3.22.29 \(\int \frac {75 e^{2 x}+5 e^x x^2}{x^2+e^x (30 x+70 x^2)+e^{2 x} (225+1050 x+1225 x^2)} \, dx\)

Optimal. Leaf size=23 \[ \frac {1}{5+\frac {3+2 x+\frac {e^{-x} x}{5}}{x}} \]

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Rubi [F]  time = 1.76, 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 {75 e^{2 x}+5 e^x x^2}{x^2+e^x \left (30 x+70 x^2\right )+e^{2 x} \left (225+1050 x+1225 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(75*E^(2*x) + 5*E^x*x^2)/(x^2 + E^x*(30*x + 70*x^2) + E^(2*x)*(225 + 1050*x + 1225*x^2)),x]

[Out]

(-15*Defer[Int][E^x/(15*E^x + x + 35*E^x*x)^2, x])/7 + 5*Defer[Int][(E^x*x^2)/(15*E^x + x + 35*E^x*x)^2, x] +
(45*Defer[Int][E^x/((3 + 7*x)*(15*E^x + x + 35*E^x*x)^2), x])/7 + 15*Defer[Int][E^x/((3 + 7*x)*(15*E^x + x + 3
5*E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {5 e^x \left (15 e^x+x^2\right )}{\left (x+5 e^x (3+7 x)\right )^2} \, dx\\ &=5 \int \frac {e^x \left (15 e^x+x^2\right )}{\left (x+5 e^x (3+7 x)\right )^2} \, dx\\ &=5 \int \left (\frac {3 e^x}{(3+7 x) \left (15 e^x+x+35 e^x x\right )}+\frac {e^x x \left (-3+3 x+7 x^2\right )}{(3+7 x) \left (15 e^x+x+35 e^x x\right )^2}\right ) \, dx\\ &=5 \int \frac {e^x x \left (-3+3 x+7 x^2\right )}{(3+7 x) \left (15 e^x+x+35 e^x x\right )^2} \, dx+15 \int \frac {e^x}{(3+7 x) \left (15 e^x+x+35 e^x x\right )} \, dx\\ &=5 \int \left (-\frac {3 e^x}{7 \left (15 e^x+x+35 e^x x\right )^2}+\frac {e^x x^2}{\left (15 e^x+x+35 e^x x\right )^2}+\frac {9 e^x}{7 (3+7 x) \left (15 e^x+x+35 e^x x\right )^2}\right ) \, dx+15 \int \frac {e^x}{(3+7 x) \left (15 e^x+x+35 e^x x\right )} \, dx\\ &=-\left (\frac {15}{7} \int \frac {e^x}{\left (15 e^x+x+35 e^x x\right )^2} \, dx\right )+5 \int \frac {e^x x^2}{\left (15 e^x+x+35 e^x x\right )^2} \, dx+\frac {45}{7} \int \frac {e^x}{(3+7 x) \left (15 e^x+x+35 e^x x\right )^2} \, dx+15 \int \frac {e^x}{(3+7 x) \left (15 e^x+x+35 e^x x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.31, size = 25, normalized size = 1.09 \begin {gather*} -\frac {15 e^x+x}{7 \left (x+5 e^x (3+7 x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(75*E^(2*x) + 5*E^x*x^2)/(x^2 + E^x*(30*x + 70*x^2) + E^(2*x)*(225 + 1050*x + 1225*x^2)),x]

[Out]

-1/7*(15*E^x + x)/(x + 5*E^x*(3 + 7*x))

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fricas [A]  time = 0.54, size = 21, normalized size = 0.91 \begin {gather*} -\frac {x + 15 \, e^{x}}{7 \, {\left (5 \, {\left (7 \, x + 3\right )} e^{x} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((75*exp(x)^2+5*exp(x)*x^2)/((1225*x^2+1050*x+225)*exp(x)^2+(70*x^2+30*x)*exp(x)+x^2),x, algorithm="f
ricas")

[Out]

-1/7*(x + 15*e^x)/(5*(7*x + 3)*e^x + x)

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giac [A]  time = 0.15, size = 21, normalized size = 0.91 \begin {gather*} -\frac {x + 15 \, e^{x}}{7 \, {\left (35 \, x e^{x} + x + 15 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((75*exp(x)^2+5*exp(x)*x^2)/((1225*x^2+1050*x+225)*exp(x)^2+(70*x^2+30*x)*exp(x)+x^2),x, algorithm="g
iac")

[Out]

-1/7*(x + 15*e^x)/(35*x*e^x + x + 15*e^x)

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maple [A]  time = 0.04, size = 19, normalized size = 0.83

method	result	size
norman	\(\frac {5 \,{\mathrm e}^{x} x}{35 \,{\mathrm e}^{x} x +15 \,{\mathrm e}^{x}+x}\)	\(19\)
risch	\(-\frac {3}{49 \left (x +\frac {3}{7}\right )}-\frac {x^{2}}{\left (7 x +3\right ) \left (35 \,{\mathrm e}^{x} x +15 \,{\mathrm e}^{x}+x \right )}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((75*exp(x)^2+5*exp(x)*x^2)/((1225*x^2+1050*x+225)*exp(x)^2+(70*x^2+30*x)*exp(x)+x^2),x,method=_RETURNVERBO
SE)

[Out]

5*exp(x)*x/(35*exp(x)*x+15*exp(x)+x)

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maxima [A]  time = 0.41, size = 21, normalized size = 0.91 \begin {gather*} -\frac {x + 15 \, e^{x}}{7 \, {\left (5 \, {\left (7 \, x + 3\right )} e^{x} + x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((75*exp(x)^2+5*exp(x)*x^2)/((1225*x^2+1050*x+225)*exp(x)^2+(70*x^2+30*x)*exp(x)+x^2),x, algorithm="m
axima")

[Out]

-1/7*(x + 15*e^x)/(5*(7*x + 3)*e^x + x)

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mupad [B]  time = 1.23, size = 34, normalized size = 1.48 \begin {gather*} \frac {x}{7\,x+3}-\frac {x^2}{\left (7\,x+3\right )\,\left (x+{\mathrm {e}}^x\,\left (35\,x+15\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((75*exp(2*x) + 5*x^2*exp(x))/(exp(2*x)*(1050*x + 1225*x^2 + 225) + exp(x)*(30*x + 70*x^2) + x^2),x)

[Out]

x/(7*x + 3) - x^2/((7*x + 3)*(x + exp(x)*(35*x + 15)))

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sympy [B]  time = 0.22, size = 32, normalized size = 1.39 \begin {gather*} - \frac {x^{2}}{7 x^{2} + 3 x + \left (245 x^{2} + 210 x + 45\right ) e^{x}} - \frac {3}{49 x + 21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((75*exp(x)**2+5*exp(x)*x**2)/((1225*x**2+1050*x+225)*exp(x)**2+(70*x**2+30*x)*exp(x)+x**2),x)

[Out]

-x**2/(7*x**2 + 3*x + (245*x**2 + 210*x + 45)*exp(x)) - 3/(49*x + 21)

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