3.51.69 \(\int \frac {e^{\frac {256-\log (x^3)}{-5+x}} (25 x-10 x^2+x^3+(-30-757 x-252 x^2-x^3) \log (6+2 x)+(3 x+x^2) \log (x^3) \log (6+2 x))}{75 x^2-5 x^3-7 x^4+x^5} \, dx\)

Optimal. Leaf size=26 \[ \frac {e^{\frac {-256+\log \left (x^3\right )}{5-x}} \log (2 (3+x))}{x} \]

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Rubi [F]  time = 21.05, 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 {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (25 x-10 x^2+x^3+\left (-30-757 x-252 x^2-x^3\right ) \log (6+2 x)+\left (3 x+x^2\right ) \log \left (x^3\right ) \log (6+2 x)\right )}{75 x^2-5 x^3-7 x^4+x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((256 - Log[x^3])/(-5 + x))*(25*x - 10*x^2 + x^3 + (-30 - 757*x - 252*x^2 - x^3)*Log[6 + 2*x] + (3*x +
x^2)*Log[x^3]*Log[6 + 2*x]))/(75*x^2 - 5*x^3 - 7*x^4 + x^5),x]

[Out]

Defer[Int][E^((256 - Log[x^3])/(-5 + x))/x, x]/3 - Defer[Int][E^((256 - Log[x^3])/(-5 + x))/(3 + x), x]/3 - (2
*Defer[Int][(E^((256 - Log[x^3])/(-5 + x))*Log[6 + 2*x])/(5 - x)^2, x])/5 - (254*Defer[Int][(E^((256 - Log[x^3
])/(-5 + x))*Log[6 + 2*x])/(-5 + x)^2, x])/5 + (253*Defer[Int][(E^((256 - Log[x^3])/(-5 + x))*Log[6 + 2*x])/(-
5 + x), x])/25 - (2*Defer[Int][(E^((256 - Log[x^3])/(-5 + x))*Log[6 + 2*x])/x^2, x])/5 - (253*Defer[Int][(E^((
256 - Log[x^3])/(-5 + x))*Log[6 + 2*x])/x, x])/25 + Defer[Int][(E^((256 - Log[x^3])/(-5 + x))*Log[x^3]*Log[6 +
2*x])/(5 - x)^2, x]/5 + (2*Defer[Int][(E^((256 - Log[x^3])/(-5 + x))*Log[x^3]*Log[6 + 2*x])/(5 - x), x])/25 +
Defer[Int][(E^((256 - Log[x^3])/(-5 + x))*Log[x^3]*Log[6 + 2*x])/(-5 + x), x]/25 + Defer[Int][(E^((256 - Log[
x^3])/(-5 + x))*Log[x^3]*Log[6 + 2*x])/x, x]/25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (25 x-10 x^2+x^3+\left (-30-757 x-252 x^2-x^3\right ) \log (6+2 x)+\left (3 x+x^2\right ) \log \left (x^3\right ) \log (6+2 x)\right )}{(5-x)^2 x^2 (3+x)} \, dx\\ &=\int \left (-\frac {10 e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{(-5+x)^2 (3+x)}+\frac {25 e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{(-5+x)^2 x (3+x)}+\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} x}{(-5+x)^2 (3+x)}+\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{(5-x)^2 x^2}\right ) \, dx\\ &=-\left (10 \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{(-5+x)^2 (3+x)} \, dx\right )+25 \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{(-5+x)^2 x (3+x)} \, dx+\int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} x}{(-5+x)^2 (3+x)} \, dx+\int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{(5-x)^2 x^2} \, dx\\ &=-\left (10 \int \left (\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{8 (-5+x)^2}-\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{64 (-5+x)}+\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{64 (3+x)}\right ) \, dx\right )+25 \int \left (\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{40 (-5+x)^2}-\frac {13 e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{1600 (-5+x)}+\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{75 x}-\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{192 (3+x)}\right ) \, dx+\int \left (\frac {5 e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{8 (-5+x)^2}+\frac {3 e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{64 (-5+x)}-\frac {3 e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{64 (3+x)}\right ) \, dx+\int \left (\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{25 (5-x)^2}+\frac {2 e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{125 (5-x)}+\frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{25 x^2}+\frac {2 e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{125 x}\right ) \, dx\\ &=\frac {2}{125} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{5-x} \, dx+\frac {2}{125} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{x} \, dx+\frac {1}{25} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{(5-x)^2} \, dx+\frac {1}{25} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}} \left (-10-249 x-x^2+x \log \left (x^3\right )\right ) \log (6+2 x)}{x^2} \, dx+\frac {3}{64} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{-5+x} \, dx-\frac {3}{64} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{3+x} \, dx-\frac {25}{192} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{3+x} \, dx+\frac {5}{32} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{-5+x} \, dx-\frac {5}{32} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{3+x} \, dx-\frac {13}{64} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{-5+x} \, dx+\frac {1}{3} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{x} \, dx+2 \left (\frac {5}{8} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{(-5+x)^2} \, dx\right )-\frac {5}{4} \int \frac {e^{\frac {256-\log \left (x^3\right )}{-5+x}}}{(-5+x)^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.95, size = 30, normalized size = 1.15 \begin {gather*} \frac {e^{\frac {256}{-5+x}} \left (x^3\right )^{-\frac {1}{-5+x}} \log (2 (3+x))}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((256 - Log[x^3])/(-5 + x))*(25*x - 10*x^2 + x^3 + (-30 - 757*x - 252*x^2 - x^3)*Log[6 + 2*x] + (
3*x + x^2)*Log[x^3]*Log[6 + 2*x]))/(75*x^2 - 5*x^3 - 7*x^4 + x^5),x]

[Out]

(E^(256/(-5 + x))*Log[2*(3 + x)])/(x*(x^3)^(-5 + x)^(-1))

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fricas [A]  time = 1.20, size = 24, normalized size = 0.92 \begin {gather*} \frac {e^{\left (-\frac {\log \left (x^{3}\right ) - 256}{x - 5}\right )} \log \left (2 \, x + 6\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+3*x)*log(2*x+6)*log(x^3)+(-x^3-252*x^2-757*x-30)*log(2*x+6)+x^3-10*x^2+25*x)*exp((-log(x^3)+25
6)/(x-5))/(x^5-7*x^4-5*x^3+75*x^2),x, algorithm="fricas")

[Out]

e^(-(log(x^3) - 256)/(x - 5))*log(2*x + 6)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + {\left (x^{2} + 3 \, x\right )} \log \left (x^{3}\right ) \log \left (2 \, x + 6\right ) - 10 \, x^{2} - {\left (x^{3} + 252 \, x^{2} + 757 \, x + 30\right )} \log \left (2 \, x + 6\right ) + 25 \, x\right )} e^{\left (-\frac {\log \left (x^{3}\right ) - 256}{x - 5}\right )}}{x^{5} - 7 \, x^{4} - 5 \, x^{3} + 75 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+3*x)*log(2*x+6)*log(x^3)+(-x^3-252*x^2-757*x-30)*log(2*x+6)+x^3-10*x^2+25*x)*exp((-log(x^3)+25
6)/(x-5))/(x^5-7*x^4-5*x^3+75*x^2),x, algorithm="giac")

[Out]

integrate((x^3 + (x^2 + 3*x)*log(x^3)*log(2*x + 6) - 10*x^2 - (x^3 + 252*x^2 + 757*x + 30)*log(2*x + 6) + 25*x
)*e^(-(log(x^3) - 256)/(x - 5))/(x^5 - 7*x^4 - 5*x^3 + 75*x^2), x)

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maple [C]  time = 0.23, size = 148, normalized size = 5.69

method	result	size
risch	\(\frac {\ln \left (2 x +6\right ) {\mathrm e}^{-\frac {-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )+i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{3}\right )^{3}+6 \ln \relax (x )-512}{2 \left (x -5\right )}}}{x}\)	\(148\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2+3*x)*ln(2*x+6)*ln(x^3)+(-x^3-252*x^2-757*x-30)*ln(2*x+6)+x^3-10*x^2+25*x)*exp((-ln(x^3)+256)/(x-5))/
(x^5-7*x^4-5*x^3+75*x^2),x,method=_RETURNVERBOSE)

[Out]

ln(2*x+6)/x*exp(-1/2*(-I*Pi*csgn(I*x^2)^3+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I*x)^2*csgn(I*x^2)-I*Pi*csg
n(I*x)*csgn(I*x^2)*csgn(I*x^3)+I*Pi*csgn(I*x^2)*csgn(I*x^3)^2+I*Pi*csgn(I*x)*csgn(I*x^3)^2-I*Pi*csgn(I*x^3)^3+
6*ln(x)-512)/(x-5))

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maxima [A]  time = 0.55, size = 29, normalized size = 1.12 \begin {gather*} \frac {{\left (\log \relax (2) + \log \left (x + 3\right )\right )} e^{\left (-\frac {3 \, \log \relax (x)}{x - 5} + \frac {256}{x - 5}\right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2+3*x)*log(2*x+6)*log(x^3)+(-x^3-252*x^2-757*x-30)*log(2*x+6)+x^3-10*x^2+25*x)*exp((-log(x^3)+25
6)/(x-5))/(x^5-7*x^4-5*x^3+75*x^2),x, algorithm="maxima")

[Out]

(log(2) + log(x + 3))*e^(-3*log(x)/(x - 5) + 256/(x - 5))/x

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\mathrm {e}}^{-\frac {\ln \left (x^3\right )-256}{x-5}}\,\left (25\,x-\ln \left (2\,x+6\right )\,\left (x^3+252\,x^2+757\,x+30\right )-10\,x^2+x^3+\ln \left (x^3\right )\,\ln \left (2\,x+6\right )\,\left (x^2+3\,x\right )\right )}{x^5-7\,x^4-5\,x^3+75\,x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(log(x^3) - 256)/(x - 5))*(25*x - log(2*x + 6)*(757*x + 252*x^2 + x^3 + 30) - 10*x^2 + x^3 + log(x^3
)*log(2*x + 6)*(3*x + x^2)))/(75*x^2 - 5*x^3 - 7*x^4 + x^5),x)

[Out]

int((exp(-(log(x^3) - 256)/(x - 5))*(25*x - log(2*x + 6)*(757*x + 252*x^2 + x^3 + 30) - 10*x^2 + x^3 + log(x^3
)*log(2*x + 6)*(3*x + x^2)))/(75*x^2 - 5*x^3 - 7*x^4 + x^5), x)

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sympy [A]  time = 0.56, size = 19, normalized size = 0.73 \begin {gather*} \frac {e^{\frac {256 - \log {\left (x^{3} \right )}}{x - 5}} \log {\left (2 x + 6 \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2+3*x)*ln(2*x+6)*ln(x**3)+(-x**3-252*x**2-757*x-30)*ln(2*x+6)+x**3-10*x**2+25*x)*exp((-ln(x**3)
+256)/(x-5))/(x**5-7*x**4-5*x**3+75*x**2),x)

[Out]

exp((256 - log(x**3))/(x - 5))*log(2*x + 6)/x

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