3.84.83 \(\int \frac {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} (10 x^3+3 x^4)+e^x (-40 x^2+8 x^3+27 x^4+7 x^5)}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} (25 x^2+30 x^3+9 x^4)+e^x (-100 x-70 x^2+84 x^3+84 x^4+18 x^5)} \, dx\)

Optimal. Leaf size=37 \[ \frac {x}{3-\frac {1}{-2-e^x+\frac {4}{x}+\frac {-2+x}{x}-x}+\frac {5}{x}} \]

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Rubi [F]  time = 4.44, 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 {40 x-28 x^2-42 x^3+12 x^4+18 x^5+3 x^6+e^{2 x} \left (10 x^3+3 x^4\right )+e^x \left (-40 x^2+8 x^3+27 x^4+7 x^5\right )}{100+20 x-179 x^2-78 x^3+75 x^4+54 x^5+9 x^6+e^{2 x} \left (25 x^2+30 x^3+9 x^4\right )+e^x \left (-100 x-70 x^2+84 x^3+84 x^4+18 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(40*x - 28*x^2 - 42*x^3 + 12*x^4 + 18*x^5 + 3*x^6 + E^(2*x)*(10*x^3 + 3*x^4) + E^x*(-40*x^2 + 8*x^3 + 27*x
^4 + 7*x^5))/(100 + 20*x - 179*x^2 - 78*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + E^(2*x)*(25*x^2 + 30*x^3 + 9*x^4) + E^
x*(-100*x - 70*x^2 + 84*x^3 + 84*x^4 + 18*x^5)),x]

[Out]

(10 + 3*x)^2/(9*(5 + 3*x)) - (4375*Defer[Int][(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^(-2), x])/729 +
(500*Defer[Int][x/(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^2, x])/243 - (25*Defer[Int][x^2/(-10 - x + 5
*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^2, x])/81 + (44*Defer[Int][x^3/(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*
x^3)^2, x])/27 + (23*Defer[Int][x^4/(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^2, x])/9 - Defer[Int][x^5/
(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^2, x]/3 - Defer[Int][x^6/(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^
2 + 3*x^3)^2, x] - (15625*Defer[Int][1/((5 + 3*x)^2*(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^2), x])/24
3 + (31250*Defer[Int][1/((5 + 3*x)*(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^2), x])/729 + (100*Defer[In
t][(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)^(-1), x])/81 + (10*Defer[Int][x/(-10 - x + 5*E^x*x + 9*x^2
+ 3*E^x*x^2 + 3*x^3), x])/27 - (8*Defer[Int][x^2/(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3), x])/9 + Defe
r[Int][x^3/(-10 - x + 5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3), x]/3 + (1250*Defer[Int][1/((5 + 3*x)^2*(-10 - x +
5*E^x*x + 9*x^2 + 3*E^x*x^2 + 3*x^3)), x])/27 - (1250*Defer[Int][1/((5 + 3*x)*(-10 - x + 5*E^x*x + 9*x^2 + 3*E
^x*x^2 + 3*x^3)), x])/81

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x \left (40-4 \left (7+10 e^x\right ) x+2 \left (-21+4 e^x+5 e^{2 x}\right ) x^2+3 \left (4+9 e^x+e^{2 x}\right ) x^3+\left (18+7 e^x\right ) x^4+3 x^5\right )}{\left (10-\left (-1+5 e^x\right ) x-3 \left (3+e^x\right ) x^2-3 x^3\right )^2} \, dx\\ &=\int \left (\frac {x (10+3 x)}{(5+3 x)^2}+\frac {x^3 \left (-15+2 x+3 x^2\right )}{(5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}-\frac {x^3 \left (-50-110 x-83 x^2+12 x^3+33 x^4+9 x^5\right )}{(5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}\right ) \, dx\\ &=\int \frac {x (10+3 x)}{(5+3 x)^2} \, dx+\int \frac {x^3 \left (-15+2 x+3 x^2\right )}{(5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )} \, dx-\int \frac {x^3 \left (-50-110 x-83 x^2+12 x^3+33 x^4+9 x^5\right )}{(5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx\\ &=\frac {(10+3 x)^2}{9 (5+3 x)}-\int \left (\frac {4375}{729 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}-\frac {500 x}{243 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}+\frac {25 x^2}{81 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}-\frac {44 x^3}{27 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}-\frac {23 x^4}{9 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}+\frac {x^5}{3 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}+\frac {x^6}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}+\frac {15625}{243 (5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}-\frac {31250}{729 (5+3 x) \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2}\right ) \, dx+\int \left (\frac {100}{81 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}+\frac {10 x}{27 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}-\frac {8 x^2}{9 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}+\frac {x^3}{3 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}+\frac {1250}{27 (5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}-\frac {1250}{81 (5+3 x) \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )}\right ) \, dx\\ &=\frac {(10+3 x)^2}{9 (5+3 x)}-\frac {25}{81} \int \frac {x^2}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx-\frac {1}{3} \int \frac {x^5}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx+\frac {1}{3} \int \frac {x^3}{-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3} \, dx+\frac {10}{27} \int \frac {x}{-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3} \, dx-\frac {8}{9} \int \frac {x^2}{-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3} \, dx+\frac {100}{81} \int \frac {1}{-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3} \, dx+\frac {44}{27} \int \frac {x^3}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx+\frac {500}{243} \int \frac {x}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx+\frac {23}{9} \int \frac {x^4}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx-\frac {4375}{729} \int \frac {1}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx-\frac {1250}{81} \int \frac {1}{(5+3 x) \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )} \, dx+\frac {31250}{729} \int \frac {1}{(5+3 x) \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx+\frac {1250}{27} \int \frac {1}{(5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )} \, dx-\frac {15625}{243} \int \frac {1}{(5+3 x)^2 \left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx-\int \frac {x^6}{\left (-10-x+5 e^x x+9 x^2+3 e^x x^2+3 x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 57, normalized size = 1.54 \begin {gather*} \frac {x}{3}+\frac {25}{9 (5+3 x)}-\frac {x^4}{(5+3 x) \left (-10+\left (-1+5 e^x\right ) x+3 \left (3+e^x\right ) x^2+3 x^3\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(40*x - 28*x^2 - 42*x^3 + 12*x^4 + 18*x^5 + 3*x^6 + E^(2*x)*(10*x^3 + 3*x^4) + E^x*(-40*x^2 + 8*x^3
+ 27*x^4 + 7*x^5))/(100 + 20*x - 179*x^2 - 78*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + E^(2*x)*(25*x^2 + 30*x^3 + 9*x^4
) + E^x*(-100*x - 70*x^2 + 84*x^3 + 84*x^4 + 18*x^5)),x]

[Out]

x/3 + 25/(9*(5 + 3*x)) - x^4/((5 + 3*x)*(-10 + (-1 + 5*E^x)*x + 3*(3 + E^x)*x^2 + 3*x^3))

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fricas [B]  time = 0.64, size = 68, normalized size = 1.84 \begin {gather*} \frac {9 \, x^{4} + 24 \, x^{3} + 27 \, x^{2} + {\left (9 \, x^{3} + 15 \, x^{2} + 25 \, x\right )} e^{x} - 5 \, x - 50}{9 \, {\left (3 \, x^{3} + 9 \, x^{2} + {\left (3 \, x^{2} + 5 \, x\right )} e^{x} - x - 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/
((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+
20*x+100),x, algorithm="fricas")

[Out]

1/9*(9*x^4 + 24*x^3 + 27*x^2 + (9*x^3 + 15*x^2 + 25*x)*e^x - 5*x - 50)/(3*x^3 + 9*x^2 + (3*x^2 + 5*x)*e^x - x
- 10)

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giac [B]  time = 0.16, size = 70, normalized size = 1.89 \begin {gather*} \frac {9 \, x^{4} + 9 \, x^{3} e^{x} + 24 \, x^{3} + 15 \, x^{2} e^{x} + 27 \, x^{2} + 25 \, x e^{x} - 5 \, x - 50}{9 \, {\left (3 \, x^{3} + 3 \, x^{2} e^{x} + 9 \, x^{2} + 5 \, x e^{x} - x - 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/
((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+
20*x+100),x, algorithm="giac")

[Out]

1/9*(9*x^4 + 9*x^3*e^x + 24*x^3 + 15*x^2*e^x + 27*x^2 + 25*x*e^x - 5*x - 50)/(3*x^3 + 3*x^2*e^x + 9*x^2 + 5*x*
e^x - x - 10)

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maple [A]  time = 0.12, size = 53, normalized size = 1.43

method	result	size
risch	\(\frac {x}{3}+\frac {25}{27 \left (x +\frac {5}{3}\right )}-\frac {x^{4}}{\left (3 x +5\right ) \left (3 \,{\mathrm e}^{x} x^{2}+3 x^{3}+5 \,{\mathrm e}^{x} x +9 x^{2}-x -10\right )}\)	\(53\)
norman	\(\frac {x^{4}-5 x^{2}+\frac {x}{3}+{\mathrm e}^{x} x^{3}-{\mathrm e}^{x} x^{2}-\frac {5 \,{\mathrm e}^{x} x}{3}+\frac {10}{3}}{3 \,{\mathrm e}^{x} x^{2}+3 x^{3}+5 \,{\mathrm e}^{x} x +9 x^{2}-x -10}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/((9*x^
4+30*x^3+25*x^2)*exp(x)^2+(18*x^5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+20*x+1
00),x,method=_RETURNVERBOSE)

[Out]

1/3*x+25/27/(x+5/3)-x^4/(3*x+5)/(3*exp(x)*x^2+3*x^3+5*exp(x)*x+9*x^2-x-10)

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maxima [B]  time = 0.43, size = 68, normalized size = 1.84 \begin {gather*} \frac {9 \, x^{4} + 24 \, x^{3} + 27 \, x^{2} + {\left (9 \, x^{3} + 15 \, x^{2} + 25 \, x\right )} e^{x} - 5 \, x - 50}{9 \, {\left (3 \, x^{3} + 9 \, x^{2} + {\left (3 \, x^{2} + 5 \, x\right )} e^{x} - x - 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^4+10*x^3)*exp(x)^2+(7*x^5+27*x^4+8*x^3-40*x^2)*exp(x)+3*x^6+18*x^5+12*x^4-42*x^3-28*x^2+40*x)/
((9*x^4+30*x^3+25*x^2)*exp(x)^2+(18*x^5+84*x^4+84*x^3-70*x^2-100*x)*exp(x)+9*x^6+54*x^5+75*x^4-78*x^3-179*x^2+
20*x+100),x, algorithm="maxima")

[Out]

1/9*(9*x^4 + 24*x^3 + 27*x^2 + (9*x^3 + 15*x^2 + 25*x)*e^x - 5*x - 50)/(3*x^3 + 9*x^2 + (3*x^2 + 5*x)*e^x - x
- 10)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {40\,x+{\mathrm {e}}^{2\,x}\,\left (3\,x^4+10\,x^3\right )+{\mathrm {e}}^x\,\left (7\,x^5+27\,x^4+8\,x^3-40\,x^2\right )-28\,x^2-42\,x^3+12\,x^4+18\,x^5+3\,x^6}{20\,x+{\mathrm {e}}^x\,\left (18\,x^5+84\,x^4+84\,x^3-70\,x^2-100\,x\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^4+30\,x^3+25\,x^2\right )-179\,x^2-78\,x^3+75\,x^4+54\,x^5+9\,x^6+100} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((40*x + exp(2*x)*(10*x^3 + 3*x^4) + exp(x)*(8*x^3 - 40*x^2 + 27*x^4 + 7*x^5) - 28*x^2 - 42*x^3 + 12*x^4 +
18*x^5 + 3*x^6)/(20*x + exp(x)*(84*x^3 - 70*x^2 - 100*x + 84*x^4 + 18*x^5) + exp(2*x)*(25*x^2 + 30*x^3 + 9*x^4
) - 179*x^2 - 78*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + 100),x)

[Out]

int((40*x + exp(2*x)*(10*x^3 + 3*x^4) + exp(x)*(8*x^3 - 40*x^2 + 27*x^4 + 7*x^5) - 28*x^2 - 42*x^3 + 12*x^4 +
18*x^5 + 3*x^6)/(20*x + exp(x)*(84*x^3 - 70*x^2 - 100*x + 84*x^4 + 18*x^5) + exp(2*x)*(25*x^2 + 30*x^3 + 9*x^4
) - 179*x^2 - 78*x^3 + 75*x^4 + 54*x^5 + 9*x^6 + 100), x)

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sympy [B]  time = 0.35, size = 49, normalized size = 1.32 \begin {gather*} - \frac {x^{4}}{9 x^{4} + 42 x^{3} + 42 x^{2} - 35 x + \left (9 x^{3} + 30 x^{2} + 25 x\right ) e^{x} - 50} + \frac {x}{3} + \frac {25}{27 x + 45} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**4+10*x**3)*exp(x)**2+(7*x**5+27*x**4+8*x**3-40*x**2)*exp(x)+3*x**6+18*x**5+12*x**4-42*x**3-28
*x**2+40*x)/((9*x**4+30*x**3+25*x**2)*exp(x)**2+(18*x**5+84*x**4+84*x**3-70*x**2-100*x)*exp(x)+9*x**6+54*x**5+
75*x**4-78*x**3-179*x**2+20*x+100),x)

[Out]

-x**4/(9*x**4 + 42*x**3 + 42*x**2 - 35*x + (9*x**3 + 30*x**2 + 25*x)*exp(x) - 50) + x/3 + 25/(27*x + 45)

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