3.57.13 \(\int \frac {(4-x)^{\frac {5 e^x}{-3+20 e^x}} (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x))+(4-x)^{\frac {10 e^x}{-3+20 e^x}} (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x))}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx\)

Optimal. Leaf size=23 \[ \left (3+(4-x)^{\frac {1}{4-\frac {3 e^{-x}}{5}}}\right )^2 \]

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Rubi [F]  time = 5.73, 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 {(4-x)^{\frac {5 e^x}{-3+20 e^x}} \left (-90 e^x+600 e^{2 x}+e^x (360-90 x) \log (4-x)\right )+(4-x)^{\frac {10 e^x}{-3+20 e^x}} \left (-30 e^x+200 e^{2 x}+e^x (120-30 x) \log (4-x)\right )}{-36+e^x (480-120 x)+9 x+e^{2 x} (-1600+400 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((4 - x)^((5*E^x)/(-3 + 20*E^x))*(-90*E^x + 600*E^(2*x) + E^x*(360 - 90*x)*Log[4 - x]) + (4 - x)^((10*E^x)
/(-3 + 20*E^x))*(-30*E^x + 200*E^(2*x) + E^x*(120 - 30*x)*Log[4 - x]))/(-36 + E^x*(480 - 120*x) + 9*x + E^(2*x
)*(-1600 + 400*x)),x]

[Out]

-90*Log[4 - x]*Defer[Int][(E^x*(4 - x)^((5*E^x)/(-3 + 20*E^x)))/(3 - 20*E^x)^2, x] - 30*Log[4 - x]*Defer[Int][
(E^x*(4 - x)^((10*E^x)/(-3 + 20*E^x)))/(3 - 20*E^x)^2, x] - 10*Defer[Int][(E^x*(4 - x)^((3 - 10*E^x)/(-3 + 20*
E^x)))/(-3 + 20*E^x), x] - 30*Defer[Int][E^x/((-3 + 20*E^x)*(4 - x)^((3*(-1 + 5*E^x))/(-3 + 20*E^x))), x] - 90
*Defer[Int][Defer[Int][(E^x*(4 - x)^((5*E^x)/(-3 + 20*E^x)))/(3 - 20*E^x)^2, x]/(4 - x), x] - 30*Defer[Int][De
fer[Int][(E^x*(4 - x)^((10*E^x)/(-3 + 20*E^x)))/(3 - 20*E^x)^2, x]/(4 - x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 e^x \left (3+(4-x)^{\frac {5 e^x}{-3+20 e^x}}\right ) (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (3-20 e^x+3 (-4+x) \log (4-x)\right )}{\left (3-20 e^x\right )^2} \, dx\\ &=10 \int \frac {e^x \left (3+(4-x)^{\frac {5 e^x}{-3+20 e^x}}\right ) (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (3-20 e^x+3 (-4+x) \log (4-x)\right )}{\left (3-20 e^x\right )^2} \, dx\\ &=10 \int \left (-\frac {3 e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (-3+20 e^x+12 \log (4-x)-3 x \log (4-x)\right )}{\left (-3+20 e^x\right )^2}-\frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (-3+20 e^x+12 \log (4-x)-3 x \log (4-x)\right )}{\left (-3+20 e^x\right )^2}\right ) \, dx\\ &=-\left (10 \int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (-3+20 e^x+12 \log (4-x)-3 x \log (4-x)\right )}{\left (-3+20 e^x\right )^2} \, dx\right )-30 \int \frac {e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \left (-3+20 e^x+12 \log (4-x)-3 x \log (4-x)\right )}{\left (-3+20 e^x\right )^2} \, dx\\ &=-\left (10 \int \frac {e^x (4-x)^{\frac {3-10 e^x}{-3+20 e^x}} \left (-3+20 e^x-3 (-4+x) \log (4-x)\right )}{\left (3-20 e^x\right )^2} \, dx\right )-30 \int \left (\frac {e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{-3+20 e^x}+\frac {3 e^x (4-x)^{1-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \log (4-x)}{\left (-3+20 e^x\right )^2}\right ) \, dx\\ &=-\left (10 \int \left (\frac {e^x (4-x)^{\frac {3-10 e^x}{-3+20 e^x}}}{-3+20 e^x}+\frac {3 e^x (4-x)^{1+\frac {3-10 e^x}{-3+20 e^x}} \log (4-x)}{\left (-3+20 e^x\right )^2}\right ) \, dx\right )-30 \int \frac {e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{-3+20 e^x} \, dx-90 \int \frac {e^x (4-x)^{1-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}} \log (4-x)}{\left (-3+20 e^x\right )^2} \, dx\\ &=-\left (10 \int \frac {e^x (4-x)^{\frac {3-10 e^x}{-3+20 e^x}}}{-3+20 e^x} \, dx\right )-30 \int \frac {e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{-3+20 e^x} \, dx-30 \int \frac {e^x (4-x)^{1+\frac {3-10 e^x}{-3+20 e^x}} \log (4-x)}{\left (-3+20 e^x\right )^2} \, dx-90 \int \frac {\int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2} \, dx}{4-x} \, dx-(90 \log (4-x)) \int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2} \, dx\\ &=-\left (10 \int \frac {e^x (4-x)^{\frac {3-10 e^x}{-3+20 e^x}}}{-3+20 e^x} \, dx\right )-30 \int \frac {e^x (4-x)^{-\frac {3 \left (-1+5 e^x\right )}{-3+20 e^x}}}{-3+20 e^x} \, dx-30 \int \frac {\int \frac {e^x (4-x)^{\frac {10 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2} \, dx}{4-x} \, dx-90 \int \frac {\int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2} \, dx}{4-x} \, dx-(30 \log (4-x)) \int \frac {e^x (4-x)^{\frac {10 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2} \, dx-(90 \log (4-x)) \int \frac {e^x (4-x)^{\frac {5 e^x}{-3+20 e^x}}}{\left (3-20 e^x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.47, size = 43, normalized size = 1.87 \begin {gather*} \left (6+(4-x)^{\frac {5 e^x}{-3+20 e^x}}\right ) (4-x)^{\frac {5 e^x}{-3+20 e^x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((4 - x)^((5*E^x)/(-3 + 20*E^x))*(-90*E^x + 600*E^(2*x) + E^x*(360 - 90*x)*Log[4 - x]) + (4 - x)^((1
0*E^x)/(-3 + 20*E^x))*(-30*E^x + 200*E^(2*x) + E^x*(120 - 30*x)*Log[4 - x]))/(-36 + E^x*(480 - 120*x) + 9*x +
E^(2*x)*(-1600 + 400*x)),x]

[Out]

(6 + (4 - x)^((5*E^x)/(-3 + 20*E^x)))*(4 - x)^((5*E^x)/(-3 + 20*E^x))

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fricas [A]  time = 0.72, size = 39, normalized size = 1.70 \begin {gather*} {\left (-x + 4\right )}^{\frac {10 \, e^{x}}{20 \, e^{x} - 3}} + 6 \, {\left (-x + 4\right )}^{\frac {5 \, e^{x}}{20 \, e^{x} - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3))^2+((-90
*x+360)*exp(x)*log(-x+4)+600*exp(x)^2-90*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2
+(-120*x+480)*exp(x)+9*x-36),x, algorithm="fricas")

[Out]

(-x + 4)^(10*e^x/(20*e^x - 3)) + 6*(-x + 4)^(5*e^x/(20*e^x - 3))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3))^2+((-90
*x+360)*exp(x)*log(-x+4)+600*exp(x)^2-90*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2
+(-120*x+480)*exp(x)+9*x-36),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{62985600000,[1,8,8,1]%%%}+%%%{503884800000,[1,8,7,1]%%%}
+%%%{-78732

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maple [A]  time = 0.25, size = 42, normalized size = 1.83

method	result	size
risch	\(\left (-x +4\right )^{\frac {10 \,{\mathrm e}^{x}}{20 \,{\mathrm e}^{x}-3}}+6 \left (-x +4\right )^{\frac {5 \,{\mathrm e}^{x}}{20 \,{\mathrm e}^{x}-3}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-30*x+120)*exp(x)*ln(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp(x)*ln(-x+4)/(20*exp(x)-3))^2+((-90*x+360)*
exp(x)*ln(-x+4)+600*exp(x)^2-90*exp(x))*exp(5*exp(x)*ln(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2+(-120*x+4
80)*exp(x)+9*x-36),x,method=_RETURNVERBOSE)

[Out]

((-x+4)^(5*exp(x)/(20*exp(x)-3)))^2+6*(-x+4)^(5*exp(x)/(20*exp(x)-3))

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maxima [A]  time = 0.50, size = 39, normalized size = 1.70 \begin {gather*} {\left (-x + 4\right )}^{\frac {10 \, e^{x}}{20 \, e^{x} - 3}} + 6 \, {\left (-x + 4\right )}^{\frac {5 \, e^{x}}{20 \, e^{x} - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*exp(x)*log(-x+4)+200*exp(x)^2-30*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3))^2+((-90
*x+360)*exp(x)*log(-x+4)+600*exp(x)^2-90*exp(x))*exp(5*exp(x)*log(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)^2
+(-120*x+480)*exp(x)+9*x-36),x, algorithm="maxima")

[Out]

(-x + 4)^(10*e^x/(20*e^x - 3)) + 6*(-x + 4)^(5*e^x/(20*e^x - 3))

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mupad [B]  time = 3.80, size = 39, normalized size = 1.70 \begin {gather*} \left ({\left (4-x\right )}^{\frac {5\,{\mathrm {e}}^x}{20\,{\mathrm {e}}^x-3}}+6\right )\,{\left (4-x\right )}^{\frac {5\,{\mathrm {e}}^x}{20\,{\mathrm {e}}^x-3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((10*exp(x)*log(4 - x))/(20*exp(x) - 3))*(30*exp(x) - 200*exp(2*x) + exp(x)*log(4 - x)*(30*x - 120))
+ exp((5*exp(x)*log(4 - x))/(20*exp(x) - 3))*(90*exp(x) - 600*exp(2*x) + exp(x)*log(4 - x)*(90*x - 360)))/(9*x
- exp(x)*(120*x - 480) + exp(2*x)*(400*x - 1600) - 36),x)

[Out]

((4 - x)^((5*exp(x))/(20*exp(x) - 3)) + 6)*(4 - x)^((5*exp(x))/(20*exp(x) - 3))

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sympy [B]  time = 0.87, size = 37, normalized size = 1.61 \begin {gather*} e^{\frac {10 e^{x} \log {\left (4 - x \right )}}{20 e^{x} - 3}} + 6 e^{\frac {5 e^{x} \log {\left (4 - x \right )}}{20 e^{x} - 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-30*x+120)*exp(x)*ln(-x+4)+200*exp(x)**2-30*exp(x))*exp(5*exp(x)*ln(-x+4)/(20*exp(x)-3))**2+((-90
*x+360)*exp(x)*ln(-x+4)+600*exp(x)**2-90*exp(x))*exp(5*exp(x)*ln(-x+4)/(20*exp(x)-3)))/((400*x-1600)*exp(x)**2
+(-120*x+480)*exp(x)+9*x-36),x)

[Out]

exp(10*exp(x)*log(4 - x)/(20*exp(x) - 3)) + 6*exp(5*exp(x)*log(4 - x)/(20*exp(x) - 3))

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