∫ ex3+(−20−15x+5x2) log(14+ex) __________________________________________ x2 (14x3+ex(−20x−15x2+6x3)+(560+210x+ex(40+15x)) log(14+ex)) ________________________________________________________________________________________________________________________________

3.10.41 \(\int \frac {e^{\frac {x^3+(-20-15 x+5 x^2) \log (14+e^x)}{x^2}} (14 x^3+e^x (-20 x-15 x^2+6 x^3)+(560+210 x+e^x (40+15 x)) \log (14+e^x))}{14 x^3+e^x x^3} \, dx\)

Optimal. Leaf size=23 \[ e^{x+\frac {5 \left (-3-\frac {4}{x}+x\right ) \log \left (14+e^x\right )}{x}} \]

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Rubi [F] time = 5.95, 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 {x^3+\left (-20-15 x+5 x^2\right ) \log \left (14+e^x\right )}{x^2}} \left (14 x^3+e^x \left (-20 x-15 x^2+6 x^3\right )+\left (560+210 x+e^x (40+15 x)\right ) \log \left (14+e^x\right )\right )}{14 x^3+e^x x^3} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(E^((x^3 + (-20 - 15*x + 5*x^2)*Log[14 + E^x])/x^2)*(14*x^3 + E^x*(-20*x - 15*x^2 + 6*x^3) + (560 + 210*x

+ E^x*(40 + 15*x))*Log[14 + E^x]))/(14*x^3 + E^x*x^3),x]

[Out]

14*Defer[Int][E^x*(14 + E^x)^(4 - 20/x^2 - 15/x), x] + 6*Defer[Int][E^(2*x)*(14 + E^x)^(4 - 20/x^2 - 15/x), x]

+ 560*Log[14 + E^x]*Defer[Int][(E^x*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^3, x] + 40*Log[14 + E^x]*Defer[Int][(E^

(2*x)*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^3, x] + 210*Log[14 + E^x]*Defer[Int][(E^x*(14 + E^x)^(4 - 20/x^2 - 15/

x))/x^2, x] - 20*Defer[Int][(E^(2*x)*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^2, x] + 15*Log[14 + E^x]*Defer[Int][(E^

(2*x)*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^2, x] - 15*Defer[Int][(E^(2*x)*(14 + E^x)^(4 - 20/x^2 - 15/x))/x, x] -

560*Defer[Int][(E^x*Defer[Int][(E^x*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^3, x])/(14 + E^x), x] - 40*Defer[Int][(

E^x*Defer[Int][(E^(2*x)*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^3, x])/(14 + E^x), x] - 210*Defer[Int][(E^x*Defer[In

t][(E^x*(14 + E^x)^(4 - 20/x^2 - 15/x))/x^2, x])/(14 + E^x), x] - 15*Defer[Int][(E^x*Defer[Int][(E^(2*x)*(14 +

E^x)^(4 - 20/x^2 - 15/x))/x^2, x])/(14 + E^x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (14 x^3+e^x x \left (-20-15 x+6 x^2\right )+5 \left (14+e^x\right ) (8+3 x) \log \left (14+e^x\right )\right )}{x^3} \, dx\\ &=\int \left (\frac {14 e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (x^3+40 \log \left (14+e^x\right )+15 x \log \left (14+e^x\right )\right )}{x^3}+\frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (-20 x-15 x^2+6 x^3+40 \log \left (14+e^x\right )+15 x \log \left (14+e^x\right )\right )}{x^3}\right ) \, dx\\ &=14 \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (x^3+40 \log \left (14+e^x\right )+15 x \log \left (14+e^x\right )\right )}{x^3} \, dx+\int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (-20 x-15 x^2+6 x^3+40 \log \left (14+e^x\right )+15 x \log \left (14+e^x\right )\right )}{x^3} \, dx\\ &=14 \int \left (e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}+\frac {5 e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} (8+3 x) \log \left (14+e^x\right )}{x^3}\right ) \, dx+\int \left (\frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (-20-15 x+6 x^2\right )}{x^2}+\frac {5 e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} (8+3 x) \log \left (14+e^x\right )}{x^3}\right ) \, dx\\ &=5 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} (8+3 x) \log \left (14+e^x\right )}{x^3} \, dx+14 \int e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \, dx+70 \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} (8+3 x) \log \left (14+e^x\right )}{x^3} \, dx+\int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \left (-20-15 x+6 x^2\right )}{x^2} \, dx\\ &=-\left (5 \int \frac {e^x \left (8 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx+3 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx\right )}{14+e^x} \, dx\right )+14 \int e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \, dx-70 \int \frac {e^x \left (8 \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx+3 \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx\right )}{14+e^x} \, dx+\left (15 \log \left (14+e^x\right )\right ) \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx+\left (40 \log \left (14+e^x\right )\right ) \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx+\left (210 \log \left (14+e^x\right )\right ) \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx+\left (560 \log \left (14+e^x\right )\right ) \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx+\int \left (6 e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}-\frac {20 e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2}-\frac {15 e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x}\right ) \, dx\\ &=-\left (5 \int \left (\frac {8 e^x \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx}{14+e^x}+\frac {3 e^x \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx}{14+e^x}\right ) \, dx\right )+6 \int e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \, dx+14 \int e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \, dx-15 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x} \, dx-20 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx-70 \int \left (\frac {8 e^x \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx}{14+e^x}+\frac {3 e^x \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx}{14+e^x}\right ) \, dx+\left (15 \log \left (14+e^x\right )\right ) \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx+\left (40 \log \left (14+e^x\right )\right ) \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx+\left (210 \log \left (14+e^x\right )\right ) \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx+\left (560 \log \left (14+e^x\right )\right ) \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx\\ &=6 \int e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \, dx+14 \int e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}} \, dx-15 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x} \, dx-15 \int \frac {e^x \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx}{14+e^x} \, dx-20 \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx-40 \int \frac {e^x \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx}{14+e^x} \, dx-210 \int \frac {e^x \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx}{14+e^x} \, dx-560 \int \frac {e^x \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx}{14+e^x} \, dx+\left (15 \log \left (14+e^x\right )\right ) \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx+\left (40 \log \left (14+e^x\right )\right ) \int \frac {e^{2 x} \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx+\left (210 \log \left (14+e^x\right )\right ) \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^2} \, dx+\left (560 \log \left (14+e^x\right )\right ) \int \frac {e^x \left (14+e^x\right )^{4-\frac {20}{x^2}-\frac {15}{x}}}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 1.45, size = 22, normalized size = 0.96 \begin {gather*} e^x \left (14+e^x\right )^{5-\frac {5 (4+3 x)}{x^2}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(E^((x^3 + (-20 - 15*x + 5*x^2)*Log[14 + E^x])/x^2)*(14*x^3 + E^x*(-20*x - 15*x^2 + 6*x^3) + (560 +

210*x + E^x*(40 + 15*x))*Log[14 + E^x]))/(14*x^3 + E^x*x^3),x]

[Out]

E^x*(14 + E^x)^(5 - (5*(4 + 3*x))/x^2)

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fricas [A] time = 1.08, size = 24, normalized size = 1.04 \begin {gather*} e^{\left (\frac {x^{3} + 5 \, {\left (x^{2} - 3 \, x - 4\right )} \log \left (e^{x} + 14\right )}{x^{2}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x+40)*exp(x)+210*x+560)*log(exp(x)+14)+(6*x^3-15*x^2-20*x)*exp(x)+14*x^3)*exp(((5*x^2-15*x-20)

*log(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+14*x^3),x, algorithm="fricas")

[Out]

e^((x^3 + 5*(x^2 - 3*x - 4)*log(e^x + 14))/x^2)

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giac [A] time = 1.58, size = 30, normalized size = 1.30 \begin {gather*} e^{\left (x - \frac {15 \, \log \left (e^{x} + 14\right )}{x} - \frac {20 \, \log \left (e^{x} + 14\right )}{x^{2}} + 5 \, \log \left (e^{x} + 14\right )\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x+40)*exp(x)+210*x+560)*log(exp(x)+14)+(6*x^3-15*x^2-20*x)*exp(x)+14*x^3)*exp(((5*x^2-15*x-20)

*log(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+14*x^3),x, algorithm="giac")

[Out]

e^(x - 15*log(e^x + 14)/x - 20*log(e^x + 14)/x^2 + 5*log(e^x + 14))

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maple [A] time = 0.03, size = 30, normalized size = 1.30


method	result	size

risch	\(\left ({\mathrm e}^{x}+14\right )^{5} \left ({\mathrm e}^{x}+14\right )^{-\frac {15}{x}} \left ({\mathrm e}^{x}+14\right )^{-\frac {20}{x^{2}}} {\mathrm e}^{x}\)	\(30\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((15*x+40)*exp(x)+210*x+560)*ln(exp(x)+14)+(6*x^3-15*x^2-20*x)*exp(x)+14*x^3)*exp(((5*x^2-15*x-20)*ln(exp

(x)+14)+x^3)/x^2)/(exp(x)*x^3+14*x^3),x,method=_RETURNVERBOSE)

[Out]

(exp(x)+14)^5*(exp(x)+14)^(-15/x)*(exp(x)+14)^(-20/x^2)*exp(x)

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maxima [B] time = 0.67, size = 56, normalized size = 2.43 \begin {gather*} {\left (e^{\left (6 \, x\right )} + 70 \, e^{\left (5 \, x\right )} + 1960 \, e^{\left (4 \, x\right )} + 27440 \, e^{\left (3 \, x\right )} + 192080 \, e^{\left (2 \, x\right )} + 537824 \, e^{x}\right )} e^{\left (-\frac {15 \, \log \left (e^{x} + 14\right )}{x} - \frac {20 \, \log \left (e^{x} + 14\right )}{x^{2}}\right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((15*x+40)*exp(x)+210*x+560)*log(exp(x)+14)+(6*x^3-15*x^2-20*x)*exp(x)+14*x^3)*exp(((5*x^2-15*x-20)

*log(exp(x)+14)+x^3)/x^2)/(exp(x)*x^3+14*x^3),x, algorithm="maxima")

[Out]

(e^(6*x) + 70*e^(5*x) + 1960*e^(4*x) + 27440*e^(3*x) + 192080*e^(2*x) + 537824*e^x)*e^(-15*log(e^x + 14)/x - 2

0*log(e^x + 14)/x^2)

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mupad [B] time = 0.86, size = 178, normalized size = 7.74 \begin {gather*} \frac {192080\,{\mathrm {e}}^{2\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {27440\,{\mathrm {e}}^{3\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {1960\,{\mathrm {e}}^{4\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {70\,{\mathrm {e}}^{5\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {{\mathrm {e}}^{6\,x}}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}}+\frac {537824\,{\mathrm {e}}^x}{{\left ({\mathrm {e}}^x+14\right )}^{15/x}\,{\left ({\mathrm {e}}^x+14\right )}^{\frac {20}{x^2}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(log(exp(x) + 14)*(15*x - 5*x^2 + 20) - x^3)/x^2)*(log(exp(x) + 14)*(210*x + exp(x)*(15*x + 40) + 56

0) + 14*x^3 - exp(x)*(20*x + 15*x^2 - 6*x^3)))/(x^3*exp(x) + 14*x^3),x)

[Out]

(192080*exp(2*x))/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (27440*exp(3*x))/((exp(x) + 14)^(15/x)*(exp(

x) + 14)^(20/x^2)) + (1960*exp(4*x))/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (70*exp(5*x))/((exp(x) +

14)^(15/x)*(exp(x) + 14)^(20/x^2)) + exp(6*x)/((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2)) + (537824*exp(x))/

((exp(x) + 14)^(15/x)*(exp(x) + 14)^(20/x^2))

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sympy [F(-1)] 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((((15*x+40)*exp(x)+210*x+560)*ln(exp(x)+14)+(6*x**3-15*x**2-20*x)*exp(x)+14*x**3)*exp(((5*x**2-15*x-

20)*ln(exp(x)+14)+x**3)/x**2)/(exp(x)*x**3+14*x**3),x)

[Out]

Timed out

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