3.16.21 \(\int \frac {84 x+e^{x/21} (42+2 x)+e^x (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4)}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx\)

Optimal. Leaf size=26 \[ e^x-2 \left (e^4+\frac {3}{x \left (e^{x/21}+x\right )}\right ) \]

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Rubi [F]  time = 0.98, 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 {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(84*x + E^(x/21)*(42 + 2*x) + E^x*(7*E^((2*x)/21)*x^2 + 14*E^(x/21)*x^3 + 7*x^4))/(7*E^((2*x)/21)*x^2 + 14
*E^(x/21)*x^3 + 7*x^4),x]

[Out]

E^x + 6*Defer[Int][1/(x*(E^(x/21) + x)^2), x] + 6*Defer[Int][1/(x^2*(E^(x/21) + x)), x] + (2*Defer[Int][1/(x*(
E^(x/21) + x)), x])/7 - 6*Defer[Subst][Defer[Int][(E^x + 21*x)^(-2), x], x, x/21]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{7 x^2 \left (e^{x/21}+x\right )^2} \, dx\\ &=\frac {1}{7} \int \frac {84 x+e^{x/21} (42+2 x)+e^x \left (7 e^{2 x/21} x^2+14 e^{x/21} x^3+7 x^4\right )}{x^2 \left (e^{x/21}+x\right )^2} \, dx\\ &=\frac {1}{7} \int \left (7 e^x-\frac {2 (-21+x)}{x \left (e^{x/21}+x\right )^2}+\frac {2 (21+x)}{x^2 \left (e^{x/21}+x\right )}\right ) \, dx\\ &=-\left (\frac {2}{7} \int \frac {-21+x}{x \left (e^{x/21}+x\right )^2} \, dx\right )+\frac {2}{7} \int \frac {21+x}{x^2 \left (e^{x/21}+x\right )} \, dx+\int e^x \, dx\\ &=e^x-\frac {2}{7} \int \left (\frac {1}{\left (e^{x/21}+x\right )^2}-\frac {21}{x \left (e^{x/21}+x\right )^2}\right ) \, dx+\frac {2}{7} \int \left (\frac {21}{x^2 \left (e^{x/21}+x\right )}+\frac {1}{x \left (e^{x/21}+x\right )}\right ) \, dx\\ &=e^x-\frac {2}{7} \int \frac {1}{\left (e^{x/21}+x\right )^2} \, dx+\frac {2}{7} \int \frac {1}{x \left (e^{x/21}+x\right )} \, dx+6 \int \frac {1}{x \left (e^{x/21}+x\right )^2} \, dx+6 \int \frac {1}{x^2 \left (e^{x/21}+x\right )} \, dx\\ &=e^x+\frac {2}{7} \int \frac {1}{x \left (e^{x/21}+x\right )} \, dx+6 \int \frac {1}{x \left (e^{x/21}+x\right )^2} \, dx+6 \int \frac {1}{x^2 \left (e^{x/21}+x\right )} \, dx-6 \operatorname {Subst}\left (\int \frac {1}{\left (e^x+21 x\right )^2} \, dx,x,\frac {x}{21}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 26, normalized size = 1.00 \begin {gather*} \frac {1}{7} \left (7 e^x-\frac {42}{x \left (e^{x/21}+x\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(84*x + E^(x/21)*(42 + 2*x) + E^x*(7*E^((2*x)/21)*x^2 + 14*E^(x/21)*x^3 + 7*x^4))/(7*E^((2*x)/21)*x^
2 + 14*E^(x/21)*x^3 + 7*x^4),x]

[Out]

(7*E^x - 42/(x*(E^(x/21) + x)))/7

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fricas [A]  time = 0.60, size = 27, normalized size = 1.04 \begin {gather*} \frac {x^{2} e^{x} + x e^{\left (\frac {22}{21} \, x\right )} - 6}{x^{2} + x e^{\left (\frac {1}{21} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^2*exp(1/21*x)^2+14*x^3*exp(1/21*x)+7*x^4)*exp(x)+(2*x+42)*exp(1/21*x)+84*x)/(7*x^2*exp(1/21*x)
^2+14*x^3*exp(1/21*x)+7*x^4),x, algorithm="fricas")

[Out]

(x^2*e^x + x*e^(22/21*x) - 6)/(x^2 + x*e^(1/21*x))

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giac [A]  time = 0.24, size = 27, normalized size = 1.04 \begin {gather*} \frac {x^{2} e^{x} + x e^{\left (\frac {22}{21} \, x\right )} - 6}{x^{2} + x e^{\left (\frac {1}{21} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^2*exp(1/21*x)^2+14*x^3*exp(1/21*x)+7*x^4)*exp(x)+(2*x+42)*exp(1/21*x)+84*x)/(7*x^2*exp(1/21*x)
^2+14*x^3*exp(1/21*x)+7*x^4),x, algorithm="giac")

[Out]

(x^2*e^x + x*e^(22/21*x) - 6)/(x^2 + x*e^(1/21*x))

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maple [A]  time = 0.04, size = 17, normalized size = 0.65

method	result	size
risch	\({\mathrm e}^{x}-\frac {6}{x \left ({\mathrm e}^{\frac {x}{21}}+x \right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((7*x^2*exp(1/21*x)^2+14*x^3*exp(1/21*x)+7*x^4)*exp(x)+(2*x+42)*exp(1/21*x)+84*x)/(7*x^2*exp(1/21*x)^2+14*
x^3*exp(1/21*x)+7*x^4),x,method=_RETURNVERBOSE)

[Out]

exp(x)-6/x/(exp(1/21*x)+x)

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maxima [A]  time = 0.46, size = 27, normalized size = 1.04 \begin {gather*} \frac {x^{2} e^{x} + x e^{\left (\frac {22}{21} \, x\right )} - 6}{x^{2} + x e^{\left (\frac {1}{21} \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x^2*exp(1/21*x)^2+14*x^3*exp(1/21*x)+7*x^4)*exp(x)+(2*x+42)*exp(1/21*x)+84*x)/(7*x^2*exp(1/21*x)
^2+14*x^3*exp(1/21*x)+7*x^4),x, algorithm="maxima")

[Out]

(x^2*e^x + x*e^(22/21*x) - 6)/(x^2 + x*e^(1/21*x))

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mupad [B]  time = 1.04, size = 16, normalized size = 0.62 \begin {gather*} {\mathrm {e}}^x-\frac {6}{x\,\left (x+{\mathrm {e}}^{x/21}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((84*x + exp(x)*(7*x^2*exp((2*x)/21) + 14*x^3*exp(x/21) + 7*x^4) + exp(x/21)*(2*x + 42))/(7*x^2*exp((2*x)/2
1) + 14*x^3*exp(x/21) + 7*x^4),x)

[Out]

exp(x) - 6/(x*(x + exp(x/21)))

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sympy [A]  time = 0.24, size = 14, normalized size = 0.54 \begin {gather*} e^{x} - \frac {6}{x^{2} + x e^{\frac {x}{21}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((7*x**2*exp(1/21*x)**2+14*x**3*exp(1/21*x)+7*x**4)*exp(x)+(2*x+42)*exp(1/21*x)+84*x)/(7*x**2*exp(1/
21*x)**2+14*x**3*exp(1/21*x)+7*x**4),x)

[Out]

exp(x) - 6/(x**2 + x*exp(x/21))

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