x + Log[25 + 9*E^4 + 18*E^2*x + 9*x^2]/3)

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fricas [A]  time = 0.57, size = 21, normalized size = 1.17 \begin {gather*} 3 \, x + \log \left (9 \, x^{2} + 18 \, x e^{2} + 9 \, e^{4} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(2)^2+(54*x+18)*exp(2)+27*x^2+18*x+75)/(9*exp(2)^2+18*exp(2)*x+9*x^2+25),x, algorithm="fricas
")

[Out]

3*x + log(9*x^2 + 18*x*e^2 + 9*e^4 + 25)

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giac [A]  time = 0.12, size = 21, normalized size = 1.17 \begin {gather*} 3 \, x + \log \left (9 \, x^{2} + 18 \, x e^{2} + 9 \, e^{4} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(2)^2+(54*x+18)*exp(2)+27*x^2+18*x+75)/(9*exp(2)^2+18*exp(2)*x+9*x^2+25),x, algorithm="giac")

[Out]

3*x + log(9*x^2 + 18*x*e^2 + 9*e^4 + 25)

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maple [A]  time = 0.26, size = 22, normalized size = 1.22

method	result	size
default	\(3 x +\ln \left (9 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +9 x^{2}+25\right )\)	\(22\)
risch	\(3 x +\ln \left (9 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +9 x^{2}+25\right )\)	\(22\)
norman	\(3 x +\ln \left (9 \,{\mathrm e}^{4}+18 \,{\mathrm e}^{2} x +9 x^{2}+25\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*exp(2)^2+(54*x+18)*exp(2)+27*x^2+18*x+75)/(9*exp(2)^2+18*exp(2)*x+9*x^2+25),x,method=_RETURNVERBOSE)

[Out]

3*x+ln(9*exp(4)+18*exp(2)*x+9*x^2+25)

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maxima [A]  time = 0.35, size = 21, normalized size = 1.17 \begin {gather*} 3 \, x + \log \left (9 \, x^{2} + 18 \, x e^{2} + 9 \, e^{4} + 25\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(2)^2+(54*x+18)*exp(2)+27*x^2+18*x+75)/(9*exp(2)^2+18*exp(2)*x+9*x^2+25),x, algorithm="maxima
")

[Out]

3*x + log(9*x^2 + 18*x*e^2 + 9*e^4 + 25)

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mupad [B]  time = 0.10, size = 17, normalized size = 0.94 \begin {gather*} 3\,x+\ln \left (x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4+\frac {25}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + 27*exp(4) + 27*x^2 + exp(2)*(54*x + 18) + 75)/(9*exp(4) + 18*x*exp(2) + 9*x^2 + 25),x)

[Out]

3*x + log(exp(4) + 2*x*exp(2) + x^2 + 25/9)

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sympy [A]  time = 0.15, size = 22, normalized size = 1.22 \begin {gather*} 3 x + \log {\left (9 x^{2} + 18 x e^{2} + 25 + 9 e^{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*exp(2)**2+(54*x+18)*exp(2)+27*x**2+18*x+75)/(9*exp(2)**2+18*exp(2)*x+9*x**2+25),x)

[Out]

3*x + log(9*x**2 + 18*x*exp(2) + 25 + 9*exp(4))

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