3.48.93 \(\int \frac {e^{\frac {1}{2} (-1-10 x)} (-17-85 x)+e^x (-3 x^2+e^{\frac {1}{2} (-1-10 x)} (3+18 x))}{3 e^{-1-10 x}-6 e^{\frac {1}{2} (-1-10 x)} x+3 x^2} \, dx\) [4793]

Optimal. Leaf size=28 \[ \frac {\left (\frac {17}{3}-e^x\right ) x}{-e^{5 \left (-\frac {1}{10}-x\right )}+x} \]

[Out]

x*(17/3-exp(x))/(x-exp(-5*x-1/2))

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Rubi [F]
time = 2.44, 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 {1}{2} (-1-10 x)} (-17-85 x)+e^x \left (-3 x^2+e^{\frac {1}{2} (-1-10 x)} (3+18 x)\right )}{3 e^{-1-10 x}-6 e^{\frac {1}{2} (-1-10 x)} x+3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-1 - 10*x)/2)*(-17 - 85*x) + E^x*(-3*x^2 + E^((-1 - 10*x)/2)*(3 + 18*x)))/(3*E^(-1 - 10*x) - 6*E^((-1
- 10*x)/2)*x + 3*x^2),x]

[Out]

(-17*Defer[Int][E^(1/2 + 5*x)/(-1 + E^(1/2 + 5*x)*x)^2, x])/3 + Defer[Int][E^(1/2 + 6*x)/(-1 + E^(1/2 + 5*x)*x
)^2, x] - (85*Defer[Int][(E^(1/2 + 5*x)*x)/(-1 + E^(1/2 + 5*x)*x)^2, x])/3 + 5*Defer[Int][(E^(1/2 + 6*x)*x)/(-
1 + E^(1/2 + 5*x)*x)^2, x] - Defer[Int][(E^(1/2 + 6*x)*x)/(-1 + E^(1/2 + 5*x)*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x-85 x+18 e^x x-3 e^{\frac {1}{2}+6 x} x^2\right )}{3 \left (1-e^{\frac {1}{2}+5 x} x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x-85 x+18 e^x x-3 e^{\frac {1}{2}+6 x} x^2\right )}{\left (1-e^{\frac {1}{2}+5 x} x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (\frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x\right ) (1+5 x)}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}-\frac {3 e^{\frac {1}{2}+6 x} x}{-1+e^{\frac {1}{2}+5 x} x}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x\right ) (1+5 x)}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx-\int \frac {e^{\frac {1}{2}+6 x} x}{-1+e^{\frac {1}{2}+5 x} x} \, dx\\ &=\frac {1}{3} \int \left (\frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x\right )}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}+\frac {5 e^{\frac {1}{2}+5 x} \left (-17+3 e^x\right ) x}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}\right ) \, dx-\int \frac {e^{\frac {1}{2}+6 x} x}{-1+e^{\frac {1}{2}+5 x} x} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x\right )}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx+\frac {5}{3} \int \frac {e^{\frac {1}{2}+5 x} \left (-17+3 e^x\right ) x}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx-\int \frac {e^{\frac {1}{2}+6 x} x}{-1+e^{\frac {1}{2}+5 x} x} \, dx\\ &=\frac {1}{3} \int \left (-\frac {17 e^{\frac {1}{2}+5 x}}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}+\frac {3 e^{\frac {1}{2}+6 x}}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}\right ) \, dx+\frac {5}{3} \int \left (-\frac {17 e^{\frac {1}{2}+5 x} x}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}+\frac {3 e^{\frac {1}{2}+6 x} x}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2}\right ) \, dx-\int \frac {e^{\frac {1}{2}+6 x} x}{-1+e^{\frac {1}{2}+5 x} x} \, dx\\ &=5 \int \frac {e^{\frac {1}{2}+6 x} x}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx-\frac {17}{3} \int \frac {e^{\frac {1}{2}+5 x}}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx-\frac {85}{3} \int \frac {e^{\frac {1}{2}+5 x} x}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx+\int \frac {e^{\frac {1}{2}+6 x}}{\left (-1+e^{\frac {1}{2}+5 x} x\right )^2} \, dx-\int \frac {e^{\frac {1}{2}+6 x} x}{-1+e^{\frac {1}{2}+5 x} x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 2.10, size = 32, normalized size = 1.14 \begin {gather*} -e^x-\frac {-17+3 e^x}{3 \left (-1+e^{\frac {1}{2}+5 x} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-1 - 10*x)/2)*(-17 - 85*x) + E^x*(-3*x^2 + E^((-1 - 10*x)/2)*(3 + 18*x)))/(3*E^(-1 - 10*x) - 6*
E^((-1 - 10*x)/2)*x + 3*x^2),x]

[Out]

-E^x - (-17 + 3*E^x)/(3*(-1 + E^(1/2 + 5*x)*x))

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Maple [A]
time = 0.59, size = 28, normalized size = 1.00

method	result	size
risch	\(-{\mathrm e}^{x}+\frac {{\mathrm e}^{-\frac {1}{2}} \left (3 \,{\mathrm e}^{x}-17\right )}{-3 x \,{\mathrm e}^{5 x}+3 \,{\mathrm e}^{-\frac {1}{2}}}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((18*x+3)*exp(-5*x-1/2)-3*x^2)*exp(x)+(-85*x-17)*exp(-5*x-1/2))/(3*exp(-5*x-1/2)^2-6*x*exp(-5*x-1/2)+3*x^
2),x,method=_RETURNVERBOSE)

[Out]

-exp(x)+1/3*exp(-1/2)*(3*exp(x)-17)/(-x*exp(5*x)+exp(-1/2))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x+3)*exp(-5*x-1/2)-3*x^2)*exp(x)+(-85*x-17)*exp(-5*x-1/2))/(3*exp(-5*x-1/2)^2-6*x*exp(-5*x-1/2
)+3*x^2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [A]
time = 0.40, size = 25, normalized size = 0.89 \begin {gather*} -\frac {3 \, x e^{\left (6 \, x + \frac {1}{2}\right )} - 17}{3 \, {\left (x e^{\left (5 \, x + \frac {1}{2}\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x+3)*exp(-5*x-1/2)-3*x^2)*exp(x)+(-85*x-17)*exp(-5*x-1/2))/(3*exp(-5*x-1/2)^2-6*x*exp(-5*x-1/2
)+3*x^2),x, algorithm="fricas")

[Out]

-1/3*(3*x*e^(6*x + 1/2) - 17)/(x*e^(5*x + 1/2) - 1)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x+3)*exp(-5*x-1/2)-3*x**2)*exp(x)+(-85*x-17)*exp(-5*x-1/2))/(3*exp(-5*x-1/2)**2-6*x*exp(-5*x-1
/2)+3*x**2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (21) = 42\).
time = 0.42, size = 48, normalized size = 1.71 \begin {gather*} -\frac {x e^{\left (6 \, x + \frac {1}{2}\right )}}{x e^{\left (5 \, x + \frac {1}{2}\right )} - 1} + \frac {170}{3 \, {\left ({\left (10 \, x + 1\right )} e^{\left (5 \, x + \frac {1}{2}\right )} - e^{\left (5 \, x + \frac {1}{2}\right )} - 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((18*x+3)*exp(-5*x-1/2)-3*x^2)*exp(x)+(-85*x-17)*exp(-5*x-1/2))/(3*exp(-5*x-1/2)^2-6*x*exp(-5*x-1/2
)+3*x^2),x, algorithm="giac")

[Out]

-x*e^(6*x + 1/2)/(x*e^(5*x + 1/2) - 1) + 170/3/((10*x + 1)*e^(5*x + 1/2) - e^(5*x + 1/2) - 10)

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Mupad [B]
time = 3.78, size = 26, normalized size = 0.93 \begin {gather*} -\frac {3\,x\,{\mathrm {e}}^{6\,x}\,\sqrt {\mathrm {e}}-17}{3\,\left (x\,{\mathrm {e}}^{5\,x}\,\sqrt {\mathrm {e}}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*(exp(- 5*x - 1/2)*(18*x + 3) - 3*x^2) - exp(- 5*x - 1/2)*(85*x + 17))/(3*exp(- 10*x - 1) - 6*x*exp
(- 5*x - 1/2) + 3*x^2),x)

[Out]

-(3*x*exp(6*x)*exp(1/2) - 17)/(3*(x*exp(5*x)*exp(1/2) - 1))

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