3.26.73 \(\int \frac {e^{\frac {2 (10+2 x)}{3+x}} (144-32 x+16 x^2)}{9+6 x+x^2} \, dx\)

Optimal. Leaf size=16 \[ 25+16 e^{4+\frac {8}{3+x}} x \]

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Rubi [F]  time = 0.26, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{\frac {2 (10+2 x)}{3+x}} \left (144-32 x+16 x^2\right )}{9+6 x+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((2*(10 + 2*x))/(3 + x))*(144 - 32*x + 16*x^2))/(9 + 6*x + x^2),x]

[Out]

-48*E^(4 + 8/(3 + x)) + 128*E^4*ExpIntegralEi[8/(3 + x)] + 16*Defer[Int][E^((2*(10 + 2*x))/(3 + x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {2 (10+2 x)}{3+x}} \left (144-32 x+16 x^2\right )}{(3+x)^2} \, dx\\ &=\int \left (16 e^{\frac {2 (10+2 x)}{3+x}}+\frac {384 e^{\frac {2 (10+2 x)}{3+x}}}{(3+x)^2}-\frac {128 e^{\frac {2 (10+2 x)}{3+x}}}{3+x}\right ) \, dx\\ &=16 \int e^{\frac {2 (10+2 x)}{3+x}} \, dx-128 \int \frac {e^{\frac {2 (10+2 x)}{3+x}}}{3+x} \, dx+384 \int \frac {e^{\frac {2 (10+2 x)}{3+x}}}{(3+x)^2} \, dx\\ &=16 \int e^{\frac {2 (10+2 x)}{3+x}} \, dx-128 \int \frac {e^{4+\frac {8}{3+x}}}{3+x} \, dx+384 \int \frac {e^{4+\frac {8}{3+x}}}{(3+x)^2} \, dx\\ &=-48 e^{4+\frac {8}{3+x}}+128 e^4 \text {Ei}\left (\frac {8}{3+x}\right )+16 \int e^{\frac {2 (10+2 x)}{3+x}} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 14, normalized size = 0.88 \begin {gather*} 16 e^{4+\frac {8}{3+x}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((2*(10 + 2*x))/(3 + x))*(144 - 32*x + 16*x^2))/(9 + 6*x + x^2),x]

[Out]

16*E^(4 + 8/(3 + x))*x

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fricas [A]  time = 0.66, size = 14, normalized size = 0.88 \begin {gather*} 16 \, x e^{\left (\frac {4 \, {\left (x + 5\right )}}{x + 3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^2-32*x+144)*exp((2*x+10)/(3+x))^2/(x^2+6*x+9),x, algorithm="fricas")

[Out]

16*x*e^(4*(x + 5)/(x + 3))

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giac [B]  time = 0.36, size = 50, normalized size = 3.12 \begin {gather*} -\frac {16 \, {\left (\frac {3 \, {\left (x + 5\right )} e^{\left (\frac {4 \, {\left (x + 5\right )}}{x + 3}\right )}}{x + 3} - 5 \, e^{\left (\frac {4 \, {\left (x + 5\right )}}{x + 3}\right )}\right )}}{\frac {x + 5}{x + 3} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^2-32*x+144)*exp((2*x+10)/(3+x))^2/(x^2+6*x+9),x, algorithm="giac")

[Out]

-16*(3*(x + 5)*e^(4*(x + 5)/(x + 3))/(x + 3) - 5*e^(4*(x + 5)/(x + 3)))/((x + 5)/(x + 3) - 1)

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maple [A]  time = 0.42, size = 15, normalized size = 0.94

method	result	size
risch	\(16 x \,{\mathrm e}^{\frac {20+4 x}{3+x}}\)	\(15\)
gosper	\(16 x \,{\mathrm e}^{\frac {20+4 x}{3+x}}\)	\(17\)
derivativedivides	\(16 \,{\mathrm e}^{\frac {8}{3+x}+4} \left (3+x \right )-48 \,{\mathrm e}^{\frac {8}{3+x}+4}\)	\(33\)
default	\(16 \,{\mathrm e}^{\frac {8}{3+x}+4} \left (3+x \right )-48 \,{\mathrm e}^{\frac {8}{3+x}+4}\)	\(33\)
norman	\(\frac {48 x \,{\mathrm e}^{\frac {20+4 x}{3+x}}+16 x^{2} {\mathrm e}^{\frac {20+4 x}{3+x}}}{3+x}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x^2-32*x+144)*exp((2*x+10)/(3+x))^2/(x^2+6*x+9),x,method=_RETURNVERBOSE)

[Out]

16*x*exp(4*(5+x)/(3+x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 16 \, x e^{\left (\frac {8}{x + 3} + 4\right )} - 18 \, e^{\left (\frac {8}{x + 3} + 4\right )} - 144 \, \int \frac {e^{\left (\frac {8}{x + 3} + 4\right )}}{x^{2} + 6 \, x + 9}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x^2-32*x+144)*exp((2*x+10)/(3+x))^2/(x^2+6*x+9),x, algorithm="maxima")

[Out]

16*x*e^(8/(x + 3) + 4) - 18*e^(8/(x + 3) + 4) - 144*integrate(e^(8/(x + 3) + 4)/(x^2 + 6*x + 9), x)

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mupad [B]  time = 0.11, size = 15, normalized size = 0.94 \begin {gather*} 16\,x\,{\mathrm {e}}^{\frac {4\,x+20}{x+3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*(2*x + 10))/(x + 3))*(16*x^2 - 32*x + 144))/(6*x + x^2 + 9),x)

[Out]

16*x*exp((4*x + 20)/(x + 3))

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sympy [A]  time = 0.14, size = 12, normalized size = 0.75 \begin {gather*} 16 x e^{\frac {2 \left (2 x + 10\right )}{x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x**2-32*x+144)*exp((2*x+10)/(3+x))**2/(x**2+6*x+9),x)

[Out]

16*x*exp(2*(2*x + 10)/(x + 3))

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