3.92.56 \(\int \frac {131250 x^6-46875 x^8+e^{-5+x} (13125 x^6-3750 x^7)}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} (12960-10800 x^2)+e^{-5+x} (129600-216000 x^2+90000 x^4)} \, dx\)

Optimal. Leaf size=33 \[ 4+\frac {x^5}{16 \left (\frac {3 \left (2+\frac {e^{-5+x}}{5}\right )}{5 x}-x\right )^2} \]

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Rubi [F]  time = 1.97, 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 {131250 x^6-46875 x^8+e^{-5+x} \left (13125 x^6-3750 x^7\right )}{432000+432 e^{-15+3 x}-1080000 x^2+900000 x^4-250000 x^6+e^{-10+2 x} \left (12960-10800 x^2\right )+e^{-5+x} \left (129600-216000 x^2+90000 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(131250*x^6 - 46875*x^8 + E^(-5 + x)*(13125*x^6 - 3750*x^7))/(432000 + 432*E^(-15 + 3*x) - 1080000*x^2 + 9
00000*x^4 - 250000*x^6 + E^(-10 + 2*x)*(12960 - 10800*x^2) + E^(-5 + x)*(129600 - 216000*x^2 + 90000*x^4)),x]

[Out]

(-9375*E^15*Defer[Int][x^7/(-30*E^5 - 3*E^x + 25*E^5*x^2)^3, x])/4 - (15625*E^15*Defer[Int][x^8/(-30*E^5 - 3*E
^x + 25*E^5*x^2)^3, x])/4 + (15625*E^15*Defer[Int][x^9/(-30*E^5 - 3*E^x + 25*E^5*x^2)^3, x])/8 + (4375*E^10*De
fer[Int][x^6/(-30*E^5 - 3*E^x + 25*E^5*x^2)^2, x])/16 - (625*E^10*Defer[Int][x^7/(-30*E^5 - 3*E^x + 25*E^5*x^2
)^2, x])/8

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {1875 e^{10} x^6 \left (-e^x (-7+2 x)-5 e^5 \left (-14+5 x^2\right )\right )}{16 \left (3 e^x-5 e^5 \left (-6+5 x^2\right )\right )^3} \, dx\\ &=\frac {1}{16} \left (1875 e^{10}\right ) \int \frac {x^6 \left (-e^x (-7+2 x)-5 e^5 \left (-14+5 x^2\right )\right )}{\left (3 e^x-5 e^5 \left (-6+5 x^2\right )\right )^3} \, dx\\ &=\frac {1}{16} \left (1875 e^{10}\right ) \int \left (\frac {10 e^5 x^7 \left (-6-10 x+5 x^2\right )}{3 \left (-30 e^5-3 e^x+25 e^5 x^2\right )^3}-\frac {x^6 (-7+2 x)}{3 \left (-30 e^5-3 e^x+25 e^5 x^2\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{16} \left (625 e^{10}\right ) \int \frac {x^6 (-7+2 x)}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^2} \, dx\right )+\frac {1}{8} \left (3125 e^{15}\right ) \int \frac {x^7 \left (-6-10 x+5 x^2\right )}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3} \, dx\\ &=-\left (\frac {1}{16} \left (625 e^{10}\right ) \int \left (-\frac {7 x^6}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^2}+\frac {2 x^7}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^2}\right ) \, dx\right )+\frac {1}{8} \left (3125 e^{15}\right ) \int \left (-\frac {6 x^7}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3}-\frac {10 x^8}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3}+\frac {5 x^9}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{8} \left (625 e^{10}\right ) \int \frac {x^7}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^2} \, dx\right )+\frac {1}{16} \left (4375 e^{10}\right ) \int \frac {x^6}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^2} \, dx+\frac {1}{8} \left (15625 e^{15}\right ) \int \frac {x^9}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3} \, dx-\frac {1}{4} \left (9375 e^{15}\right ) \int \frac {x^7}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3} \, dx-\frac {1}{4} \left (15625 e^{15}\right ) \int \frac {x^8}{\left (-30 e^5-3 e^x+25 e^5 x^2\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 31, normalized size = 0.94 \begin {gather*} \frac {625 e^{10} x^7}{16 \left (30 e^5+3 e^x-25 e^5 x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(131250*x^6 - 46875*x^8 + E^(-5 + x)*(13125*x^6 - 3750*x^7))/(432000 + 432*E^(-15 + 3*x) - 1080000*x
^2 + 900000*x^4 - 250000*x^6 + E^(-10 + 2*x)*(12960 - 10800*x^2) + E^(-5 + x)*(129600 - 216000*x^2 + 90000*x^4
)),x]

[Out]

(625*E^10*x^7)/(16*(30*E^5 + 3*E^x - 25*E^5*x^2)^2)

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fricas [A]  time = 0.66, size = 40, normalized size = 1.21 \begin {gather*} \frac {625 \, x^{7}}{16 \, {\left (625 \, x^{4} - 1500 \, x^{2} - 30 \, {\left (5 \, x^{2} - 6\right )} e^{\left (x - 5\right )} + 9 \, e^{\left (2 \, x - 10\right )} + 900\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3750*x^7+13125*x^6)*exp(x-5)-46875*x^8+131250*x^6)/(432*exp(x-5)^3+(-10800*x^2+12960)*exp(x-5)^2+
(90000*x^4-216000*x^2+129600)*exp(x-5)-250000*x^6+900000*x^4-1080000*x^2+432000),x, algorithm="fricas")

[Out]

625/16*x^7/(625*x^4 - 1500*x^2 - 30*(5*x^2 - 6)*e^(x - 5) + 9*e^(2*x - 10) + 900)

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giac [B]  time = 0.16, size = 49, normalized size = 1.48 \begin {gather*} \frac {625 \, x^{7} e^{10}}{16 \, {\left (625 \, x^{4} e^{10} - 1500 \, x^{2} e^{10} - 150 \, x^{2} e^{\left (x + 5\right )} + 900 \, e^{10} + 9 \, e^{\left (2 \, x\right )} + 180 \, e^{\left (x + 5\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3750*x^7+13125*x^6)*exp(x-5)-46875*x^8+131250*x^6)/(432*exp(x-5)^3+(-10800*x^2+12960)*exp(x-5)^2+
(90000*x^4-216000*x^2+129600)*exp(x-5)-250000*x^6+900000*x^4-1080000*x^2+432000),x, algorithm="giac")

[Out]

625/16*x^7*e^10/(625*x^4*e^10 - 1500*x^2*e^10 - 150*x^2*e^(x + 5) + 900*e^10 + 9*e^(2*x) + 180*e^(x + 5))

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maple [A]  time = 0.53, size = 21, normalized size = 0.64

method	result	size
risch	\(\frac {625 x^{7}}{16 \left (25 x^{2}-3 \,{\mathrm e}^{x -5}-30\right )^{2}}\)	\(21\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-3750*x^7+13125*x^6)*exp(x-5)-46875*x^8+131250*x^6)/(432*exp(x-5)^3+(-10800*x^2+12960)*exp(x-5)^2+(90000
*x^4-216000*x^2+129600)*exp(x-5)-250000*x^6+900000*x^4-1080000*x^2+432000),x,method=_RETURNVERBOSE)

[Out]

625/16*x^7/(25*x^2-3*exp(x-5)-30)^2

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maxima [B]  time = 0.42, size = 50, normalized size = 1.52 \begin {gather*} \frac {625 \, x^{7} e^{10}}{16 \, {\left (625 \, x^{4} e^{10} - 1500 \, x^{2} e^{10} - 30 \, {\left (5 \, x^{2} e^{5} - 6 \, e^{5}\right )} e^{x} + 900 \, e^{10} + 9 \, e^{\left (2 \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3750*x^7+13125*x^6)*exp(x-5)-46875*x^8+131250*x^6)/(432*exp(x-5)^3+(-10800*x^2+12960)*exp(x-5)^2+
(90000*x^4-216000*x^2+129600)*exp(x-5)-250000*x^6+900000*x^4-1080000*x^2+432000),x, algorithm="maxima")

[Out]

625/16*x^7*e^10/(625*x^4*e^10 - 1500*x^2*e^10 - 30*(5*x^2*e^5 - 6*e^5)*e^x + 900*e^10 + 9*e^(2*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{x-5}\,\left (13125\,x^6-3750\,x^7\right )+131250\,x^6-46875\,x^8}{432\,{\mathrm {e}}^{3\,x-15}+{\mathrm {e}}^{x-5}\,\left (90000\,x^4-216000\,x^2+129600\right )-{\mathrm {e}}^{2\,x-10}\,\left (10800\,x^2-12960\right )-1080000\,x^2+900000\,x^4-250000\,x^6+432000} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - 5)*(13125*x^6 - 3750*x^7) + 131250*x^6 - 46875*x^8)/(432*exp(3*x - 15) + exp(x - 5)*(90000*x^4 -
216000*x^2 + 129600) - exp(2*x - 10)*(10800*x^2 - 12960) - 1080000*x^2 + 900000*x^4 - 250000*x^6 + 432000),x)

[Out]

int((exp(x - 5)*(13125*x^6 - 3750*x^7) + 131250*x^6 - 46875*x^8)/(432*exp(3*x - 15) + exp(x - 5)*(90000*x^4 -
216000*x^2 + 129600) - exp(2*x - 10)*(10800*x^2 - 12960) - 1080000*x^2 + 900000*x^4 - 250000*x^6 + 432000), x)

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sympy [A]  time = 0.23, size = 36, normalized size = 1.09 \begin {gather*} \frac {625 x^{7}}{10000 x^{4} - 24000 x^{2} + \left (2880 - 2400 x^{2}\right ) e^{x - 5} + 144 e^{2 x - 10} + 14400} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-3750*x**7+13125*x**6)*exp(x-5)-46875*x**8+131250*x**6)/(432*exp(x-5)**3+(-10800*x**2+12960)*exp(x
-5)**2+(90000*x**4-216000*x**2+129600)*exp(x-5)-250000*x**6+900000*x**4-1080000*x**2+432000),x)

[Out]

625*x**7/(10000*x**4 - 24000*x**2 + (2880 - 2400*x**2)*exp(x - 5) + 144*exp(2*x - 10) + 14400)

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