3.31.7 \(\int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+(-10 x+14 x^2) \log (2)+x^2 \log ^2(2)}{x^2}} (-400+560 x+16 x^3+80 x \log (2))}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+(-10 x+14 x^2) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx\)

Optimal. Leaf size=29 \[ \log \left (\left (\frac {13}{4}+e^{2 x+\left (-7+\frac {5}{x}-\log (2)\right )^2}-x\right )^2\right ) \]

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Rubi [F]  time = 8.25, 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 {-8 x^3+\exp \left (\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}\right ) \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 \exp \left (\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}\right ) x^3-4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-8*x^3 + E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/x^2)*(-400 + 560*x + 16
*x^3 + 80*x*Log[2]))/(13*x^3 + 4*E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/x^2)
*x^3 - 4*x^4),x]

[Out]

50/x^2 + 4*x - (20*(7 + Log[2]))/x + 60*Defer[Int][E^((70 + 10*Log[2])/x)/(-13*4^(5/x)*E^(70/x) - 65536*E^(49
+ 25/x^2 + 2*x + Log[2]^2) + 4^(1 + 5/x)*E^(70/x)*x), x] - 1300*Defer[Int][E^((70 + 10*Log[2])/x)/(x^3*(-13*4^
(5/x)*E^(70/x) - 65536*E^(49 + 25/x^2 + 2*x + Log[2]^2) + 4^(1 + 5/x)*E^(70/x)*x)), x] + 20*(111 + Log[8192])*
Defer[Int][E^((70 + 10*Log[2])/x)/(x^2*(-13*4^(5/x)*E^(70/x) - 65536*E^(49 + 25/x^2 + 2*x + Log[2]^2) + 4^(1 +
5/x)*E^(70/x)*x)), x] - 16*Defer[Int][(E^((70 + 10*Log[2])/x)*x)/(-13*4^(5/x)*E^(70/x) - 65536*E^(49 + 25/x^2
+ 2*x + Log[2]^2) + 4^(1 + 5/x)*E^(70/x)*x), x] - (80*(7 + Log[2])*Defer[Subst][Defer[Int][E^((10*(4^((5 + x)
/x)*E^(70/x) - x)*(7 + Log[2]))/(13*4^(5/x)*E^(70/x) + 65536*E^(49 + 25/x^2 + 2*x + Log[2]^2)))/x, x], x, (-13
*4^(5/x)*E^(70/x) - 65536*E^(49 + 25/x^2 + 2*x + Log[2]^2) + 4^(1 + 5/x)*E^(70/x)*x)/x])/(13*4^(5/x)*E^(70/x)
+ 65536*E^(49 + 25/x^2 + 2*x + Log[2]^2))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 \left (-25+x^3+5 x (7+\log (2))\right )}{x^3}+\frac {2^{2+\frac {10}{x}} e^{70/x} \left (325-15 x^3+4 x^4+20 x^2 (7+\log (2))-5 x (111+\log (8192))\right )}{x^3 \left (13\ 2^{10/x} e^{70/x}+65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}-2^{2+\frac {10}{x}} e^{70/x} x\right )}\right ) \, dx\\ &=4 \int \frac {-25+x^3+5 x (7+\log (2))}{x^3} \, dx+\int \frac {2^{2+\frac {10}{x}} e^{70/x} \left (325-15 x^3+4 x^4+20 x^2 (7+\log (2))-5 x (111+\log (8192))\right )}{x^3 \left (13\ 2^{10/x} e^{70/x}+65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}-2^{2+\frac {10}{x}} e^{70/x} x\right )} \, dx\\ &=4 \int \left (1-\frac {25}{x^3}+\frac {5 (7+\log (2))}{x^2}\right ) \, dx+\int \frac {4 e^{\frac {70+10 \log (2)}{x}} \left (325-15 x^3+4 x^4+20 x^2 (7+\log (2))-5 x (111+\log (8192))\right )}{x^3 \left (13\ 2^{10/x} e^{70/x}+65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}-2^{2+\frac {10}{x}} e^{70/x} x\right )} \, dx\\ &=\frac {50}{x^2}+4 x-\frac {20 (7+\log (2))}{x}+4 \int \frac {e^{\frac {70+10 \log (2)}{x}} \left (325-15 x^3+4 x^4+20 x^2 (7+\log (2))-5 x (111+\log (8192))\right )}{x^3 \left (13\ 2^{10/x} e^{70/x}+65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}-2^{2+\frac {10}{x}} e^{70/x} x\right )} \, dx\\ &=\frac {50}{x^2}+4 x-\frac {20 (7+\log (2))}{x}+4 \int \left (\frac {15 e^{\frac {70+10 \log (2)}{x}}}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x}-\frac {325 e^{\frac {70+10 \log (2)}{x}}}{x^3 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )}-\frac {4 e^{\frac {70+10 \log (2)}{x}} x}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x}-\frac {20 e^{\frac {70+10 \log (2)}{x}} (7+\log (2))}{x \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )}+\frac {5 e^{\frac {70+10 \log (2)}{x}} (111+\log (8192))}{x^2 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )}\right ) \, dx\\ &=\frac {50}{x^2}+4 x-\frac {20 (7+\log (2))}{x}-16 \int \frac {e^{\frac {70+10 \log (2)}{x}} x}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x} \, dx+60 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x} \, dx-1300 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^3 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx-(80 (7+\log (2))) \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx+(20 (111+\log (8192))) \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^2 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx\\ &=\frac {50}{x^2}+4 x-\frac {20 (7+\log (2))}{x}-16 \int \frac {e^{\frac {70+10 \log (2)}{x}} x}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x} \, dx+60 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x} \, dx-1300 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^3 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx+\frac {(80 (7+\log (2))) \operatorname {Subst}\left (\int \frac {\exp \left (-\frac {4^{1+\frac {5}{x}} e^{70/x} (70+10 \log (2))}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}}+\frac {x (70+10 \log (2))}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}}\right )}{x} \, dx,x,\frac {-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x}{x}\right )}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}}+(20 (111+\log (8192))) \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^2 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx\\ &=\frac {50}{x^2}+4 x-\frac {20 (7+\log (2))}{x}-16 \int \frac {e^{\frac {70+10 \log (2)}{x}} x}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x} \, dx+60 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x} \, dx-1300 \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^3 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx+\frac {(80 (7+\log (2))) \operatorname {Subst}\left (\int \frac {\exp \left (\frac {10 \left (4^{\frac {5+x}{x}} e^{70/x}-x\right ) (7+\log (2))}{13\ 4^{5/x} e^{70/x}+65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}}\right )}{x} \, dx,x,\frac {-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x}{x}\right )}{-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}}+(20 (111+\log (8192))) \int \frac {e^{\frac {70+10 \log (2)}{x}}}{x^2 \left (-13 4^{5/x} e^{70/x}-65536 e^{49+\frac {25}{x^2}+2 x+\log ^2(2)}+4^{1+\frac {5}{x}} e^{70/x} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.65, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-8 x^3+e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} \left (-400+560 x+16 x^3+80 x \log (2)\right )}{13 x^3+4 e^{\frac {25-70 x+49 x^2+2 x^3+\left (-10 x+14 x^2\right ) \log (2)+x^2 \log ^2(2)}{x^2}} x^3-4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-8*x^3 + E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/x^2)*(-400 + 560*
x + 16*x^3 + 80*x*Log[2]))/(13*x^3 + 4*E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2
)/x^2)*x^3 - 4*x^4),x]

[Out]

Integrate[(-8*x^3 + E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2)/x^2)*(-400 + 560*
x + 16*x^3 + 80*x*Log[2]))/(13*x^3 + 4*E^((25 - 70*x + 49*x^2 + 2*x^3 + (-10*x + 14*x^2)*Log[2] + x^2*Log[2]^2
)/x^2)*x^3 - 4*x^4), x]

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fricas [A]  time = 0.58, size = 51, normalized size = 1.76 \begin {gather*} 2 \, \log \left (-4 \, x + 4 \, e^{\left (\frac {x^{2} \log \relax (2)^{2} + 2 \, x^{3} + 49 \, x^{2} + 2 \, {\left (7 \, x^{2} - 5 \, x\right )} \log \relax (2) - 70 \, x + 25}{x^{2}}\right )} + 13\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*
x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x, algorithm="fric
as")

[Out]

2*log(-4*x + 4*e^((x^2*log(2)^2 + 2*x^3 + 49*x^2 + 2*(7*x^2 - 5*x)*log(2) - 70*x + 25)/x^2) + 13)

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giac [A]  time = 0.37, size = 50, normalized size = 1.72 \begin {gather*} 2 \, \log \left (4 \, x - 4 \, e^{\left (\frac {x^{2} \log \relax (2)^{2} + 2 \, x^{3} + 14 \, x^{2} \log \relax (2) + 49 \, x^{2} - 10 \, x \log \relax (2) - 70 \, x + 25}{x^{2}}\right )} - 13\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*
x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x, algorithm="giac
")

[Out]

2*log(4*x - 4*e^((x^2*log(2)^2 + 2*x^3 + 14*x^2*log(2) + 49*x^2 - 10*x*log(2) - 70*x + 25)/x^2) - 13)

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maple [A]  time = 0.23, size = 51, normalized size = 1.76

method	result	size
norman	\(2 \ln \left (4 x -4 \,{\mathrm e}^{\frac {x^{2} \ln \relax (2)^{2}+\left (14 x^{2}-10 x \right ) \ln \relax (2)+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}-13\right )\)	\(51\)
risch	\(4 x +\frac {\left (-20 \ln \relax (2)-140\right ) x +50}{x^{2}}-\frac {2 \left (x^{2} \ln \relax (2)^{2}+\left (14 x^{2}-10 x \right ) \ln \relax (2)+2 x^{3}+49 x^{2}-70 x +25\right )}{x^{2}}+2 \ln \left (-x +16384 \left (\frac {1}{1024}\right )^{\frac {1}{x}} {\mathrm e}^{\frac {x^{2} \ln \relax (2)^{2}+2 x^{3}+49 x^{2}-70 x +25}{x^{2}}}+\frac {13}{4}\right )\)	\(102\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((80*x*ln(2)+16*x^3+560*x-400)*exp((x^2*ln(2)^2+(14*x^2-10*x)*ln(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*x^3)/(4*x
^3*exp((x^2*ln(2)^2+(14*x^2-10*x)*ln(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x,method=_RETURNVERBOSE)

[Out]

2*ln(4*x-4*exp((x^2*ln(2)^2+(14*x^2-10*x)*ln(2)+2*x^3+49*x^2-70*x+25)/x^2)-13)

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maxima [B]  time = 0.91, size = 71, normalized size = 2.45 \begin {gather*} -\frac {20 \, \log \relax (2)}{x} + 2 \, \log \left (4 \, x - 13\right ) + 2 \, \log \left (\frac {{\left ({\left (4 \, x - 13\right )} e^{\left (\frac {10 \, \log \relax (2)}{x} + \frac {70}{x}\right )} - 65536 \, e^{\left (\log \relax (2)^{2} + 2 \, x + \frac {25}{x^{2}} + 49\right )}\right )} e^{\left (-\frac {70}{x}\right )}}{4 \, x - 13}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x*log(2)+16*x^3+560*x-400)*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-8*
x^3)/(4*x^3*exp((x^2*log(2)^2+(14*x^2-10*x)*log(2)+2*x^3+49*x^2-70*x+25)/x^2)-4*x^4+13*x^3),x, algorithm="maxi
ma")

[Out]

-20*log(2)/x + 2*log(4*x - 13) + 2*log(((4*x - 13)*e^(10*log(2)/x + 70/x) - 65536*e^(log(2)^2 + 2*x + 25/x^2 +
49))*e^(-70/x)/(4*x - 13))

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mupad [B]  time = 1.94, size = 56, normalized size = 1.93 \begin {gather*} 2\,\ln \left (13\,2^{10/x}-4\,2^{10/x}\,x+65536\,{\mathrm {e}}^{{\ln \relax (2)}^2}\,{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{49}\,{\mathrm {e}}^{\frac {25}{x^2}}\,{\mathrm {e}}^{-\frac {70}{x}}\right )-\frac {20\,\ln \relax (2)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x^2*log(2)^2 - 70*x - log(2)*(10*x - 14*x^2) + 49*x^2 + 2*x^3 + 25)/x^2)*(560*x + 80*x*log(2) + 16*x
^3 - 400) - 8*x^3)/(4*x^3*exp((x^2*log(2)^2 - 70*x - log(2)*(10*x - 14*x^2) + 49*x^2 + 2*x^3 + 25)/x^2) + 13*x
^3 - 4*x^4),x)

[Out]

2*log(13*2^(10/x) - 4*2^(10/x)*x + 65536*exp(log(2)^2)*exp(2*x)*exp(49)*exp(25/x^2)*exp(-70/x)) - (20*log(2))/
x

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sympy [B]  time = 0.34, size = 48, normalized size = 1.66 \begin {gather*} 2 \log {\left (- x + e^{\frac {2 x^{3} + x^{2} \log {\relax (2 )}^{2} + 49 x^{2} - 70 x + \left (14 x^{2} - 10 x\right ) \log {\relax (2 )} + 25}{x^{2}}} + \frac {13}{4} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((80*x*ln(2)+16*x**3+560*x-400)*exp((x**2*ln(2)**2+(14*x**2-10*x)*ln(2)+2*x**3+49*x**2-70*x+25)/x**2
)-8*x**3)/(4*x**3*exp((x**2*ln(2)**2+(14*x**2-10*x)*ln(2)+2*x**3+49*x**2-70*x+25)/x**2)-4*x**4+13*x**3),x)

[Out]

2*log(-x + exp((2*x**3 + x**2*log(2)**2 + 49*x**2 - 70*x + (14*x**2 - 10*x)*log(2) + 25)/x**2) + 13/4)

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