3.35.93 \(\int \frac {e^{\frac {20 x^4+36 x^3 \log (2 x+25 x^2)+54 x^2 \log ^2(2 x+25 x^2)+36 x \log ^3(2 x+25 x^2)+9 \log ^4(2 x+25 x^2)}{9 x^4}} (8 x^3+200 x^4+(24 x^2+592 x^3-100 x^4) \log (2 x+25 x^2)+(24 x+576 x^2-300 x^3) \log ^2(2 x+25 x^2)+(8+176 x-300 x^2) \log ^3(2 x+25 x^2)+(-8-100 x) \log ^4(2 x+25 x^2))}{2 x^5+25 x^6} \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {11}{9}+\frac {\left (x+\log \left (2 x+25 x^2\right )\right )^4}{x^4}} \]

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Rubi [F]  time = 19.48, 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 {\exp \left (\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}\right ) \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{2 x^5+25 x^6} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((20*x^4 + 36*x^3*Log[2*x + 25*x^2] + 54*x^2*Log[2*x + 25*x^2]^2 + 36*x*Log[2*x + 25*x^2]^3 + 9*Log[2*x
+ 25*x^2]^4)/(9*x^4))*(8*x^3 + 200*x^4 + (24*x^2 + 592*x^3 - 100*x^4)*Log[2*x + 25*x^2] + (24*x + 576*x^2 - 3
00*x^3)*Log[2*x + 25*x^2]^2 + (8 + 176*x - 300*x^2)*Log[2*x + 25*x^2]^3 + (-8 - 100*x)*Log[2*x + 25*x^2]^4))/(
2*x^5 + 25*x^6),x]

[Out]

200*Defer[Int][E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*
(x*(2 + 25*x))^(-1 + 4/x), x] + 8*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/
x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x))/x, x] - 100*Defer[Int][E^(20/9 + (6*Log[x*(2 + 25*x)
]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)],
x] + 24*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4
/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)])/x^2, x] + 592*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)
/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)])/x,
x] + 24*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/
x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^2)/x^3, x] + 576*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2
)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^2)/
x^2, x] - 300*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*
x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^2)/x, x] + 8*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]
^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^3
)/x^4, x] + 176*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 2
5*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25*x)]^3)/x^3, x] - 300*Defer[Int][(E^(20/9 + (6*Log[x*(2 +
25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^(-1 + 4/x)*Log[x*(2 + 25
*x)]^3)/x^2, x] - 4*Defer[Int][(E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2
+ 25*x)]^4/x^4)*(x*(2 + 25*x))^(4/x)*Log[x*(2 + 25*x)]^4)/x^5, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {20 x^4+36 x^3 \log \left (2 x+25 x^2\right )+54 x^2 \log ^2\left (2 x+25 x^2\right )+36 x \log ^3\left (2 x+25 x^2\right )+9 \log ^4\left (2 x+25 x^2\right )}{9 x^4}\right ) \left (8 x^3+200 x^4+\left (24 x^2+592 x^3-100 x^4\right ) \log \left (2 x+25 x^2\right )+\left (24 x+576 x^2-300 x^3\right ) \log ^2\left (2 x+25 x^2\right )+\left (8+176 x-300 x^2\right ) \log ^3\left (2 x+25 x^2\right )+(-8-100 x) \log ^4\left (2 x+25 x^2\right )\right )}{x^5 (2+25 x)} \, dx\\ &=\int \frac {4 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} (x+\log (x (2+25 x)))^3 (2+50 x-(2+25 x) \log (x (2+25 x)))}{x^4} \, dx\\ &=4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} (x+\log (x (2+25 x)))^3 (2+50 x-(2+25 x) \log (x (2+25 x)))}{x^4} \, dx\\ &=4 \int \left (\frac {2 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (1+25 x) (x (2+25 x))^{-1+\frac {4}{x}}}{x}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-6-148 x+25 x^2\right ) \log (x (2+25 x))}{x^2}-\frac {3 \exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-48 x+25 x^2\right ) \log ^2(x (2+25 x))}{x^3}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-44 x+75 x^2\right ) \log ^3(x (2+25 x))}{x^4}-\frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (2+25 x) (x (2+25 x))^{-1+\frac {4}{x}} \log ^4(x (2+25 x))}{x^4}\right ) \, dx\\ &=-\left (4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-6-148 x+25 x^2\right ) \log (x (2+25 x))}{x^2} \, dx\right )-4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-44 x+75 x^2\right ) \log ^3(x (2+25 x))}{x^4} \, dx-4 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (2+25 x) (x (2+25 x))^{-1+\frac {4}{x}} \log ^4(x (2+25 x))}{x^4} \, dx+8 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (1+25 x) (x (2+25 x))^{-1+\frac {4}{x}}}{x} \, dx-12 \int \frac {\exp \left (\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}\right ) (x (2+25 x))^{-1+\frac {4}{x}} \left (-2-48 x+25 x^2\right ) \log ^2(x (2+25 x))}{x^3} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.14, size = 64, normalized size = 2.67 \begin {gather*} e^{\frac {20}{9}+\frac {6 \log ^2(x (2+25 x))}{x^2}+\frac {4 \log ^3(x (2+25 x))}{x^3}+\frac {\log ^4(x (2+25 x))}{x^4}} (x (2+25 x))^{4/x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((20*x^4 + 36*x^3*Log[2*x + 25*x^2] + 54*x^2*Log[2*x + 25*x^2]^2 + 36*x*Log[2*x + 25*x^2]^3 + 9*L
og[2*x + 25*x^2]^4)/(9*x^4))*(8*x^3 + 200*x^4 + (24*x^2 + 592*x^3 - 100*x^4)*Log[2*x + 25*x^2] + (24*x + 576*x
^2 - 300*x^3)*Log[2*x + 25*x^2]^2 + (8 + 176*x - 300*x^2)*Log[2*x + 25*x^2]^3 + (-8 - 100*x)*Log[2*x + 25*x^2]
^4))/(2*x^5 + 25*x^6),x]

[Out]

E^(20/9 + (6*Log[x*(2 + 25*x)]^2)/x^2 + (4*Log[x*(2 + 25*x)]^3)/x^3 + Log[x*(2 + 25*x)]^4/x^4)*(x*(2 + 25*x))^
(4/x)

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fricas [B]  time = 0.50, size = 73, normalized size = 3.04 \begin {gather*} e^{\left (\frac {20 \, x^{4} + 36 \, x^{3} \log \left (25 \, x^{2} + 2 \, x\right ) + 54 \, x^{2} \log \left (25 \, x^{2} + 2 \, x\right )^{2} + 36 \, x \log \left (25 \, x^{2} + 2 \, x\right )^{3} + 9 \, \log \left (25 \, x^{2} + 2 \, x\right )^{4}}{9 \, x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x-8)*log(25*x^2+2*x)^4+(-300*x^2+176*x+8)*log(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*log(25*x^
2+2*x)^2+(-100*x^4+592*x^3+24*x^2)*log(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*log(25*x^2+2*x)^4+36*x*log(25*x^2
+2*x)^3+54*x^2*log(25*x^2+2*x)^2+36*x^3*log(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x, algorithm="fricas")

[Out]

e^(1/9*(20*x^4 + 36*x^3*log(25*x^2 + 2*x) + 54*x^2*log(25*x^2 + 2*x)^2 + 36*x*log(25*x^2 + 2*x)^3 + 9*log(25*x
^2 + 2*x)^4)/x^4)

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giac [B]  time = 14.66, size = 68, normalized size = 2.83 \begin {gather*} e^{\left (\frac {4 \, \log \left (25 \, x^{2} + 2 \, x\right )}{x} + \frac {6 \, \log \left (25 \, x^{2} + 2 \, x\right )^{2}}{x^{2}} + \frac {4 \, \log \left (25 \, x^{2} + 2 \, x\right )^{3}}{x^{3}} + \frac {\log \left (25 \, x^{2} + 2 \, x\right )^{4}}{x^{4}} + \frac {20}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x-8)*log(25*x^2+2*x)^4+(-300*x^2+176*x+8)*log(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*log(25*x^
2+2*x)^2+(-100*x^4+592*x^3+24*x^2)*log(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*log(25*x^2+2*x)^4+36*x*log(25*x^2
+2*x)^3+54*x^2*log(25*x^2+2*x)^2+36*x^3*log(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x, algorithm="giac")

[Out]

e^(4*log(25*x^2 + 2*x)/x + 6*log(25*x^2 + 2*x)^2/x^2 + 4*log(25*x^2 + 2*x)^3/x^3 + log(25*x^2 + 2*x)^4/x^4 + 2
0/9)

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maple [B]  time = 0.07, size = 74, normalized size = 3.08

method	result	size
risch	\({\mathrm e}^{\frac {9 \ln \left (25 x^{2}+2 x \right )^{4}+36 x \ln \left (25 x^{2}+2 x \right )^{3}+54 x^{2} \ln \left (25 x^{2}+2 x \right )^{2}+36 x^{3} \ln \left (25 x^{2}+2 x \right )+20 x^{4}}{9 x^{4}}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-100*x-8)*ln(25*x^2+2*x)^4+(-300*x^2+176*x+8)*ln(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*ln(25*x^2+2*x)^2+
(-100*x^4+592*x^3+24*x^2)*ln(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*ln(25*x^2+2*x)^4+36*x*ln(25*x^2+2*x)^3+54*x
^2*ln(25*x^2+2*x)^2+36*x^3*ln(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x,method=_RETURNVERBOSE)

[Out]

exp(1/9*(9*ln(25*x^2+2*x)^4+36*x*ln(25*x^2+2*x)^3+54*x^2*ln(25*x^2+2*x)^2+36*x^3*ln(25*x^2+2*x)+20*x^4)/x^4)

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maxima [B]  time = 1.25, size = 175, normalized size = 7.29 \begin {gather*} e^{\left (\frac {4 \, \log \left (25 \, x + 2\right )}{x} + \frac {6 \, \log \left (25 \, x + 2\right )^{2}}{x^{2}} + \frac {4 \, \log \left (25 \, x + 2\right )^{3}}{x^{3}} + \frac {\log \left (25 \, x + 2\right )^{4}}{x^{4}} + \frac {4 \, \log \relax (x)}{x} + \frac {12 \, \log \left (25 \, x + 2\right ) \log \relax (x)}{x^{2}} + \frac {12 \, \log \left (25 \, x + 2\right )^{2} \log \relax (x)}{x^{3}} + \frac {4 \, \log \left (25 \, x + 2\right )^{3} \log \relax (x)}{x^{4}} + \frac {6 \, \log \relax (x)^{2}}{x^{2}} + \frac {12 \, \log \left (25 \, x + 2\right ) \log \relax (x)^{2}}{x^{3}} + \frac {6 \, \log \left (25 \, x + 2\right )^{2} \log \relax (x)^{2}}{x^{4}} + \frac {4 \, \log \relax (x)^{3}}{x^{3}} + \frac {4 \, \log \left (25 \, x + 2\right ) \log \relax (x)^{3}}{x^{4}} + \frac {\log \relax (x)^{4}}{x^{4}} + \frac {20}{9}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x-8)*log(25*x^2+2*x)^4+(-300*x^2+176*x+8)*log(25*x^2+2*x)^3+(-300*x^3+576*x^2+24*x)*log(25*x^
2+2*x)^2+(-100*x^4+592*x^3+24*x^2)*log(25*x^2+2*x)+200*x^4+8*x^3)*exp(1/9*(9*log(25*x^2+2*x)^4+36*x*log(25*x^2
+2*x)^3+54*x^2*log(25*x^2+2*x)^2+36*x^3*log(25*x^2+2*x)+20*x^4)/x^4)/(25*x^6+2*x^5),x, algorithm="maxima")

[Out]

e^(4*log(25*x + 2)/x + 6*log(25*x + 2)^2/x^2 + 4*log(25*x + 2)^3/x^3 + log(25*x + 2)^4/x^4 + 4*log(x)/x + 12*l
og(25*x + 2)*log(x)/x^2 + 12*log(25*x + 2)^2*log(x)/x^3 + 4*log(25*x + 2)^3*log(x)/x^4 + 6*log(x)^2/x^2 + 12*l
og(25*x + 2)*log(x)^2/x^3 + 6*log(25*x + 2)^2*log(x)^2/x^4 + 4*log(x)^3/x^3 + 4*log(25*x + 2)*log(x)^3/x^4 + l
og(x)^4/x^4 + 20/9)

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mupad [B]  time = 2.31, size = 71, normalized size = 2.96 \begin {gather*} {\mathrm {e}}^{20/9}\,{\mathrm {e}}^{\frac {{\ln \left (25\,x^2+2\,x\right )}^4}{x^4}}\,{\mathrm {e}}^{\frac {6\,{\ln \left (25\,x^2+2\,x\right )}^2}{x^2}}\,{\mathrm {e}}^{\frac {4\,{\ln \left (25\,x^2+2\,x\right )}^3}{x^3}}\,{\left (25\,x^2+2\,x\right )}^{4/x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((4*x*log(2*x + 25*x^2)^3 + 4*x^3*log(2*x + 25*x^2) + log(2*x + 25*x^2)^4 + 6*x^2*log(2*x + 25*x^2)^2
+ (20*x^4)/9)/x^4)*(log(2*x + 25*x^2)^3*(176*x - 300*x^2 + 8) + log(2*x + 25*x^2)*(24*x^2 + 592*x^3 - 100*x^4)
+ log(2*x + 25*x^2)^2*(24*x + 576*x^2 - 300*x^3) - log(2*x + 25*x^2)^4*(100*x + 8) + 8*x^3 + 200*x^4))/(2*x^5
+ 25*x^6),x)

[Out]

exp(20/9)*exp(log(2*x + 25*x^2)^4/x^4)*exp((6*log(2*x + 25*x^2)^2)/x^2)*exp((4*log(2*x + 25*x^2)^3)/x^3)*(2*x
+ 25*x^2)^(4/x)

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sympy [B]  time = 0.87, size = 70, normalized size = 2.92 \begin {gather*} e^{\frac {\frac {20 x^{4}}{9} + 4 x^{3} \log {\left (25 x^{2} + 2 x \right )} + 6 x^{2} \log {\left (25 x^{2} + 2 x \right )}^{2} + 4 x \log {\left (25 x^{2} + 2 x \right )}^{3} + \log {\left (25 x^{2} + 2 x \right )}^{4}}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-100*x-8)*ln(25*x**2+2*x)**4+(-300*x**2+176*x+8)*ln(25*x**2+2*x)**3+(-300*x**3+576*x**2+24*x)*ln(2
5*x**2+2*x)**2+(-100*x**4+592*x**3+24*x**2)*ln(25*x**2+2*x)+200*x**4+8*x**3)*exp(1/9*(9*ln(25*x**2+2*x)**4+36*
x*ln(25*x**2+2*x)**3+54*x**2*ln(25*x**2+2*x)**2+36*x**3*ln(25*x**2+2*x)+20*x**4)/x**4)/(25*x**6+2*x**5),x)

[Out]

exp((20*x**4/9 + 4*x**3*log(25*x**2 + 2*x) + 6*x**2*log(25*x**2 + 2*x)**2 + 4*x*log(25*x**2 + 2*x)**3 + log(25
*x**2 + 2*x)**4)/x**4)

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