3.4.20 \(\int \frac {274848547600-4294737920 x+25165376 x^2-65536 x^3+64 x^4+e^{\frac {131067 x-1024 x^2+2 x^3}{131065-1024 x+2 x^2}} (17178296355 x-268421120 x^2+1572832 x^3-4096 x^4+4 x^5)}{17178034225 x-268421120 x^2+1572836 x^3-4096 x^4+4 x^5} \, dx\)

Optimal. Leaf size=24 \[ e^{x+\frac {x}{-\frac {7}{2}+(256-x)^2}}+16 \log (x) \]

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Rubi [F]  time = 5.19, 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 {274848547600-4294737920 x+25165376 x^2-65536 x^3+64 x^4+e^{\frac {131067 x-1024 x^2+2 x^3}{131065-1024 x+2 x^2}} \left (17178296355 x-268421120 x^2+1572832 x^3-4096 x^4+4 x^5\right )}{17178034225 x-268421120 x^2+1572836 x^3-4096 x^4+4 x^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(274848547600 - 4294737920*x + 25165376*x^2 - 65536*x^3 + 64*x^4 + E^((131067*x - 1024*x^2 + 2*x^3)/(13106
5 - 1024*x + 2*x^2))*(17178296355*x - 268421120*x^2 + 1572832*x^3 - 4096*x^4 + 4*x^5))/(17178034225*x - 268421
120*x^2 + 1572836*x^3 - 4096*x^4 + 4*x^5),x]

[Out]

(4096*x)/7 + (6291344*(131065 - 512*x))/(7*(131065 - 1024*x + 2*x^2)) + (1048520*(131079 - 512*x))/(7*(131065
- 1024*x + 2*x^2)) - (2147368960*(256 - x))/(7*(131065 - 1024*x + 2*x^2)) - (16384*(131065 - 512*x)*x)/(7*(131
065 - 1024*x + 2*x^2)) + (16*(131065 - 512*x)*x^2)/(7*(131065 - 1024*x + 2*x^2)) + (536784896*Sqrt[2/7]*ArcTan
h[Sqrt[2/7]*(256 - x)])/7 - (8*(49 - 16774528*Sqrt[14])*Log[512 - Sqrt[14] - 2*x])/49 + (8*(49 + 16774528*Sqrt
[14])*Log[512 - Sqrt[14] - 2*x])/49 + (8*(49 - 16774528*Sqrt[14])*Log[512 + Sqrt[14] - 2*x])/49 - (8*(49 + 167
74528*Sqrt[14])*Log[512 + Sqrt[14] - 2*x])/49 + 16*Log[x] + Defer[Int][E^((x*(131067 - 1024*x + 2*x^2))/(13106
5 - 1024*x + 2*x^2)), x] + (1048520*Defer[Int][E^((x*(131067 - 1024*x + 2*x^2))/(131065 - 1024*x + 2*x^2))/(10
24 + 2*Sqrt[14] - 4*x)^2, x])/7 - (2048*(512 + Sqrt[14])*Defer[Int][E^((x*(131067 - 1024*x + 2*x^2))/(131065 -
1024*x + 2*x^2))/(1024 + 2*Sqrt[14] - 4*x)^2, x])/7 + (1048520*Defer[Int][E^((x*(131067 - 1024*x + 2*x^2))/(1
31065 - 1024*x + 2*x^2))/(-1024 + 2*Sqrt[14] + 4*x)^2, x])/7 - (2048*(512 - Sqrt[14])*Defer[Int][E^((x*(131067
- 1024*x + 2*x^2))/(131065 - 1024*x + 2*x^2))/(-1024 + 2*Sqrt[14] + 4*x)^2, x])/7

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {274848547600-4294737920 x+25165376 x^2-65536 x^3+64 x^4+e^{\frac {131067 x-1024 x^2+2 x^3}{131065-1024 x+2 x^2}} \left (17178296355 x-268421120 x^2+1572832 x^3-4096 x^4+4 x^5\right )}{x \left (131065-1024 x+2 x^2\right )^2} \, dx\\ &=\int \left (-\frac {4294737920}{\left (131065-1024 x+2 x^2\right )^2}+\frac {274848547600}{x \left (131065-1024 x+2 x^2\right )^2}+\frac {25165376 x}{\left (131065-1024 x+2 x^2\right )^2}-\frac {65536 x^2}{\left (131065-1024 x+2 x^2\right )^2}+\frac {64 x^3}{\left (131065-1024 x+2 x^2\right )^2}+\frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} \left (17178296355-268421120 x+1572832 x^2-4096 x^3+4 x^4\right )}{\left (131065-1024 x+2 x^2\right )^2}\right ) \, dx\\ &=64 \int \frac {x^3}{\left (131065-1024 x+2 x^2\right )^2} \, dx-65536 \int \frac {x^2}{\left (131065-1024 x+2 x^2\right )^2} \, dx+25165376 \int \frac {x}{\left (131065-1024 x+2 x^2\right )^2} \, dx-4294737920 \int \frac {1}{\left (131065-1024 x+2 x^2\right )^2} \, dx+274848547600 \int \frac {1}{x \left (131065-1024 x+2 x^2\right )^2} \, dx+\int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} \left (17178296355-268421120 x+1572832 x^2-4096 x^3+4 x^4\right )}{\left (131065-1024 x+2 x^2\right )^2} \, dx\\ &=\frac {6291344 (131065-512 x)}{7 \left (131065-1024 x+2 x^2\right )}+\frac {1048520 (131079-512 x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {2147368960 (256-x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {16384 (131065-512 x) x}{7 \left (131065-1024 x+2 x^2\right )}+\frac {16 (131065-512 x) x^2}{7 \left (131065-1024 x+2 x^2\right )}-\frac {8}{7} \int \frac {(524260-1024 x) x}{131065-1024 x+2 x^2} \, dx-\frac {262130}{7} \int \frac {-56+2048 x}{x \left (131065-1024 x+2 x^2\right )} \, dx+2 \left (\frac {2147368960}{7} \int \frac {1}{131065-1024 x+2 x^2} \, dx\right )-\frac {3221168128}{7} \int \frac {1}{131065-1024 x+2 x^2} \, dx+\int \left (e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}-\frac {4 e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} (-131065+512 x)}{\left (131065-1024 x+2 x^2\right )^2}-\frac {2 e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{131065-1024 x+2 x^2}\right ) \, dx\\ &=\frac {4096 x}{7}+\frac {6291344 (131065-512 x)}{7 \left (131065-1024 x+2 x^2\right )}+\frac {1048520 (131079-512 x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {2147368960 (256-x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {16384 (131065-512 x) x}{7 \left (131065-1024 x+2 x^2\right )}+\frac {16 (131065-512 x) x^2}{7 \left (131065-1024 x+2 x^2\right )}-\frac {4}{7} \int \frac {134210560-56 x}{131065-1024 x+2 x^2} \, dx-2 \int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{131065-1024 x+2 x^2} \, dx-4 \int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} (-131065+512 x)}{\left (131065-1024 x+2 x^2\right )^2} \, dx-\frac {262130}{7} \int \left (-\frac {56}{131065 x}+\frac {16 (16772736+7 x)}{131065 \left (131065-1024 x+2 x^2\right )}\right ) \, dx-2 \left (\frac {4294737920}{7} \operatorname {Subst}\left (\int \frac {1}{56-x^2} \, dx,x,-1024+4 x\right )\right )+\frac {6442336256}{7} \operatorname {Subst}\left (\int \frac {1}{56-x^2} \, dx,x,-1024+4 x\right )+\int e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} \, dx\\ &=\frac {4096 x}{7}+\frac {6291344 (131065-512 x)}{7 \left (131065-1024 x+2 x^2\right )}+\frac {1048520 (131079-512 x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {2147368960 (256-x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {16384 (131065-512 x) x}{7 \left (131065-1024 x+2 x^2\right )}+\frac {16 (131065-512 x) x^2}{7 \left (131065-1024 x+2 x^2\right )}+\frac {536784896}{7} \sqrt {\frac {2}{7}} \tanh ^{-1}\left (\sqrt {\frac {2}{7}} (256-x)\right )+16 \log (x)-2 \int \left (-\frac {\sqrt {\frac {2}{7}} e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{1024+2 \sqrt {14}-4 x}-\frac {\sqrt {\frac {2}{7}} e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{-1024+2 \sqrt {14}+4 x}\right ) \, dx-4 \int \left (-\frac {131065 e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{\left (131065-1024 x+2 x^2\right )^2}+\frac {512 e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} x}{\left (131065-1024 x+2 x^2\right )^2}\right ) \, dx-\frac {32}{7} \int \frac {16772736+7 x}{131065-1024 x+2 x^2} \, dx+\frac {1}{49} \left (16 \left (49-16774528 \sqrt {14}\right )\right ) \int \frac {1}{-512-\sqrt {14}+2 x} \, dx+\frac {1}{49} \left (16 \left (49+16774528 \sqrt {14}\right )\right ) \int \frac {1}{-512+\sqrt {14}+2 x} \, dx+\int e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} \, dx\\ &=\frac {4096 x}{7}+\frac {6291344 (131065-512 x)}{7 \left (131065-1024 x+2 x^2\right )}+\frac {1048520 (131079-512 x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {2147368960 (256-x)}{7 \left (131065-1024 x+2 x^2\right )}-\frac {16384 (131065-512 x) x}{7 \left (131065-1024 x+2 x^2\right )}+\frac {16 (131065-512 x) x^2}{7 \left (131065-1024 x+2 x^2\right )}+\frac {536784896}{7} \sqrt {\frac {2}{7}} \tanh ^{-1}\left (\sqrt {\frac {2}{7}} (256-x)\right )+\frac {8}{49} \left (49+16774528 \sqrt {14}\right ) \log \left (512-\sqrt {14}-2 x\right )+\frac {8}{49} \left (49-16774528 \sqrt {14}\right ) \log \left (512+\sqrt {14}-2 x\right )+16 \log (x)-2048 \int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} x}{\left (131065-1024 x+2 x^2\right )^2} \, dx+524260 \int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{\left (131065-1024 x+2 x^2\right )^2} \, dx+\left (2 \sqrt {\frac {2}{7}}\right ) \int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{1024+2 \sqrt {14}-4 x} \, dx+\left (2 \sqrt {\frac {2}{7}}\right ) \int \frac {e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}}}{-1024+2 \sqrt {14}+4 x} \, dx-\frac {1}{49} \left (16 \left (49-16774528 \sqrt {14}\right )\right ) \int \frac {1}{-512+\sqrt {14}+2 x} \, dx-\frac {1}{49} \left (16 \left (49+16774528 \sqrt {14}\right )\right ) \int \frac {1}{-512-\sqrt {14}+2 x} \, dx+\int e^{\frac {x \left (131067-1024 x+2 x^2\right )}{131065-1024 x+2 x^2}} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 24, normalized size = 1.00 \begin {gather*} e^{x+\frac {2 x}{131065-1024 x+2 x^2}}+16 \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(274848547600 - 4294737920*x + 25165376*x^2 - 65536*x^3 + 64*x^4 + E^((131067*x - 1024*x^2 + 2*x^3)/
(131065 - 1024*x + 2*x^2))*(17178296355*x - 268421120*x^2 + 1572832*x^3 - 4096*x^4 + 4*x^5))/(17178034225*x -
268421120*x^2 + 1572836*x^3 - 4096*x^4 + 4*x^5),x]

[Out]

E^(x + (2*x)/(131065 - 1024*x + 2*x^2)) + 16*Log[x]

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fricas [A]  time = 0.65, size = 33, normalized size = 1.38 \begin {gather*} e^{\left (\frac {2 \, x^{3} - 1024 \, x^{2} + 131067 \, x}{2 \, x^{2} - 1024 \, x + 131065}\right )} + 16 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5-4096*x^4+1572832*x^3-268421120*x^2+17178296355*x)*exp((2*x^3-1024*x^2+131067*x)/(2*x^2-1024*
x+131065))+64*x^4-65536*x^3+25165376*x^2-4294737920*x+274848547600)/(4*x^5-4096*x^4+1572836*x^3-268421120*x^2+
17178034225*x),x, algorithm="fricas")

[Out]

e^((2*x^3 - 1024*x^2 + 131067*x)/(2*x^2 - 1024*x + 131065)) + 16*log(x)

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giac [A]  time = 0.64, size = 33, normalized size = 1.38 \begin {gather*} e^{\left (\frac {2 \, x^{3} - 1024 \, x^{2} + 131067 \, x}{2 \, x^{2} - 1024 \, x + 131065}\right )} + 16 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5-4096*x^4+1572832*x^3-268421120*x^2+17178296355*x)*exp((2*x^3-1024*x^2+131067*x)/(2*x^2-1024*
x+131065))+64*x^4-65536*x^3+25165376*x^2-4294737920*x+274848547600)/(4*x^5-4096*x^4+1572836*x^3-268421120*x^2+
17178034225*x),x, algorithm="giac")

[Out]

e^((2*x^3 - 1024*x^2 + 131067*x)/(2*x^2 - 1024*x + 131065)) + 16*log(x)

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maple [A]  time = 10.01, size = 31, normalized size = 1.29

method	result	size
risch	\(16 \ln \relax (x )+{\mathrm e}^{\frac {x \left (2 x^{2}-1024 x +131067\right )}{2 x^{2}-1024 x +131065}}\)	\(31\)
norman	\(\frac {-1024 x \,{\mathrm e}^{\frac {2 x^{3}-1024 x^{2}+131067 x}{2 x^{2}-1024 x +131065}}+2 x^{2} {\mathrm e}^{\frac {2 x^{3}-1024 x^{2}+131067 x}{2 x^{2}-1024 x +131065}}+131065 \,{\mathrm e}^{\frac {2 x^{3}-1024 x^{2}+131067 x}{2 x^{2}-1024 x +131065}}}{2 x^{2}-1024 x +131065}+16 \ln \relax (x )\)	\(114\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^5-4096*x^4+1572832*x^3-268421120*x^2+17178296355*x)*exp((2*x^3-1024*x^2+131067*x)/(2*x^2-1024*x+1310
65))+64*x^4-65536*x^3+25165376*x^2-4294737920*x+274848547600)/(4*x^5-4096*x^4+1572836*x^3-268421120*x^2+171780
34225*x),x,method=_RETURNVERBOSE)

[Out]

16*ln(x)+exp(x*(2*x^2-1024*x+131067)/(2*x^2-1024*x+131065))

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maxima [B]  time = 21.13, size = 116, normalized size = 4.83 \begin {gather*} -\frac {8 \, {\left (67119616 \, x - 17179869135\right )}}{7 \, {\left (2 \, x^{2} - 1024 \, x + 131065\right )}} + \frac {16384 \, {\left (131079 \, x - 33552640\right )}}{7 \, {\left (2 \, x^{2} - 1024 \, x + 131065\right )}} - \frac {6291344 \, {\left (512 \, x - 131065\right )}}{7 \, {\left (2 \, x^{2} - 1024 \, x + 131065\right )}} - \frac {1048520 \, {\left (512 \, x - 131079\right )}}{7 \, {\left (2 \, x^{2} - 1024 \, x + 131065\right )}} + \frac {2147368960 \, {\left (x - 256\right )}}{7 \, {\left (2 \, x^{2} - 1024 \, x + 131065\right )}} + e^{\left (x + \frac {2 \, x}{2 \, x^{2} - 1024 \, x + 131065}\right )} + 16 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^5-4096*x^4+1572832*x^3-268421120*x^2+17178296355*x)*exp((2*x^3-1024*x^2+131067*x)/(2*x^2-1024*
x+131065))+64*x^4-65536*x^3+25165376*x^2-4294737920*x+274848547600)/(4*x^5-4096*x^4+1572836*x^3-268421120*x^2+
17178034225*x),x, algorithm="maxima")

[Out]

-8/7*(67119616*x - 17179869135)/(2*x^2 - 1024*x + 131065) + 16384/7*(131079*x - 33552640)/(2*x^2 - 1024*x + 13
1065) - 6291344/7*(512*x - 131065)/(2*x^2 - 1024*x + 131065) - 1048520/7*(512*x - 131079)/(2*x^2 - 1024*x + 13
1065) + 2147368960/7*(x - 256)/(2*x^2 - 1024*x + 131065) + e^(x + 2*x/(2*x^2 - 1024*x + 131065)) + 16*log(x)

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mupad [B]  time = 0.53, size = 58, normalized size = 2.42 \begin {gather*} 16\,\ln \relax (x)+{\mathrm {e}}^{\frac {131067\,x}{2\,x^2-1024\,x+131065}}\,{\mathrm {e}}^{\frac {2\,x^3}{2\,x^2-1024\,x+131065}}\,{\mathrm {e}}^{-\frac {1024\,x^2}{2\,x^2-1024\,x+131065}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((131067*x - 1024*x^2 + 2*x^3)/(2*x^2 - 1024*x + 131065))*(17178296355*x - 268421120*x^2 + 1572832*x^3
- 4096*x^4 + 4*x^5) - 4294737920*x + 25165376*x^2 - 65536*x^3 + 64*x^4 + 274848547600)/(17178034225*x - 26842
1120*x^2 + 1572836*x^3 - 4096*x^4 + 4*x^5),x)

[Out]

16*log(x) + exp((131067*x)/(2*x^2 - 1024*x + 131065))*exp((2*x^3)/(2*x^2 - 1024*x + 131065))*exp(-(1024*x^2)/(
2*x^2 - 1024*x + 131065))

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sympy [A]  time = 0.27, size = 29, normalized size = 1.21 \begin {gather*} e^{\frac {2 x^{3} - 1024 x^{2} + 131067 x}{2 x^{2} - 1024 x + 131065}} + 16 \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**5-4096*x**4+1572832*x**3-268421120*x**2+17178296355*x)*exp((2*x**3-1024*x**2+131067*x)/(2*x**
2-1024*x+131065))+64*x**4-65536*x**3+25165376*x**2-4294737920*x+274848547600)/(4*x**5-4096*x**4+1572836*x**3-2
68421120*x**2+17178034225*x),x)

[Out]

exp((2*x**3 - 1024*x**2 + 131067*x)/(2*x**2 - 1024*x + 131065)) + 16*log(x)

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