3.102.99 \(\int \frac {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750-120 x+6 x^2)+e^{\frac {1}{3} (-3 x+\log (25-x))} (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x))}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} (-750+30 x+e^3 (-150+6 x))} \, dx\)

Optimal. Leaf size=28 \[ x+\frac {x^2}{5+e^3+e^{-x} \sqrt [3]{25-x}} \]

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Rubi [F]  time = 14.78, 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 {-1875-675 x+30 x^2+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 \left (-750-120 x+6 x^2\right )+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )}{-1875+75 x+e^6 (-75+3 x)+e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)+e^3 (-750+30 x)+e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750+30 x+e^3 (-150+6 x)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1875 - 675*x + 30*x^2 + E^6*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 - 120*x
+ 6*x^2) + E^((-3*x + Log[25 - x])/3)*(-750 - 120*x - 70*x^2 + 3*x^3 + E^3*(-150 + 6*x)))/(-1875 + 75*x + E^6
*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 + 30*x) + E^((-3*x + Log[25 - x])/3)*(-7
50 + 30*x + E^3*(-150 + 6*x))),x]

[Out]

x - 6*E^25*(5 + E^3)*Defer[Subst][Defer[Int][x^2/(-5*E^25*(1 + E^3/5) - E^x^3*x), x], x, (25 - x)^(1/3)] - 155
*E^25*Defer[Subst][Defer[Int][x^5/(-5*E^25*(1 + E^3/5) - E^x^3*x), x], x, (25 - x)^(1/3)] + 625*E^50*(5 + E^3)
*Defer[Subst][Defer[Int][1/(x*(5*E^25*(1 + E^3/5) + E^x^3*x)^2), x], x, (25 - x)^(1/3)] - 3*E^50*(5 + E^3)^2*D
efer[Subst][Defer[Int][x^2/(5*E^25*(1 + E^3/5) + E^x^3*x)^2, x], x, (25 - x)^(1/3)] - 3*E^50*(5 + E^3)*(55 + E
^3)*Defer[Subst][Defer[Int][x^2/(5*E^25*(1 + E^3/5) + E^x^3*x)^2, x], x, (25 - x)^(1/3)] + E^50*(10025 + 2035*
E^3 + 6*E^6)*Defer[Subst][Defer[Int][x^2/(5*E^25*(1 + E^3/5) + E^x^3*x)^2, x], x, (25 - x)^(1/3)] - 149*E^50*(
5 + E^3)*Defer[Subst][Defer[Int][x^5/(5*E^25*(1 + E^3/5) + E^x^3*x)^2, x], x, (25 - x)^(1/3)] + 3*E^50*(5 + E^
3)*Defer[Subst][Defer[Int][x^8/(5*E^25*(1 + E^3/5) + E^x^3*x)^2, x], x, (25 - x)^(1/3)] - 625*E^25*Defer[Subst
][Defer[Int][1/(x*(5*E^25*(1 + E^3/5) + E^x^3*x)), x], x, (25 - x)^(1/3)] - E^25*(2005 + 6*E^3)*Defer[Subst][D
efer[Int][x^2/(5*E^25*(1 + E^3/5) + E^x^3*x), x], x, (25 - x)^(1/3)] - 3*E^25*Defer[Subst][Defer[Int][x^8/(5*E
^25*(1 + E^3/5) + E^x^3*x), x], x, (25 - x)^(1/3)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (1875+675 x-30 x^2-e^6 (-75+3 x)-e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)-e^3 \left (-750-120 x+6 x^2\right )-e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )\right )}{3 \left (5 e^x \left (1+\frac {e^3}{5}\right )+\sqrt [3]{25-x}\right )^2 (25-x)} \, dx\\ &=\frac {1}{3} \int \frac {e^{2 x} \left (1875+675 x-30 x^2-e^6 (-75+3 x)-e^{\frac {2}{3} (-3 x+\log (25-x))} (-75+3 x)-e^3 \left (-750-120 x+6 x^2\right )-e^{\frac {1}{3} (-3 x+\log (25-x))} \left (-750-120 x-70 x^2+3 x^3+e^3 (-150+6 x)\right )\right )}{\left (5 e^x \left (1+\frac {e^3}{5}\right )+\sqrt [3]{25-x}\right )^2 (25-x)} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {3 e^{56} x^3+6 e^{28+x^3} \left (x^3\right )^{4/3}+3 e^{2 x^3} \left (x^3\right )^{5/3}-6 e^{53} x^3 \left (-30+x^3\right )+e^{50} \left (825 x^3-30 x^6\right )+e^{25+x^3} \sqrt [3]{x^3} \left (625+2005 x^3-155 x^6+3 x^9\right )}{x \left (5 e^{25}+e^{28}+e^{x^3} x\right )^2} \, dx,x,\sqrt [3]{25-x}\right )\\ &=-\operatorname {Subst}\left (\int \frac {3 e^{56} x^3+6 e^{28+x^3} \left (x^3\right )^{4/3}+3 e^{2 x^3} \left (x^3\right )^{5/3}-6 e^{53} x^3 \left (-30+x^3\right )+e^{50} \left (825 x^3-30 x^6\right )+e^{25+x^3} \sqrt [3]{x^3} \left (625+2005 x^3-155 x^6+3 x^9\right )}{x \left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2} \, dx,x,\sqrt [3]{25-x}\right )\\ &=-\operatorname {Subst}\left (\int \left (3 \left (x^3\right )^{2/3}+\frac {155 e^{25} x^4 \sqrt [3]{x^3}}{-5 e^{25} \left (1+\frac {e^3}{5}\right )-e^{x^3} x}+\frac {30 e^{25} \left (1+\frac {e^3}{5}\right ) \left (x^3\right )^{2/3}}{-5 e^{25} \left (1+\frac {e^3}{5}\right )-e^{x^3} x}-\frac {3125 e^{50} \left (1+\frac {e^3}{5}\right ) \sqrt [3]{x^3}}{x^2 \left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2}-\frac {10025 e^{50} \left (1+\frac {e^3 \left (2035+6 e^3\right )}{10025}\right ) x \sqrt [3]{x^3}}{\left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2}+\frac {775 e^{50} \left (1+\frac {e^3}{5}\right ) x^4 \sqrt [3]{x^3}}{\left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2}-\frac {15 e^{50} \left (1+\frac {e^3}{5}\right ) x^7 \sqrt [3]{x^3}}{\left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2}+\frac {75 e^{50} \left (1+\frac {1}{25} e^3 \left (10+e^3\right )\right ) \left (x^3\right )^{2/3}}{\left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2}+\frac {625 e^{25} \sqrt [3]{x^3}}{x^2 \left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )}+\frac {2005 e^{25} \left (1+\frac {6 e^3}{2005}\right ) x \sqrt [3]{x^3}}{5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x}+\frac {3 e^{25} x^7 \sqrt [3]{x^3}}{5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x}+\frac {3 e^{50} \left (5+e^3\right ) x^2 \left (55+e^3-2 x^3\right )}{\left (5 e^{25} \left (1+\frac {e^3}{5}\right )+e^{x^3} x\right )^2}\right ) \, dx,x,\sqrt [3]{25-x}\right )\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.33, size = 138, normalized size = 4.93 \begin {gather*} \frac {x \left (25+e^{9+3 x}-x-5 e^{2 x} \sqrt [3]{25-x} x-e^{3+2 x} \sqrt [3]{25-x} x+e^x (25-x)^{2/3} x+25 e^{3 x} (5+x)+e^{6+3 x} (15+x)+5 e^{3+3 x} (15+2 x)\right )}{25+125 e^{3 x}+75 e^{3+3 x}+15 e^{6+3 x}+e^{9+3 x}-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1875 - 675*x + 30*x^2 + E^6*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 -
120*x + 6*x^2) + E^((-3*x + Log[25 - x])/3)*(-750 - 120*x - 70*x^2 + 3*x^3 + E^3*(-150 + 6*x)))/(-1875 + 75*x
+ E^6*(-75 + 3*x) + E^((2*(-3*x + Log[25 - x]))/3)*(-75 + 3*x) + E^3*(-750 + 30*x) + E^((-3*x + Log[25 - x])/
3)*(-750 + 30*x + E^3*(-150 + 6*x))),x]

[Out]

(x*(25 + E^(9 + 3*x) - x - 5*E^(2*x)*(25 - x)^(1/3)*x - E^(3 + 2*x)*(25 - x)^(1/3)*x + E^x*(25 - x)^(2/3)*x +
25*E^(3*x)*(5 + x) + E^(6 + 3*x)*(15 + x) + 5*E^(3 + 3*x)*(15 + 2*x)))/(25 + 125*E^(3*x) + 75*E^(3 + 3*x) + 15
*E^(6 + 3*x) + E^(9 + 3*x) - x)

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fricas [A]  time = 0.98, size = 46, normalized size = 1.64 \begin {gather*} \frac {x^{2} + x e^{3} + x e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + 5 \, x}{e^{3} + e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2-120*x-750)*exp(1/3*log(-x+25)-x)+(3
*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp(3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3
)+30*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1875),x, algorithm="fricas")

[Out]

(x^2 + x*e^3 + x*e^(-x + 1/3*log(-x + 25)) + 5*x)/(e^3 + e^(-x + 1/3*log(-x + 25)) + 5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {30 \, x^{2} + 3 \, {\left (x - 25\right )} e^{6} + 6 \, {\left (x^{2} - 20 \, x - 125\right )} e^{3} + {\left (3 \, x^{3} - 70 \, x^{2} + 6 \, {\left (x - 25\right )} e^{3} - 120 \, x - 750\right )} e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + 3 \, {\left (x - 25\right )} e^{\left (-2 \, x + \frac {2}{3} \, \log \left (-x + 25\right )\right )} - 675 \, x - 1875}{3 \, {\left ({\left (x - 25\right )} e^{6} + 10 \, {\left (x - 25\right )} e^{3} + 2 \, {\left ({\left (x - 25\right )} e^{3} + 5 \, x - 125\right )} e^{\left (-x + \frac {1}{3} \, \log \left (-x + 25\right )\right )} + {\left (x - 25\right )} e^{\left (-2 \, x + \frac {2}{3} \, \log \left (-x + 25\right )\right )} + 25 \, x - 625\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2-120*x-750)*exp(1/3*log(-x+25)-x)+(3
*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp(3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3
)+30*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1875),x, algorithm="giac")

[Out]

integrate(1/3*(30*x^2 + 3*(x - 25)*e^6 + 6*(x^2 - 20*x - 125)*e^3 + (3*x^3 - 70*x^2 + 6*(x - 25)*e^3 - 120*x -
750)*e^(-x + 1/3*log(-x + 25)) + 3*(x - 25)*e^(-2*x + 2/3*log(-x + 25)) - 675*x - 1875)/((x - 25)*e^6 + 10*(x
- 25)*e^3 + 2*((x - 25)*e^3 + 5*x - 125)*e^(-x + 1/3*log(-x + 25)) + (x - 25)*e^(-2*x + 2/3*log(-x + 25)) + 2
5*x - 625), x)

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maple [A]  time = 0.47, size = 25, normalized size = 0.89

method	result	size
risch	\(\frac {x^{2}}{\left (-x +25\right )^{\frac {1}{3}} {\mathrm e}^{-x}+{\mathrm e}^{3}+5}+x\)	\(25\)
norman	\(\frac {x^{2}+\left ({\mathrm e}^{3}+5\right ) x +{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x} x}{{\mathrm e}^{\frac {\ln \left (-x +25\right )}{3}-x}+{\mathrm e}^{3}+5}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x-75)*exp(1/3*ln(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2-120*x-750)*exp(1/3*ln(-x+25)-x)+(3*x-75)*e
xp(3)^2+(6*x^2-120*x-750)*exp(3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*ln(-x+25)-x)^2+((6*x-150)*exp(3)+30*x-75
0)*exp(1/3*ln(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1875),x,method=_RETURNVERBOSE)

[Out]

x^2/((-x+25)^(1/3)*exp(-x)+exp(3)+5)+x

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maxima [B]  time = 0.41, size = 63, normalized size = 2.25 \begin {gather*} \frac {{\left (x - 25\right )}^{2} e^{25} - {\left (-x + 25\right )}^{\frac {4}{3}} e^{\left (-x + 25\right )} + {\left (x - 25\right )} {\left (e^{28} + 55 \, e^{25}\right )} + 625 \, e^{25}}{{\left (-x + 25\right )}^{\frac {1}{3}} e^{\left (-x + 25\right )} + e^{28} + 5 \, e^{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3)+3*x^3-70*x^2-120*x-750)*exp(1/3*log(-x+25)-x)+(3
*x-75)*exp(3)^2+(6*x^2-120*x-750)*exp(3)+30*x^2-675*x-1875)/((3*x-75)*exp(1/3*log(-x+25)-x)^2+((6*x-150)*exp(3
)+30*x-750)*exp(1/3*log(-x+25)-x)+(3*x-75)*exp(3)^2+(30*x-750)*exp(3)+75*x-1875),x, algorithm="maxima")

[Out]

((x - 25)^2*e^25 - (-x + 25)^(4/3)*e^(-x + 25) + (x - 25)*(e^28 + 55*e^25) + 625*e^25)/((-x + 25)^(1/3)*e^(-x
+ 25) + e^28 + 5*e^25)

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mupad [B]  time = 6.99, size = 60, normalized size = 2.14 \begin {gather*} x-\frac {76\,x^2\,{\mathrm {e}}^3-3\,x^3\,{\mathrm {e}}^3+380\,x^2-15\,x^3}{\left (3\,x-76\right )\,\left ({\mathrm {e}}^3+5\right )\,\left ({\mathrm {e}}^3+{\mathrm {e}}^{-x}\,{\left (25-x\right )}^{1/3}+5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(675*x + exp(3)*(120*x - 6*x^2 + 750) - exp((2*log(25 - x))/3 - 2*x)*(3*x - 75) + exp(log(25 - x)/3 - x)*
(120*x + 70*x^2 - 3*x^3 - exp(3)*(6*x - 150) + 750) - 30*x^2 - exp(6)*(3*x - 75) + 1875)/(75*x + exp(log(25 -
x)/3 - x)*(30*x + exp(3)*(6*x - 150) - 750) + exp((2*log(25 - x))/3 - 2*x)*(3*x - 75) + exp(6)*(3*x - 75) + ex
p(3)*(30*x - 750) - 1875),x)

[Out]

x - (76*x^2*exp(3) - 3*x^3*exp(3) + 380*x^2 - 15*x^3)/((3*x - 76)*(exp(3) + 5)*(exp(3) + exp(-x)*(25 - x)^(1/3
) + 5))

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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(((3*x-75)*exp(1/3*ln(-x+25)-x)**2+((6*x-150)*exp(3)+3*x**3-70*x**2-120*x-750)*exp(1/3*ln(-x+25)-x)+(
3*x-75)*exp(3)**2+(6*x**2-120*x-750)*exp(3)+30*x**2-675*x-1875)/((3*x-75)*exp(1/3*ln(-x+25)-x)**2+((6*x-150)*e
xp(3)+30*x-750)*exp(1/3*ln(-x+25)-x)+(3*x-75)*exp(3)**2+(30*x-750)*exp(3)+75*x-1875),x)

[Out]

Timed out

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