∫ 250000−45000x5−187500x6+e5x(−1562500x3+1171875x9)+e5x(−187500x2+140625x8) log(−4+3x6)+e5x(−7500x+5625x7) log2(−4+3x6)+e5x(−100+75x6) log3(−4+3x6)__________________________________________________________________________________________________________________________ −62500x3+46875x9+(−7500x2+5625x8) log(−4+3x6)+(−300x+225x7) log2(−4+3x6)+(−4+3x6) log3(−4+3x6) dx

3.45.38 \(\int \frac {250000-45000 x^5-187500 x^6+e^{5 x} (-1562500 x^3+1171875 x^9)+e^{5 x} (-187500 x^2+140625 x^8) \log (-4+3 x^6)+e^{5 x} (-7500 x+5625 x^7) \log ^2(-4+3 x^6)+e^{5 x} (-100+75 x^6) \log ^3(-4+3 x^6)}{-62500 x^3+46875 x^9+(-7500 x^2+5625 x^8) \log (-4+3 x^6)+(-300 x+225 x^7) \log ^2(-4+3 x^6)+(-4+3 x^6) \log ^3(-4+3 x^6)} \, dx\)

Optimal. Leaf size=26 \[ 5 e^{5 x}+\frac {2}{\left (x+\frac {1}{25} \log \left (-4+3 x^6\right )\right )^2} \]

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Rubi [F] time = 4.71, 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 {250000-45000 x^5-187500 x^6+e^{5 x} \left (-1562500 x^3+1171875 x^9\right )+e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )+e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )+e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{-62500 x^3+46875 x^9+\left (-7500 x^2+5625 x^8\right ) \log \left (-4+3 x^6\right )+\left (-300 x+225 x^7\right ) \log ^2\left (-4+3 x^6\right )+\left (-4+3 x^6\right ) \log ^3\left (-4+3 x^6\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(250000 - 45000*x^5 - 187500*x^6 + E^(5*x)*(-1562500*x^3 + 1171875*x^9) + E^(5*x)*(-187500*x^2 + 140625*x^

8)*Log[-4 + 3*x^6] + E^(5*x)*(-7500*x + 5625*x^7)*Log[-4 + 3*x^6]^2 + E^(5*x)*(-100 + 75*x^6)*Log[-4 + 3*x^6]^

3)/(-62500*x^3 + 46875*x^9 + (-7500*x^2 + 5625*x^8)*Log[-4 + 3*x^6] + (-300*x + 225*x^7)*Log[-4 + 3*x^6]^2 + (

-4 + 3*x^6)*Log[-4 + 3*x^6]^3),x]

[Out]

5*E^(5*x) - 62500*Defer[Int][(25*x + Log[-4 + 3*x^6])^(-3), x] + 2500*3^(1/6)*Defer[Int][1/((-(-2)^(1/3) - 3^(

1/6)*x)*(25*x + Log[-4 + 3*x^6])^3), x] + 2500*3^(1/6)*Defer[Int][1/((2^(1/3) - 3^(1/6)*x)*(25*x + Log[-4 + 3*

x^6])^3), x] + 2500*3^(1/6)*Defer[Int][1/(((-1)^(2/3)*2^(1/3) - 3^(1/6)*x)*(25*x + Log[-4 + 3*x^6])^3), x] - 2

500*3^(1/6)*Defer[Int][1/((-(-2)^(1/3) + 3^(1/6)*x)*(25*x + Log[-4 + 3*x^6])^3), x] - 2500*3^(1/6)*Defer[Int][

1/((2^(1/3) + 3^(1/6)*x)*(25*x + Log[-4 + 3*x^6])^3), x] - 2500*3^(1/6)*Defer[Int][1/(((-1)^(2/3)*2^(1/3) + 3^

(1/6)*x)*(25*x + Log[-4 + 3*x^6])^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-250000+45000 x^5+187500 x^6-e^{5 x} \left (-1562500 x^3+1171875 x^9\right )-e^{5 x} \left (-187500 x^2+140625 x^8\right ) \log \left (-4+3 x^6\right )-e^{5 x} \left (-7500 x+5625 x^7\right ) \log ^2\left (-4+3 x^6\right )-e^{5 x} \left (-100+75 x^6\right ) \log ^3\left (-4+3 x^6\right )}{\left (4-3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx\\ &=\int \left (25 e^{5 x}+\frac {250000}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {45000 x^5}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {187500 x^6}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx\\ &=25 \int e^{5 x} \, dx-45000 \int \frac {x^5}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-187500 \int \frac {x^6}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx+250000 \int \frac {1}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx\\ &=5 e^{5 x}-45000 \int \left (\frac {x^2}{2 \left (-2 \sqrt {3}+3 x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}+\frac {x^2}{2 \left (2 \sqrt {3}+3 x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx-187500 \int \left (\frac {1}{3 \left (25 x+\log \left (-4+3 x^6\right )\right )^3}+\frac {4}{3 \left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx+250000 \int \left (-\frac {1}{4 \left (2-\sqrt {3} x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {1}{4 \left (2+\sqrt {3} x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx\\ &=5 e^{5 x}-22500 \int \frac {x^2}{\left (-2 \sqrt {3}+3 x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-22500 \int \frac {x^2}{\left (2 \sqrt {3}+3 x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-62500 \int \frac {1}{\left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-62500 \int \frac {1}{\left (2-\sqrt {3} x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-62500 \int \frac {1}{\left (2+\sqrt {3} x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-250000 \int \frac {1}{\left (-4+3 x^6\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx\\ &=5 e^{5 x}-22500 \int \left (-\frac {1}{3\ 3^{5/6} \left (-\sqrt [3]{-2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {1}{3\ 3^{5/6} \left (\sqrt [3]{2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {1}{3\ 3^{5/6} \left ((-1)^{2/3} \sqrt [3]{2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx-22500 \int \left (\frac {1}{3\ 3^{5/6} \left (-\sqrt [3]{-2}+\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}+\frac {1}{3\ 3^{5/6} \left (\sqrt [3]{2}+\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}+\frac {1}{3\ 3^{5/6} \left ((-1)^{2/3} \sqrt [3]{2}+\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx-62500 \int \frac {1}{\left (25 x+\log \left (-4+3 x^6\right )\right )^3} \, dx-62500 \int \left (-\frac {1}{3\ 2^{2/3} \left (-\sqrt [3]{2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {1}{3\ 2^{2/3} \left (-\sqrt [3]{2}+\sqrt [3]{-1} \sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {1}{3\ 2^{2/3} \left (-\sqrt [3]{2}-(-1)^{2/3} \sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx-62500 \int \left (\frac {1}{3\ 2^{2/3} \left (\sqrt [3]{2}-\sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}+\frac {1}{3\ 2^{2/3} \left (\sqrt [3]{2}+\sqrt [3]{-1} \sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}+\frac {1}{3\ 2^{2/3} \left (\sqrt [3]{2}-(-1)^{2/3} \sqrt [6]{3} x\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx-250000 \int \left (-\frac {1}{4 \left (2-\sqrt {3} x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}-\frac {1}{4 \left (2+\sqrt {3} x^3\right ) \left (25 x+\log \left (-4+3 x^6\right )\right )^3}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A] time = 0.08, size = 28, normalized size = 1.08 \begin {gather*} 25 \left (\frac {e^{5 x}}{5}+\frac {50}{\left (25 x+\log \left (-4+3 x^6\right )\right )^2}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(250000 - 45000*x^5 - 187500*x^6 + E^(5*x)*(-1562500*x^3 + 1171875*x^9) + E^(5*x)*(-187500*x^2 + 140

625*x^8)*Log[-4 + 3*x^6] + E^(5*x)*(-7500*x + 5625*x^7)*Log[-4 + 3*x^6]^2 + E^(5*x)*(-100 + 75*x^6)*Log[-4 + 3

*x^6]^3)/(-62500*x^3 + 46875*x^9 + (-7500*x^2 + 5625*x^8)*Log[-4 + 3*x^6] + (-300*x + 225*x^7)*Log[-4 + 3*x^6]

^2 + (-4 + 3*x^6)*Log[-4 + 3*x^6]^3),x]

[Out]

25*(E^(5*x)/5 + 50/(25*x + Log[-4 + 3*x^6])^2)

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fricas [B] time = 0.48, size = 72, normalized size = 2.77 \begin {gather*} \frac {5 \, {\left (625 \, x^{2} e^{\left (5 \, x\right )} + 50 \, x e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right ) + e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right )^{2} + 250\right )}}{625 \, x^{2} + 50 \, x \log \left (3 \, x^{6} - 4\right ) + \log \left (3 \, x^{6} - 4\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*log(3*x^6-4)^2+(140625*x^8-187500*x

^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6

-4)^3+(225*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875*x^9-62500*x^3),x, algorithm="frica

s")

[Out]

5*(625*x^2*e^(5*x) + 50*x*e^(5*x)*log(3*x^6 - 4) + e^(5*x)*log(3*x^6 - 4)^2 + 250)/(625*x^2 + 50*x*log(3*x^6 -

4) + log(3*x^6 - 4)^2)

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giac [B] time = 6.15, size = 72, normalized size = 2.77 \begin {gather*} \frac {5 \, {\left (625 \, x^{2} e^{\left (5 \, x\right )} + 50 \, x e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right ) + e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right )^{2} + 250\right )}}{625 \, x^{2} + 50 \, x \log \left (3 \, x^{6} - 4\right ) + \log \left (3 \, x^{6} - 4\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*log(3*x^6-4)^2+(140625*x^8-187500*x

^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6

-4)^3+(225*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875*x^9-62500*x^3),x, algorithm="giac"

)

[Out]

5*(625*x^2*e^(5*x) + 50*x*e^(5*x)*log(3*x^6 - 4) + e^(5*x)*log(3*x^6 - 4)^2 + 250)/(625*x^2 + 50*x*log(3*x^6 -

4) + log(3*x^6 - 4)^2)

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maple [A] time = 0.04, size = 24, normalized size = 0.92


method	result	size

risch	\(5 \,{\mathrm e}^{5 x}+\frac {1250}{\left (\ln \left (3 x^{6}-4\right )+25 x \right )^{2}}\)	\(24\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(((75*x^6-100)*exp(5*x)*ln(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*ln(3*x^6-4)^2+(140625*x^8-187500*x^2)*exp(

5*x)*ln(3*x^6-4)+(1171875*x^9-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*ln(3*x^6-4)^3+(225

*x^7-300*x)*ln(3*x^6-4)^2+(5625*x^8-7500*x^2)*ln(3*x^6-4)+46875*x^9-62500*x^3),x,method=_RETURNVERBOSE)

[Out]

5*exp(5*x)+1250/(ln(3*x^6-4)+25*x)^2

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maxima [B] time = 0.49, size = 72, normalized size = 2.77 \begin {gather*} \frac {5 \, {\left (625 \, x^{2} e^{\left (5 \, x\right )} + 50 \, x e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right ) + e^{\left (5 \, x\right )} \log \left (3 \, x^{6} - 4\right )^{2} + 250\right )}}{625 \, x^{2} + 50 \, x \log \left (3 \, x^{6} - 4\right ) + \log \left (3 \, x^{6} - 4\right )^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x^6-100)*exp(5*x)*log(3*x^6-4)^3+(5625*x^7-7500*x)*exp(5*x)*log(3*x^6-4)^2+(140625*x^8-187500*x

^2)*exp(5*x)*log(3*x^6-4)+(1171875*x^9-1562500*x^3)*exp(5*x)-187500*x^6-45000*x^5+250000)/((3*x^6-4)*log(3*x^6

-4)^3+(225*x^7-300*x)*log(3*x^6-4)^2+(5625*x^8-7500*x^2)*log(3*x^6-4)+46875*x^9-62500*x^3),x, algorithm="maxim

a")

[Out]

5*(625*x^2*e^(5*x) + 50*x*e^(5*x)*log(3*x^6 - 4) + e^(5*x)*log(3*x^6 - 4)^2 + 250)/(625*x^2 + 50*x*log(3*x^6 -

4) + log(3*x^6 - 4)^2)

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mupad [B] time = 3.17, size = 23, normalized size = 0.88 \begin {gather*} 5\,{\mathrm {e}}^{5\,x}+\frac {1250}{{\left (25\,x+\ln \left (3\,x^6-4\right )\right )}^2} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((exp(5*x)*(1562500*x^3 - 1171875*x^9) + 45000*x^5 + 187500*x^6 + log(3*x^6 - 4)^2*exp(5*x)*(7500*x - 5625*

x^7) + log(3*x^6 - 4)*exp(5*x)*(187500*x^2 - 140625*x^8) - log(3*x^6 - 4)^3*exp(5*x)*(75*x^6 - 100) - 250000)/

(log(3*x^6 - 4)^2*(300*x - 225*x^7) + log(3*x^6 - 4)*(7500*x^2 - 5625*x^8) - log(3*x^6 - 4)^3*(3*x^6 - 4) + 62

500*x^3 - 46875*x^9),x)

[Out]

5*exp(5*x) + 1250/(25*x + log(3*x^6 - 4))^2

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sympy [A] time = 0.47, size = 34, normalized size = 1.31 \begin {gather*} 5 e^{5 x} + \frac {1250}{625 x^{2} + 50 x \log {\left (3 x^{6} - 4 \right )} + \log {\left (3 x^{6} - 4 \right )}^{2}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(((75*x**6-100)*exp(5*x)*ln(3*x**6-4)**3+(5625*x**7-7500*x)*exp(5*x)*ln(3*x**6-4)**2+(140625*x**8-187

500*x**2)*exp(5*x)*ln(3*x**6-4)+(1171875*x**9-1562500*x**3)*exp(5*x)-187500*x**6-45000*x**5+250000)/((3*x**6-4

)*ln(3*x**6-4)**3+(225*x**7-300*x)*ln(3*x**6-4)**2+(5625*x**8-7500*x**2)*ln(3*x**6-4)+46875*x**9-62500*x**3),x

)

[Out]

5*exp(5*x) + 1250/(625*x**2 + 50*x*log(3*x**6 - 4) + log(3*x**6 - 4)**2)

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