3.99.81 \(\int (e^{2 x} (7+14 x)+x^{-3+6 x} (e^{2 x} (-2+8 x)+6 e^{2 x} x \log (x))) \, dx\)

Optimal. Leaf size=21 \[ -9+e^{2 x} \left (x+x \left (6+x^{-3+6 x}\right )\right ) \]

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Rubi [F]  time = 0.30, 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 \left (e^{2 x} (7+14 x)+x^{-3+6 x} \left (e^{2 x} (-2+8 x)+6 e^{2 x} x \log (x)\right )\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[E^(2*x)*(7 + 14*x) + x^(-3 + 6*x)*(E^(2*x)*(-2 + 8*x) + 6*E^(2*x)*x*Log[x]),x]

[Out]

(-7*E^(2*x))/2 + (7*E^(2*x)*(1 + 2*x))/2 - 2*Defer[Int][E^(2*x)*x^(-3 + 6*x), x] + 8*Defer[Int][E^(2*x)*x^(-2
+ 6*x), x] + 6*Log[x]*Defer[Int][E^(2*x)*x^(-2 + 6*x), x] - 6*Defer[Int][Defer[Int][E^(2*x)*x^(-2 + 6*x), x]/x
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int e^{2 x} (7+14 x) \, dx+\int x^{-3+6 x} \left (e^{2 x} (-2+8 x)+6 e^{2 x} x \log (x)\right ) \, dx\\ &=\frac {7}{2} e^{2 x} (1+2 x)-7 \int e^{2 x} \, dx+\int 2 e^{2 x} x^{-3+6 x} (-1+4 x+3 x \log (x)) \, dx\\ &=-\frac {7 e^{2 x}}{2}+\frac {7}{2} e^{2 x} (1+2 x)+2 \int e^{2 x} x^{-3+6 x} (-1+4 x+3 x \log (x)) \, dx\\ &=-\frac {7 e^{2 x}}{2}+\frac {7}{2} e^{2 x} (1+2 x)+2 \int \left (-e^{2 x} x^{-3+6 x}+4 e^{2 x} x^{-2+6 x}+3 e^{2 x} x^{-2+6 x} \log (x)\right ) \, dx\\ &=-\frac {7 e^{2 x}}{2}+\frac {7}{2} e^{2 x} (1+2 x)-2 \int e^{2 x} x^{-3+6 x} \, dx+6 \int e^{2 x} x^{-2+6 x} \log (x) \, dx+8 \int e^{2 x} x^{-2+6 x} \, dx\\ &=-\frac {7 e^{2 x}}{2}+\frac {7}{2} e^{2 x} (1+2 x)-2 \int e^{2 x} x^{-3+6 x} \, dx-6 \int \frac {\int e^{2 x} x^{-2+6 x} \, dx}{x} \, dx+8 \int e^{2 x} x^{-2+6 x} \, dx+(6 \log (x)) \int e^{2 x} x^{-2+6 x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 20, normalized size = 0.95 \begin {gather*} \frac {e^{2 x} \left (7 x^3+x^{6 x}\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(2*x)*(7 + 14*x) + x^(-3 + 6*x)*(E^(2*x)*(-2 + 8*x) + 6*E^(2*x)*x*Log[x]),x]

[Out]

(E^(2*x)*(7*x^3 + x^(6*x)))/x^2

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fricas [A]  time = 0.85, size = 21, normalized size = 1.00 \begin {gather*} x x^{6 \, x - 3} e^{\left (2 \, x\right )} + 7 \, x e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x)^2*log(x)+(8*x-2)*exp(x)^2)*exp((6*x-3)*log(x))+(14*x+7)*exp(x)^2,x, algorithm="fricas")

[Out]

x*x^(6*x - 3)*e^(2*x) + 7*x*e^(2*x)

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giac [A]  time = 0.29, size = 22, normalized size = 1.05 \begin {gather*} 7 \, x e^{\left (2 \, x\right )} + e^{\left (6 \, x \log \relax (x) + 2 \, x - 2 \, \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x)^2*log(x)+(8*x-2)*exp(x)^2)*exp((6*x-3)*log(x))+(14*x+7)*exp(x)^2,x, algorithm="giac")

[Out]

7*x*e^(2*x) + e^(6*x*log(x) + 2*x - 2*log(x))

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maple [A]  time = 0.07, size = 22, normalized size = 1.05

method	result	size
risch	\(x \,{\mathrm e}^{2 x} x^{6 x -3}+7 x \,{\mathrm e}^{2 x}\)	\(22\)
default	\(7 x \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} {\mathrm e}^{\left (6 x -3\right ) \ln \relax (x )} x\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*exp(x)^2*ln(x)+(8*x-2)*exp(x)^2)*exp((6*x-3)*ln(x))+(14*x+7)*exp(x)^2,x,method=_RETURNVERBOSE)

[Out]

x*exp(2*x)*x^(6*x-3)+7*x*exp(2*x)

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maxima [A]  time = 0.36, size = 32, normalized size = 1.52 \begin {gather*} \frac {7}{2} \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + \frac {e^{\left (6 \, x \log \relax (x) + 2 \, x\right )}}{x^{2}} + \frac {7}{2} \, e^{\left (2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x)^2*log(x)+(8*x-2)*exp(x)^2)*exp((6*x-3)*log(x))+(14*x+7)*exp(x)^2,x, algorithm="maxima")

[Out]

7/2*(2*x - 1)*e^(2*x) + e^(6*x*log(x) + 2*x)/x^2 + 7/2*e^(2*x)

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mupad [B]  time = 6.01, size = 19, normalized size = 0.90 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (x^{6\,x}+7\,x^3\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(log(x)*(6*x - 3))*(exp(2*x)*(8*x - 2) + 6*x*exp(2*x)*log(x)) + exp(2*x)*(14*x + 7),x)

[Out]

(exp(2*x)*(x^(6*x) + 7*x^3))/x^2

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sympy [A]  time = 5.59, size = 24, normalized size = 1.14 \begin {gather*} x e^{2 x} e^{\left (6 x - 3\right ) \log {\relax (x )}} + 7 x e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((6*x*exp(x)**2*ln(x)+(8*x-2)*exp(x)**2)*exp((6*x-3)*ln(x))+(14*x+7)*exp(x)**2,x)

[Out]

x*exp(2*x)*exp((6*x - 3)*log(x)) + 7*x*exp(2*x)

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