3.83.32 \(\int \frac {18 x+13 x^2+4 x^3-3 x^4+x^5+e^{25} (36 x^2+16 x^3-8 x^4+4 x^5)+(18 x+8 x^2-4 x^3+2 x^4+e^{25} (72 x+32 x^2-16 x^3+8 x^4)) \log (\frac {9+4 x-2 x^2+x^3}{x^2})+e^{25} (36+16 x-8 x^2+4 x^3) \log ^2(\frac {9+4 x-2 x^2+x^3}{x^2})}{e^{25} (9 x^2+4 x^3-2 x^4+x^5)+e^{25} (18 x+8 x^2-4 x^3+2 x^4) \log (\frac {9+4 x-2 x^2+x^3}{x^2})+e^{25} (9+4 x-2 x^2+x^3) \log ^2(\frac {9+4 x-2 x^2+x^3}{x^2})} \, dx\)

Optimal. Leaf size=26 \[ x \left (4+\frac {x}{e^{25} \left (x+\log \left (-2+x+\frac {9+4 x}{x^2}\right )\right )}\right ) \]

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Rubi [F]  time = 1.70, 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 {18 x+13 x^2+4 x^3-3 x^4+x^5+e^{25} \left (36 x^2+16 x^3-8 x^4+4 x^5\right )+\left (18 x+8 x^2-4 x^3+2 x^4+e^{25} \left (72 x+32 x^2-16 x^3+8 x^4\right )\right ) \log \left (\frac {9+4 x-2 x^2+x^3}{x^2}\right )+e^{25} \left (36+16 x-8 x^2+4 x^3\right ) \log ^2\left (\frac {9+4 x-2 x^2+x^3}{x^2}\right )}{e^{25} \left (9 x^2+4 x^3-2 x^4+x^5\right )+e^{25} \left (18 x+8 x^2-4 x^3+2 x^4\right ) \log \left (\frac {9+4 x-2 x^2+x^3}{x^2}\right )+e^{25} \left (9+4 x-2 x^2+x^3\right ) \log ^2\left (\frac {9+4 x-2 x^2+x^3}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(18*x + 13*x^2 + 4*x^3 - 3*x^4 + x^5 + E^25*(36*x^2 + 16*x^3 - 8*x^4 + 4*x^5) + (18*x + 8*x^2 - 4*x^3 + 2*
x^4 + E^25*(72*x + 32*x^2 - 16*x^3 + 8*x^4))*Log[(9 + 4*x - 2*x^2 + x^3)/x^2] + E^25*(36 + 16*x - 8*x^2 + 4*x^
3)*Log[(9 + 4*x - 2*x^2 + x^3)/x^2]^2)/(E^25*(9*x^2 + 4*x^3 - 2*x^4 + x^5) + E^25*(18*x + 8*x^2 - 4*x^3 + 2*x^
4)*Log[(9 + 4*x - 2*x^2 + x^3)/x^2] + E^25*(9 + 4*x - 2*x^2 + x^3)*Log[(9 + 4*x - 2*x^2 + x^3)/x^2]^2),x]

[Out]

4*x - (2*Defer[Int][(x + Log[-2 + 9/x^2 + 4/x + x])^(-2), x])/E^25 - Defer[Int][x/(x + Log[-2 + 9/x^2 + 4/x +
x])^2, x]/E^25 - Defer[Int][x^2/(x + Log[-2 + 9/x^2 + 4/x + x])^2, x]/E^25 + (18*Defer[Int][1/((9 + 4*x - 2*x^
2 + x^3)*(x + Log[-2 + 9/x^2 + 4/x + x])^2), x])/E^25 + (35*Defer[Int][x/((9 + 4*x - 2*x^2 + x^3)*(x + Log[-2
+ 9/x^2 + 4/x + x])^2), x])/E^25 + (4*Defer[Int][x^2/((9 + 4*x - 2*x^2 + x^3)*(x + Log[-2 + 9/x^2 + 4/x + x])^
2), x])/E^25 + (2*Defer[Int][x/(x + Log[-2 + 9/x^2 + 4/x + x]), x])/E^25

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {18 x+13 x^2+4 x^3-3 x^4+x^5+4 e^{25} x^2 \left (9+4 x-2 x^2+x^3\right )+2 \left (1+4 e^{25}\right ) x \left (9+4 x-2 x^2+x^3\right ) \log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )+4 e^{25} \left (9+4 x-2 x^2+x^3\right ) \log ^2\left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )}{e^{25} \left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx\\ &=\frac {\int \frac {18 x+13 x^2+4 x^3-3 x^4+x^5+4 e^{25} x^2 \left (9+4 x-2 x^2+x^3\right )+2 \left (1+4 e^{25}\right ) x \left (9+4 x-2 x^2+x^3\right ) \log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )+4 e^{25} \left (9+4 x-2 x^2+x^3\right ) \log ^2\left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}\\ &=\frac {\int \left (4 e^{25}-\frac {x \left (-18+5 x+4 x^2-x^3+x^4\right )}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}+\frac {2 x}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )}\right ) \, dx}{e^{25}}\\ &=4 x-\frac {\int \frac {x \left (-18+5 x+4 x^2-x^3+x^4\right )}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}+\frac {2 \int \frac {x}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )} \, dx}{e^{25}}\\ &=4 x-\frac {\int \left (\frac {2}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}+\frac {x}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}+\frac {x^2}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}+\frac {-18-35 x-4 x^2}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}\right ) \, dx}{e^{25}}+\frac {2 \int \frac {x}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )} \, dx}{e^{25}}\\ &=4 x-\frac {\int \frac {x}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {\int \frac {x^2}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {\int \frac {-18-35 x-4 x^2}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {2 \int \frac {1}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}+\frac {2 \int \frac {x}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )} \, dx}{e^{25}}\\ &=4 x-\frac {\int \frac {x}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {\int \frac {x^2}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {\int \left (-\frac {18}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}-\frac {35 x}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}-\frac {4 x^2}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2}\right ) \, dx}{e^{25}}-\frac {2 \int \frac {1}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}+\frac {2 \int \frac {x}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )} \, dx}{e^{25}}\\ &=4 x-\frac {\int \frac {x}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {\int \frac {x^2}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}-\frac {2 \int \frac {1}{\left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}+\frac {2 \int \frac {x}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )} \, dx}{e^{25}}+\frac {4 \int \frac {x^2}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}+\frac {18 \int \frac {1}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}+\frac {35 \int \frac {x}{\left (9+4 x-2 x^2+x^3\right ) \left (x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )\right )^2} \, dx}{e^{25}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.08, size = 33, normalized size = 1.27 \begin {gather*} \frac {4 e^{25} x+\frac {x^2}{x+\log \left (-2+\frac {9}{x^2}+\frac {4}{x}+x\right )}}{e^{25}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(18*x + 13*x^2 + 4*x^3 - 3*x^4 + x^5 + E^25*(36*x^2 + 16*x^3 - 8*x^4 + 4*x^5) + (18*x + 8*x^2 - 4*x^
3 + 2*x^4 + E^25*(72*x + 32*x^2 - 16*x^3 + 8*x^4))*Log[(9 + 4*x - 2*x^2 + x^3)/x^2] + E^25*(36 + 16*x - 8*x^2
+ 4*x^3)*Log[(9 + 4*x - 2*x^2 + x^3)/x^2]^2)/(E^25*(9*x^2 + 4*x^3 - 2*x^4 + x^5) + E^25*(18*x + 8*x^2 - 4*x^3
+ 2*x^4)*Log[(9 + 4*x - 2*x^2 + x^3)/x^2] + E^25*(9 + 4*x - 2*x^2 + x^3)*Log[(9 + 4*x - 2*x^2 + x^3)/x^2]^2),x
]

[Out]

(4*E^25*x + x^2/(x + Log[-2 + 9/x^2 + 4/x + x]))/E^25

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fricas [B]  time = 0.65, size = 63, normalized size = 2.42 \begin {gather*} \frac {4 \, x^{2} e^{25} + 4 \, x e^{25} \log \left (\frac {x^{3} - 2 \, x^{2} + 4 \, x + 9}{x^{2}}\right ) + x^{2}}{x e^{25} + e^{25} \log \left (\frac {x^{3} - 2 \, x^{2} + 4 \, x + 9}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-8*x^2+16*x+36)*exp(25)*log((x^3-2*x^2+4*x+9)/x^2)^2+((8*x^4-16*x^3+32*x^2+72*x)*exp(25)+2*x^
4-4*x^3+8*x^2+18*x)*log((x^3-2*x^2+4*x+9)/x^2)+(4*x^5-8*x^4+16*x^3+36*x^2)*exp(25)+x^5-3*x^4+4*x^3+13*x^2+18*x
)/((x^3-2*x^2+4*x+9)*exp(25)*log((x^3-2*x^2+4*x+9)/x^2)^2+(2*x^4-4*x^3+8*x^2+18*x)*exp(25)*log((x^3-2*x^2+4*x+
9)/x^2)+(x^5-2*x^4+4*x^3+9*x^2)*exp(25)),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-8*x^2+16*x+36)*exp(25)*log((x^3-2*x^2+4*x+9)/x^2)^2+((8*x^4-16*x^3+32*x^2+72*x)*exp(25)+2*x^
4-4*x^3+8*x^2+18*x)*log((x^3-2*x^2+4*x+9)/x^2)+(4*x^5-8*x^4+16*x^3+36*x^2)*exp(25)+x^5-3*x^4+4*x^3+13*x^2+18*x
)/((x^3-2*x^2+4*x+9)*exp(25)*log((x^3-2*x^2+4*x+9)/x^2)^2+(2*x^4-4*x^3+8*x^2+18*x)*exp(25)*log((x^3-2*x^2+4*x+
9)/x^2)+(x^5-2*x^4+4*x^3+9*x^2)*exp(25)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Evaluation time: 0.57sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument
Value

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maple [A]  time = 0.25, size = 33, normalized size = 1.27

method	result	size
risch	\(4 x +\frac {x^{2} {\mathrm e}^{-25}}{\ln \left (\frac {x^{3}-2 x^{2}+4 x +9}{x^{2}}\right )+x}\)	\(33\)
norman	\(\frac {\left (4 \,{\mathrm e}^{25}+1\right ) {\mathrm e}^{-25} x^{2}+4 \ln \left (\frac {x^{3}-2 x^{2}+4 x +9}{x^{2}}\right ) x}{\ln \left (\frac {x^{3}-2 x^{2}+4 x +9}{x^{2}}\right )+x}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-8*x^2+16*x+36)*exp(25)*ln((x^3-2*x^2+4*x+9)/x^2)^2+((8*x^4-16*x^3+32*x^2+72*x)*exp(25)+2*x^4-4*x^3
+8*x^2+18*x)*ln((x^3-2*x^2+4*x+9)/x^2)+(4*x^5-8*x^4+16*x^3+36*x^2)*exp(25)+x^5-3*x^4+4*x^3+13*x^2+18*x)/((x^3-
2*x^2+4*x+9)*exp(25)*ln((x^3-2*x^2+4*x+9)/x^2)^2+(2*x^4-4*x^3+8*x^2+18*x)*exp(25)*ln((x^3-2*x^2+4*x+9)/x^2)+(x
^5-2*x^4+4*x^3+9*x^2)*exp(25)),x,method=_RETURNVERBOSE)

[Out]

4*x+x^2*exp(-25)/(ln((x^3-2*x^2+4*x+9)/x^2)+x)

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maxima [B]  time = 0.42, size = 68, normalized size = 2.62 \begin {gather*} \frac {x^{2} {\left (4 \, e^{25} + 1\right )} + 4 \, x e^{25} \log \left (x^{3} - 2 \, x^{2} + 4 \, x + 9\right ) - 8 \, x e^{25} \log \relax (x)}{x e^{25} + e^{25} \log \left (x^{3} - 2 \, x^{2} + 4 \, x + 9\right ) - 2 \, e^{25} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-8*x^2+16*x+36)*exp(25)*log((x^3-2*x^2+4*x+9)/x^2)^2+((8*x^4-16*x^3+32*x^2+72*x)*exp(25)+2*x^
4-4*x^3+8*x^2+18*x)*log((x^3-2*x^2+4*x+9)/x^2)+(4*x^5-8*x^4+16*x^3+36*x^2)*exp(25)+x^5-3*x^4+4*x^3+13*x^2+18*x
)/((x^3-2*x^2+4*x+9)*exp(25)*log((x^3-2*x^2+4*x+9)/x^2)^2+(2*x^4-4*x^3+8*x^2+18*x)*exp(25)*log((x^3-2*x^2+4*x+
9)/x^2)+(x^5-2*x^4+4*x^3+9*x^2)*exp(25)),x, algorithm="maxima")

[Out]

(x^2*(4*e^25 + 1) + 4*x*e^25*log(x^3 - 2*x^2 + 4*x + 9) - 8*x*e^25*log(x))/(x*e^25 + e^25*log(x^3 - 2*x^2 + 4*
x + 9) - 2*e^25*log(x))

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mupad [B]  time = 5.82, size = 82, normalized size = 3.15 \begin {gather*} \frac {{\mathrm {e}}^{-25}\,\left (2\,x+2\,\ln \left (\frac {x^3-2\,x^2+4\,x+9}{x^2}\right )+4\,x^2\,{\mathrm {e}}^{25}+x^2+4\,x\,{\mathrm {e}}^{25}\,\ln \left (\frac {x^3-2\,x^2+4\,x+9}{x^2}\right )\right )}{x+\ln \left (\frac {x^3-2\,x^2+4\,x+9}{x^2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((18*x + log((4*x - 2*x^2 + x^3 + 9)/x^2)*(18*x + exp(25)*(72*x + 32*x^2 - 16*x^3 + 8*x^4) + 8*x^2 - 4*x^3
+ 2*x^4) + 13*x^2 + 4*x^3 - 3*x^4 + x^5 + exp(25)*(36*x^2 + 16*x^3 - 8*x^4 + 4*x^5) + exp(25)*log((4*x - 2*x^2
+ x^3 + 9)/x^2)^2*(16*x - 8*x^2 + 4*x^3 + 36))/(exp(25)*(9*x^2 + 4*x^3 - 2*x^4 + x^5) + exp(25)*log((4*x - 2*
x^2 + x^3 + 9)/x^2)^2*(4*x - 2*x^2 + x^3 + 9) + exp(25)*log((4*x - 2*x^2 + x^3 + 9)/x^2)*(18*x + 8*x^2 - 4*x^3
+ 2*x^4)),x)

[Out]

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

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sympy [A]  time = 0.28, size = 32, normalized size = 1.23 \begin {gather*} \frac {x^{2}}{x e^{25} + e^{25} \log {\left (\frac {x^{3} - 2 x^{2} + 4 x + 9}{x^{2}} \right )}} + 4 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-8*x**2+16*x+36)*exp(25)*ln((x**3-2*x**2+4*x+9)/x**2)**2+((8*x**4-16*x**3+32*x**2+72*x)*exp(
25)+2*x**4-4*x**3+8*x**2+18*x)*ln((x**3-2*x**2+4*x+9)/x**2)+(4*x**5-8*x**4+16*x**3+36*x**2)*exp(25)+x**5-3*x**
4+4*x**3+13*x**2+18*x)/((x**3-2*x**2+4*x+9)*exp(25)*ln((x**3-2*x**2+4*x+9)/x**2)**2+(2*x**4-4*x**3+8*x**2+18*x
)*exp(25)*ln((x**3-2*x**2+4*x+9)/x**2)+(x**5-2*x**4+4*x**3+9*x**2)*exp(25)),x)

[Out]

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

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