3.40.91 \(\int \frac {x+10 x^2+x^4-25 x^6+e^{2 x} (x^2+2 x^3-17 x^4-10 x^5)+e^x (-2 x^3-2 x^4+42 x^5+10 x^6)+(-1-4 x+5 x^2) \log (x+5 x^2)}{-x^6-5 x^7+e^{2 x} (-x^4-5 x^5)+e^x (2 x^5+10 x^6)+(x^2+5 x^3) \log (x+5 x^2)} \, dx\)

Optimal. Leaf size=31 \[ \frac {1}{x}+\log (x)+\log \left (x^2 \left (-e^x+x\right )^2-\log \left (x+5 x^2\right )\right ) \]

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Rubi [F]  time = 27.22, 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 {x+10 x^2+x^4-25 x^6+e^{2 x} \left (x^2+2 x^3-17 x^4-10 x^5\right )+e^x \left (-2 x^3-2 x^4+42 x^5+10 x^6\right )+\left (-1-4 x+5 x^2\right ) \log \left (x+5 x^2\right )}{-x^6-5 x^7+e^{2 x} \left (-x^4-5 x^5\right )+e^x \left (2 x^5+10 x^6\right )+\left (x^2+5 x^3\right ) \log \left (x+5 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x + 10*x^2 + x^4 - 25*x^6 + E^(2*x)*(x^2 + 2*x^3 - 17*x^4 - 10*x^5) + E^x*(-2*x^3 - 2*x^4 + 42*x^5 + 10*x
^6) + (-1 - 4*x + 5*x^2)*Log[x + 5*x^2])/(-x^6 - 5*x^7 + E^(2*x)*(-x^4 - 5*x^5) + E^x*(2*x^5 + 10*x^6) + (x^2
+ 5*x^3)*Log[x + 5*x^2]),x]

[Out]

x^(-1) + 2*x + 3*Log[x] - 2*Defer[Int][(E^x*x^2)/(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 - Log[x*(1 + 5*x)]), x] + 2*De
fer[Int][x^3/(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 - Log[x*(1 + 5*x)]), x] + 2*Defer[Int][(E^x*x^3)/(E^(2*x)*x^2 - 2*
E^x*x^3 + x^4 - Log[x*(1 + 5*x)]), x] - 2*Defer[Int][x^4/(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 - Log[x*(1 + 5*x)]), x
] - 5*Defer[Int][1/((1 + 5*x)*(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 - Log[x*(1 + 5*x)])), x] + 2*Defer[Int][Log[x*(1
+ 5*x)]/(E^(2*x)*x^2 - 2*E^x*x^3 + x^4 - Log[x*(1 + 5*x)]), x] - Defer[Int][((E^x - x)^2*x^3 - x*Log[x*(1 + 5*
x)])^(-1), x] + 2*Defer[Int][Log[x*(1 + 5*x)]/((E^x - x)^2*x^3 - x*Log[x*(1 + 5*x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-x-10 x^2-x^4+25 x^6-e^{2 x} \left (x^2+2 x^3-17 x^4-10 x^5\right )-e^x \left (-2 x^3-2 x^4+42 x^5+10 x^6\right )-\left (-1-4 x+5 x^2\right ) \log \left (x+5 x^2\right )}{x^2 (1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx\\ &=\int \left (\frac {-1+3 x+2 x^2}{x^2}-\frac {1+10 x+2 e^x x^3-2 x^4+8 e^x x^4-8 x^5-10 e^x x^5+10 x^6-2 \log (x (1+5 x))-12 x \log (x (1+5 x))-10 x^2 \log (x (1+5 x))}{x (1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}\right ) \, dx\\ &=\int \frac {-1+3 x+2 x^2}{x^2} \, dx-\int \frac {1+10 x+2 e^x x^3-2 x^4+8 e^x x^4-8 x^5-10 e^x x^5+10 x^6-2 \log (x (1+5 x))-12 x \log (x (1+5 x))-10 x^2 \log (x (1+5 x))}{x (1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx\\ &=\int \left (2-\frac {1}{x^2}+\frac {3}{x}\right ) \, dx-\int \left (\frac {1+10 x+2 e^x x^3-2 x^4+8 e^x x^4-8 x^5-10 e^x x^5+10 x^6-2 \log (x (1+5 x))-12 x \log (x (1+5 x))-10 x^2 \log (x (1+5 x))}{x \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {5 \left (1+10 x+2 e^x x^3-2 x^4+8 e^x x^4-8 x^5-10 e^x x^5+10 x^6-2 \log (x (1+5 x))-12 x \log (x (1+5 x))-10 x^2 \log (x (1+5 x))\right )}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}\right ) \, dx\\ &=\frac {1}{x}+2 x+3 \log (x)+5 \int \frac {1+10 x+2 e^x x^3-2 x^4+8 e^x x^4-8 x^5-10 e^x x^5+10 x^6-2 \log (x (1+5 x))-12 x \log (x (1+5 x))-10 x^2 \log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-\int \frac {1+10 x+2 e^x x^3-2 x^4+8 e^x x^4-8 x^5-10 e^x x^5+10 x^6-2 \log (x (1+5 x))-12 x \log (x (1+5 x))-10 x^2 \log (x (1+5 x))}{x \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx\\ &=\frac {1}{x}+2 x+3 \log (x)+5 \int \left (\frac {1}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}+\frac {10 x}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}+\frac {2 e^x x^3}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {2 x^4}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}+\frac {8 e^x x^4}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {8 x^5}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {10 e^x x^5}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}+\frac {10 x^6}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {2 \log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {12 x \log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {10 x^2 \log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}\right ) \, dx-\int \left (\frac {10}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}+\frac {1}{x \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}+\frac {2 e^x x^2}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}-\frac {2 x^3}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}+\frac {8 e^x x^3}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}-\frac {8 x^4}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}-\frac {10 e^x x^4}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}+\frac {10 x^5}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}-\frac {12 \log (x (1+5 x))}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}-\frac {2 \log (x (1+5 x))}{x \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )}-\frac {10 x \log (x (1+5 x))}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))}\right ) \, dx\\ &=\frac {1}{x}+2 x+3 \log (x)-2 \int \frac {e^x x^2}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx+2 \int \frac {x^3}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx+2 \int \frac {\log (x (1+5 x))}{x \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx+5 \int \frac {1}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-8 \int \frac {e^x x^3}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx+8 \int \frac {x^4}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx-10 \int \frac {1}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx+10 \int \frac {e^x x^4}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx-10 \int \frac {x^5}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx+10 \int \frac {e^x x^3}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-10 \int \frac {x^4}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx+10 \int \frac {x \log (x (1+5 x))}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx-10 \int \frac {\log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx+12 \int \frac {\log (x (1+5 x))}{e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))} \, dx+40 \int \frac {e^x x^4}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-40 \int \frac {x^5}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx+50 \int \frac {x}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-50 \int \frac {e^x x^5}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx+50 \int \frac {x^6}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-50 \int \frac {x^2 \log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-60 \int \frac {x \log (x (1+5 x))}{(1+5 x) \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx-\int \frac {1}{x \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 38, normalized size = 1.23 \begin {gather*} \frac {1}{x}+\log (x)+\log \left (e^{2 x} x^2-2 e^x x^3+x^4-\log (x (1+5 x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x + 10*x^2 + x^4 - 25*x^6 + E^(2*x)*(x^2 + 2*x^3 - 17*x^4 - 10*x^5) + E^x*(-2*x^3 - 2*x^4 + 42*x^5
+ 10*x^6) + (-1 - 4*x + 5*x^2)*Log[x + 5*x^2])/(-x^6 - 5*x^7 + E^(2*x)*(-x^4 - 5*x^5) + E^x*(2*x^5 + 10*x^6) +
(x^2 + 5*x^3)*Log[x + 5*x^2]),x]

[Out]

x^(-1) + Log[x] + Log[E^(2*x)*x^2 - 2*E^x*x^3 + x^4 - Log[x*(1 + 5*x)]]

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fricas [A]  time = 0.69, size = 43, normalized size = 1.39 \begin {gather*} \frac {x \log \left (-x^{4} + 2 \, x^{3} e^{x} - x^{2} e^{\left (2 \, x\right )} + \log \left (5 \, x^{2} + x\right )\right ) + x \log \relax (x) + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2-4*x-1)*log(5*x^2+x)+(-10*x^5-17*x^4+2*x^3+x^2)*exp(x)^2+(10*x^6+42*x^5-2*x^4-2*x^3)*exp(x)-2
5*x^6+x^4+10*x^2+x)/((5*x^3+x^2)*log(5*x^2+x)+(-5*x^5-x^4)*exp(x)^2+(10*x^6+2*x^5)*exp(x)-5*x^7-x^6),x, algori
thm="fricas")

[Out]

(x*log(-x^4 + 2*x^3*e^x - x^2*e^(2*x) + log(5*x^2 + x)) + x*log(x) + 1)/x

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giac [A]  time = 0.20, size = 43, normalized size = 1.39 \begin {gather*} \frac {x \log \left (-x^{4} + 2 \, x^{3} e^{x} - x^{2} e^{\left (2 \, x\right )} + \log \left (5 \, x^{2} + x\right )\right ) + x \log \relax (x) + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2-4*x-1)*log(5*x^2+x)+(-10*x^5-17*x^4+2*x^3+x^2)*exp(x)^2+(10*x^6+42*x^5-2*x^4-2*x^3)*exp(x)-2
5*x^6+x^4+10*x^2+x)/((5*x^3+x^2)*log(5*x^2+x)+(-5*x^5-x^4)*exp(x)^2+(10*x^6+2*x^5)*exp(x)-5*x^7-x^6),x, algori
thm="giac")

[Out]

(x*log(-x^4 + 2*x^3*e^x - x^2*e^(2*x) + log(5*x^2 + x)) + x*log(x) + 1)/x

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maple [C]  time = 0.07, size = 123, normalized size = 3.97

method	result	size
risch	\(\frac {x \ln \relax (x )+1}{x}+\ln \left (\ln \left (\frac {1}{5}+x \right )-\frac {i \left (-2 i x^{4}+4 i x^{3} {\mathrm e}^{x}-2 i x^{2} {\mathrm e}^{2 x}+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (\frac {1}{5}+x \right )\right ) \mathrm {csgn}\left (i x \left (\frac {1}{5}+x \right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (\frac {1}{5}+x \right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (\frac {1}{5}+x \right )\right ) \mathrm {csgn}\left (i x \left (\frac {1}{5}+x \right )\right )^{2}+\pi \mathrm {csgn}\left (i x \left (\frac {1}{5}+x \right )\right )^{3}+2 i \ln \relax (x )\right )}{2}\right )\)	\(123\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^2-4*x-1)*ln(5*x^2+x)+(-10*x^5-17*x^4+2*x^3+x^2)*exp(x)^2+(10*x^6+42*x^5-2*x^4-2*x^3)*exp(x)-25*x^6+x
^4+10*x^2+x)/((5*x^3+x^2)*ln(5*x^2+x)+(-5*x^5-x^4)*exp(x)^2+(10*x^6+2*x^5)*exp(x)-5*x^7-x^6),x,method=_RETURNV
ERBOSE)

[Out]

(x*ln(x)+1)/x+ln(ln(1/5+x)-1/2*I*(-2*I*x^4+4*I*x^3*exp(x)-2*I*x^2*exp(2*x)+Pi*csgn(I*x)*csgn(I*(1/5+x))*csgn(I
*x*(1/5+x))-Pi*csgn(I*x)*csgn(I*x*(1/5+x))^2-Pi*csgn(I*(1/5+x))*csgn(I*x*(1/5+x))^2+Pi*csgn(I*x*(1/5+x))^3+2*I
*ln(x)))

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maxima [A]  time = 0.44, size = 37, normalized size = 1.19 \begin {gather*} \frac {1}{x} + \log \left (-x^{4} + 2 \, x^{3} e^{x} - x^{2} e^{\left (2 \, x\right )} + \log \left (5 \, x + 1\right ) + \log \relax (x)\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2-4*x-1)*log(5*x^2+x)+(-10*x^5-17*x^4+2*x^3+x^2)*exp(x)^2+(10*x^6+42*x^5-2*x^4-2*x^3)*exp(x)-2
5*x^6+x^4+10*x^2+x)/((5*x^3+x^2)*log(5*x^2+x)+(-5*x^5-x^4)*exp(x)^2+(10*x^6+2*x^5)*exp(x)-5*x^7-x^6),x, algori
thm="maxima")

[Out]

1/x + log(-x^4 + 2*x^3*e^x - x^2*e^(2*x) + log(5*x + 1) + log(x)) + log(x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {x-\ln \left (5\,x^2+x\right )\,\left (-5\,x^2+4\,x+1\right )+{\mathrm {e}}^{2\,x}\,\left (-10\,x^5-17\,x^4+2\,x^3+x^2\right )-{\mathrm {e}}^x\,\left (-10\,x^6-42\,x^5+2\,x^4+2\,x^3\right )+10\,x^2+x^4-25\,x^6}{{\mathrm {e}}^{2\,x}\,\left (5\,x^5+x^4\right )-{\mathrm {e}}^x\,\left (10\,x^6+2\,x^5\right )-\ln \left (5\,x^2+x\right )\,\left (5\,x^3+x^2\right )+x^6+5\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x - log(x + 5*x^2)*(4*x - 5*x^2 + 1) + exp(2*x)*(x^2 + 2*x^3 - 17*x^4 - 10*x^5) - exp(x)*(2*x^3 + 2*x^4
- 42*x^5 - 10*x^6) + 10*x^2 + x^4 - 25*x^6)/(exp(2*x)*(x^4 + 5*x^5) - exp(x)*(2*x^5 + 10*x^6) - log(x + 5*x^2)
*(x^2 + 5*x^3) + x^6 + 5*x^7),x)

[Out]

int(-(x - log(x + 5*x^2)*(4*x - 5*x^2 + 1) + exp(2*x)*(x^2 + 2*x^3 - 17*x^4 - 10*x^5) - exp(x)*(2*x^3 + 2*x^4
- 42*x^5 - 10*x^6) + 10*x^2 + x^4 - 25*x^6)/(exp(2*x)*(x^4 + 5*x^5) - exp(x)*(2*x^5 + 10*x^6) - log(x + 5*x^2)
*(x^2 + 5*x^3) + x^6 + 5*x^7), x)

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sympy [A]  time = 0.90, size = 36, normalized size = 1.16 \begin {gather*} 3 \log {\relax (x )} + \log {\left (- 2 x e^{x} + e^{2 x} + \frac {x^{4} - \log {\left (5 x^{2} + x \right )}}{x^{2}} \right )} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**2-4*x-1)*ln(5*x**2+x)+(-10*x**5-17*x**4+2*x**3+x**2)*exp(x)**2+(10*x**6+42*x**5-2*x**4-2*x**3
)*exp(x)-25*x**6+x**4+10*x**2+x)/((5*x**3+x**2)*ln(5*x**2+x)+(-5*x**5-x**4)*exp(x)**2+(10*x**6+2*x**5)*exp(x)-
5*x**7-x**6),x)

[Out]

3*log(x) + log(-2*x*exp(x) + exp(2*x) + (x**4 - log(5*x**2 + x))/x**2) + 1/x

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