3.8.84 \(\int \frac {5^{e x} (\frac {x^4}{6+6 x^2})^{e x} (1+x^2+e (4 x+2 x^3)+e (x+x^3) \log (\frac {5 x^4}{6+6 x^2}))}{1+x^2} \, dx\)

Optimal. Leaf size=26 \[ \left (\frac {5}{3}\right )^{e x} x \left (\frac {x^4}{2+2 x^2}\right )^{e x} \]

________________________________________________________________________________________

Rubi [F]  time = 2.30, 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 \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \left (1+x^2+e \left (4 x+2 x^3\right )+e \left (x+x^3\right ) \log \left (\frac {5 x^4}{6+6 x^2}\right )\right )}{1+x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(5^(E*x)*(x^4/(6 + 6*x^2))^(E*x)*(1 + x^2 + E*(4*x + 2*x^3) + E*(x + x^3)*Log[(5*x^4)/(6 + 6*x^2)]))/(1 +
x^2),x]

[Out]

Defer[Int][5^(E*x)*(x^4/(6 + 6*x^2))^(E*x), x] - E*Defer[Int][(5^(E*x)*(x^4/(6 + 6*x^2))^(E*x))/(I - x), x] +
2*E*Defer[Int][5^(E*x)*x*(x^4/(6 + 6*x^2))^(E*x), x] + E*Log[(5*x^4)/(6*(1 + x^2))]*Defer[Int][5^(E*x)*x*(x^4/
(6 + 6*x^2))^(E*x), x] + E*Defer[Int][(5^(E*x)*(x^4/(6 + 6*x^2))^(E*x))/(I + x), x] - E*Defer[Int][Defer[Int][
5^(E*x)*x*(x^4/(6 + 6*x^2))^(E*x), x]/(I - x), x] - 4*E*Defer[Int][Defer[Int][5^(E*x)*x*(x^4/(6 + 6*x^2))^(E*x
), x]/x, x] + E*Defer[Int][Defer[Int][5^(E*x)*x*(x^4/(6 + 6*x^2))^(E*x), x]/(I + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \left (1+4 e x+x^2+2 e x^3\right )}{1+x^2}+5^{e x} e x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \, dx\\ &=e \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \log \left (\frac {5 x^4}{6+6 x^2}\right ) \, dx+\int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \left (1+4 e x+x^2+2 e x^3\right )}{1+x^2} \, dx\\ &=-\left (e \int \frac {2 \left (2+x^2\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{x \left (1+x^2\right )} \, dx\right )+\left (e \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+\int \left (5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}+2\ 5^{e x} e x \left (\frac {x^4}{6+6 x^2}\right )^{e x}+\frac {2\ 5^{e x} e x \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{1+x^2}\right ) \, dx\\ &=(2 e) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+(2 e) \int \frac {5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{1+x^2} \, dx-(2 e) \int \frac {\left (2+x^2\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{x \left (1+x^2\right )} \, dx+\left (e \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+\int 5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx\\ &=(2 e) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+(2 e) \int \left (-\frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{2 (i-x)}+\frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{2 (i+x)}\right ) \, dx-(2 e) \int \left (\frac {2 \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{x}-\frac {x \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{1+x^2}\right ) \, dx+\left (e \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+\int 5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx\\ &=-\left (e \int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{i-x} \, dx\right )+e \int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{i+x} \, dx+(2 e) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+(2 e) \int \frac {x \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{1+x^2} \, dx-(4 e) \int \frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{x} \, dx+\left (e \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+\int 5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx\\ &=-\left (e \int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{i-x} \, dx\right )+e \int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{i+x} \, dx+(2 e) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+(2 e) \int \left (-\frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{2 (i-x)}+\frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{2 (i+x)}\right ) \, dx-(4 e) \int \frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{x} \, dx+\left (e \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+\int 5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx\\ &=-\left (e \int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{i-x} \, dx\right )+e \int \frac {5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x}}{i+x} \, dx-e \int \frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{i-x} \, dx+e \int \frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{i+x} \, dx+(2 e) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx-(4 e) \int \frac {\int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx}{x} \, dx+\left (e \log \left (\frac {5 x^4}{6+6 x^2}\right )\right ) \int 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx+\int 5^{e x} \left (\frac {x^4}{6+6 x^2}\right )^{e x} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.93, size = 24, normalized size = 0.92 \begin {gather*} 5^{e x} x \left (\frac {x^4}{6+6 x^2}\right )^{e x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(5^(E*x)*(x^4/(6 + 6*x^2))^(E*x)*(1 + x^2 + E*(4*x + 2*x^3) + E*(x + x^3)*Log[(5*x^4)/(6 + 6*x^2)]))
/(1 + x^2),x]

[Out]

5^(E*x)*x*(x^4/(6 + 6*x^2))^(E*x)

________________________________________________________________________________________

fricas [A]  time = 1.14, size = 19, normalized size = 0.73 \begin {gather*} \left (\frac {5 \, x^{4}}{6 \, {\left (x^{2} + 1\right )}}\right )^{x e} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x)*exp(1)*log(5*x^4/(6*x^2+6))+(2*x^3+4*x)*exp(1)+x^2+1)*exp(x*exp(1)*log(5*x^4/(6*x^2+6)))/(x
^2+1),x, algorithm="fricas")

[Out]

(5/6*x^4/(x^2 + 1))^(x*e)*x

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{3} + x\right )} e \log \left (\frac {5 \, x^{4}}{6 \, {\left (x^{2} + 1\right )}}\right ) + x^{2} + 2 \, {\left (x^{3} + 2 \, x\right )} e + 1\right )} \left (\frac {5 \, x^{4}}{6 \, {\left (x^{2} + 1\right )}}\right )^{x e}}{x^{2} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x)*exp(1)*log(5*x^4/(6*x^2+6))+(2*x^3+4*x)*exp(1)+x^2+1)*exp(x*exp(1)*log(5*x^4/(6*x^2+6)))/(x
^2+1),x, algorithm="giac")

[Out]

integrate(((x^3 + x)*e*log(5/6*x^4/(x^2 + 1)) + x^2 + 2*(x^3 + 2*x)*e + 1)*(5/6*x^4/(x^2 + 1))^(x*e)/(x^2 + 1)
, x)

________________________________________________________________________________________

maple [A]  time = 0.40, size = 22, normalized size = 0.85

method	result	size
risch	\(x \left (\frac {5 x^{4}}{6 x^{2}+6}\right )^{x \,{\mathrm e}}\)	\(22\)
norman	\(x \,{\mathrm e}^{x \,{\mathrm e} \ln \left (\frac {5 x^{4}}{6 x^{2}+6}\right )}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3+x)*exp(1)*ln(5*x^4/(6*x^2+6))+(2*x^3+4*x)*exp(1)+x^2+1)*exp(x*exp(1)*ln(5*x^4/(6*x^2+6)))/(x^2+1),x,
method=_RETURNVERBOSE)

[Out]

x*(5*x^4/(6*x^2+6))^(x*exp(1))

________________________________________________________________________________________

maxima [B]  time = 0.79, size = 42, normalized size = 1.62 \begin {gather*} x e^{\left (x e \log \relax (5) - x e \log \relax (3) - x e \log \relax (2) - x e \log \left (x^{2} + 1\right ) + 4 \, x e \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^3+x)*exp(1)*log(5*x^4/(6*x^2+6))+(2*x^3+4*x)*exp(1)+x^2+1)*exp(x*exp(1)*log(5*x^4/(6*x^2+6)))/(x
^2+1),x, algorithm="maxima")

[Out]

x*e^(x*e*log(5) - x*e*log(3) - x*e*log(2) - x*e*log(x^2 + 1) + 4*x*e*log(x))

________________________________________________________________________________________

mupad [B]  time = 0.92, size = 21, normalized size = 0.81 \begin {gather*} x\,{\left (\frac {5\,x^4}{6\,x^2+6}\right )}^{x\,\mathrm {e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x*exp(1)*log((5*x^4)/(6*x^2 + 6)))*(exp(1)*(4*x + 2*x^3) + x^2 + exp(1)*log((5*x^4)/(6*x^2 + 6))*(x +
x^3) + 1))/(x^2 + 1),x)

[Out]

x*((5*x^4)/(6*x^2 + 6))^(x*exp(1))

________________________________________________________________________________________

sympy [A]  time = 0.55, size = 20, normalized size = 0.77 \begin {gather*} x e^{e x \log {\left (\frac {5 x^{4}}{6 x^{2} + 6} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**3+x)*exp(1)*ln(5*x**4/(6*x**2+6))+(2*x**3+4*x)*exp(1)+x**2+1)*exp(x*exp(1)*ln(5*x**4/(6*x**2+6)
))/(x**2+1),x)

[Out]

x*exp(E*x*log(5*x**4/(6*x**2 + 6)))

________________________________________________________________________________________