3.61.1 \(\int \frac {-3744 x+6 x^2+e^{\frac {2 (15+x)}{3}} (-1250+2 x)+e^{\frac {15+x}{3}} (-3744-1244 x+2 x^2)+(-3744+6 x+e^{\frac {15+x}{3}} (-1250+2 x)) \log (625-x)}{-1875+3 x} \, dx\)

Optimal. Leaf size=19 \[ \left (e^{5+\frac {x}{3}}+x+\log (625-x)\right )^2 \]

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Rubi [A]  time = 0.25, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 79, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {6688, 12, 6686} \begin {gather*} \left (x+e^{\frac {x}{3}+5}+\log (625-x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3744*x + 6*x^2 + E^((2*(15 + x))/3)*(-1250 + 2*x) + E^((15 + x)/3)*(-3744 - 1244*x + 2*x^2) + (-3744 + 6
*x + E^((15 + x)/3)*(-1250 + 2*x))*Log[625 - x])/(-1875 + 3*x),x]

[Out]

(E^(5 + x/3) + x + Log[625 - x])^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (e^{5+\frac {x}{3}} (-625+x)+3 (-624+x)\right ) \left (-e^{5+\frac {x}{3}}-x-\log (625-x)\right )}{3 (625-x)} \, dx\\ &=\frac {2}{3} \int \frac {\left (e^{5+\frac {x}{3}} (-625+x)+3 (-624+x)\right ) \left (-e^{5+\frac {x}{3}}-x-\log (625-x)\right )}{625-x} \, dx\\ &=\left (e^{5+\frac {x}{3}}+x+\log (625-x)\right )^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 1.00 \begin {gather*} \left (e^{5+\frac {x}{3}}+x+\log (625-x)\right )^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3744*x + 6*x^2 + E^((2*(15 + x))/3)*(-1250 + 2*x) + E^((15 + x)/3)*(-3744 - 1244*x + 2*x^2) + (-37
44 + 6*x + E^((15 + x)/3)*(-1250 + 2*x))*Log[625 - x])/(-1875 + 3*x),x]

[Out]

(E^(5 + x/3) + x + Log[625 - x])^2

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fricas [B]  time = 0.50, size = 43, normalized size = 2.26 \begin {gather*} x^{2} + 2 \, x e^{\left (\frac {1}{3} \, x + 5\right )} + 2 \, {\left (x + e^{\left (\frac {1}{3} \, x + 5\right )}\right )} \log \left (-x + 625\right ) + \log \left (-x + 625\right )^{2} + e^{\left (\frac {2}{3} \, x + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1250)*exp(1/3*x+5)+6*x-3744)*log(-x+625)+(2*x-1250)*exp(1/3*x+5)^2+(2*x^2-1244*x-3744)*exp(1/
3*x+5)+6*x^2-3744*x)/(3*x-1875),x, algorithm="fricas")

[Out]

x^2 + 2*x*e^(1/3*x + 5) + 2*(x + e^(1/3*x + 5))*log(-x + 625) + log(-x + 625)^2 + e^(2/3*x + 10)

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giac [B]  time = 0.13, size = 50, normalized size = 2.63 \begin {gather*} x^{2} + 2 \, x e^{\left (\frac {1}{3} \, x + 5\right )} + 2 \, x \log \left (-x + 625\right ) + 2 \, e^{\left (\frac {1}{3} \, x + 5\right )} \log \left (-x + 625\right ) + \log \left (-x + 625\right )^{2} + e^{\left (\frac {2}{3} \, x + 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1250)*exp(1/3*x+5)+6*x-3744)*log(-x+625)+(2*x-1250)*exp(1/3*x+5)^2+(2*x^2-1244*x-3744)*exp(1/
3*x+5)+6*x^2-3744*x)/(3*x-1875),x, algorithm="giac")

[Out]

x^2 + 2*x*e^(1/3*x + 5) + 2*x*log(-x + 625) + 2*e^(1/3*x + 5)*log(-x + 625) + log(-x + 625)^2 + e^(2/3*x + 10)

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maple [B]  time = 0.29, size = 47, normalized size = 2.47

method	result	size
risch	\(\ln \left (-x +625\right )^{2}+\left (2 x +2 \,{\mathrm e}^{\frac {x}{3}+5}\right ) \ln \left (-x +625\right )+x^{2}+2 \,{\mathrm e}^{\frac {x}{3}+5} x +{\mathrm e}^{\frac {2 x}{3}+10}\)	\(47\)
norman	\(x^{2}+{\mathrm e}^{\frac {2 x}{3}+10}+\ln \left (-x +625\right )^{2}+2 \,{\mathrm e}^{\frac {x}{3}+5} x +2 \,{\mathrm e}^{\frac {x}{3}+5} \ln \left (-x +625\right )+2 \ln \left (-x +625\right ) x\)	\(53\)
default	\(2 \,{\mathrm e}^{\frac {x}{3}+5} x +2 \,{\mathrm e}^{\frac {x}{3}+5} \ln \left (-x +625\right )+{\mathrm e}^{\frac {2 x}{3}+10}+x^{2}+1250 \ln \left (x -625\right )-2 \left (-x +625\right ) \ln \left (-x +625\right )+1250+\ln \left (-x +625\right )^{2}\)	\(64\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x-1250)*exp(1/3*x+5)+6*x-3744)*ln(-x+625)+(2*x-1250)*exp(1/3*x+5)^2+(2*x^2-1244*x-3744)*exp(1/3*x+5)+
6*x^2-3744*x)/(3*x-1875),x,method=_RETURNVERBOSE)

[Out]

ln(-x+625)^2+(2*x+2*exp(1/3*x+5))*ln(-x+625)+x^2+2*exp(1/3*x+5)*x+exp(2/3*x+10)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} x^{2} + 1248 \, e^{\frac {640}{3}} E_{1}\left (-\frac {1}{3} \, x + \frac {625}{3}\right ) - 625 \, \log \left (x - 625\right )^{2} + 2 \, {\left (x + 625 \, \log \left (x - 625\right )\right )} \log \left (-x + 625\right ) + 2 \, e^{\left (\frac {1}{3} \, x + 5\right )} \log \left (-x + 625\right ) - 624 \, \log \left (-x + 625\right )^{2} + e^{\left (\frac {2}{3} \, x + 10\right )} + \frac {2}{3} \, \int \frac {{\left (x^{2} e^{5} - 622 \, x e^{5} - 3 \, e^{5}\right )} e^{\left (\frac {1}{3} \, x\right )}}{x - 625}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1250)*exp(1/3*x+5)+6*x-3744)*log(-x+625)+(2*x-1250)*exp(1/3*x+5)^2+(2*x^2-1244*x-3744)*exp(1/
3*x+5)+6*x^2-3744*x)/(3*x-1875),x, algorithm="maxima")

[Out]

x^2 + 1248*e^(640/3)*exp_integral_e(1, -1/3*x + 625/3) - 625*log(x - 625)^2 + 2*(x + 625*log(x - 625))*log(-x
+ 625) + 2*e^(1/3*x + 5)*log(-x + 625) - 624*log(-x + 625)^2 + e^(2/3*x + 10) + 2/3*integrate((x^2*e^5 - 622*x
*e^5 - 3*e^5)*e^(1/3*x)/(x - 625), x)

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mupad [B]  time = 0.23, size = 46, normalized size = 2.42 \begin {gather*} {\mathrm {e}}^{\frac {2\,x}{3}+10}+{\ln \left (625-x\right )}^2+\ln \left (625-x\right )\,\left (2\,x+2\,{\mathrm {e}}^{\frac {x}{3}+5}\right )+2\,x\,{\mathrm {e}}^{\frac {x}{3}+5}+x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((2*x)/3 + 10)*(2*x - 1250) - 3744*x - exp(x/3 + 5)*(1244*x - 2*x^2 + 3744) + log(625 - x)*(6*x + exp(
x/3 + 5)*(2*x - 1250) - 3744) + 6*x^2)/(3*x - 1875),x)

[Out]

exp((2*x)/3 + 10) + log(625 - x)^2 + log(625 - x)*(2*x + 2*exp(x/3 + 5)) + 2*x*exp(x/3 + 5) + x^2

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sympy [B]  time = 0.45, size = 42, normalized size = 2.21 \begin {gather*} x^{2} + 2 x \log {\left (625 - x \right )} + \left (2 x + 2 \log {\left (625 - x \right )}\right ) e^{\frac {x}{3} + 5} + e^{\frac {2 x}{3} + 10} + \log {\left (625 - x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1250)*exp(1/3*x+5)+6*x-3744)*ln(-x+625)+(2*x-1250)*exp(1/3*x+5)**2+(2*x**2-1244*x-3744)*exp(1
/3*x+5)+6*x**2-3744*x)/(3*x-1875),x)

[Out]

x**2 + 2*x*log(625 - x) + (2*x + 2*log(625 - x))*exp(x/3 + 5) + exp(2*x/3 + 10) + log(625 - x)**2

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