\(\int (e^{5 x^3} (32-1920 x^2+480 x^3)+e^{10 x^3} (-128+32 x+7680 x^2-3840 x^3+480 x^4)+(64 x+e^{5 x^3} (-256 x+64 x^2)) \log (x)+(64 x+e^{5 x^3} (-256 x+96 x^2-1920 x^4+480 x^5)) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)) \, dx\) [961]

Optimal result

Integrand size = 127, antiderivative size = 25 \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=16 \left (1+e^{5 x^3} (-4+x)+x^2 \log ^2(x)\right )^2 \] Output:

16*(1+(-4+x)*exp(5*x^3)+x^2*ln(x)^2)^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.45 (sec) , antiderivative size = 316, normalized size of antiderivative = 12.64 \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=32 \left (\frac {1}{2}-4 e^{5 x^3}+8 e^{10 x^3}+e^{5 x^3} x-4 e^{10 x^3} x+\frac {1}{2} e^{10 x^3} x^2-2 x^2 \, _3F_3\left (\frac {2}{3},\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3},\frac {5}{3};5 x^3\right )-\frac {24}{25} x^5 \, _3F_3\left (\frac {5}{3},\frac {5}{3},\frac {5}{3};\frac {8}{3},\frac {8}{3},\frac {8}{3};5 x^3\right )+\frac {8 \sqrt [3]{-x^3} \Gamma \left (\frac {2}{3},0,-5 x^3\right ) \log (x)}{3\ 5^{2/3} x}+\frac {24}{5} x^5 \, _2F_2\left (\frac {5}{3},\frac {5}{3};\frac {8}{3},\frac {8}{3};5 x^3\right ) \log (x)+x^2 \log ^2(x)+e^{5 x^3} x^3 \log ^2(x)+\frac {8 \sqrt [3]{-x^3} \Gamma \left (\frac {2}{3},0,-5 x^3\right ) \log ^2(x)}{3\ 5^{2/3} x}+\frac {4 x^2 \Gamma \left (\frac {5}{3},0,-5 x^3\right ) \log ^2(x)}{5^{2/3} \left (-x^3\right )^{2/3}}+\frac {1}{2} x^4 \log ^4(x)+2 x^2 \, _2F_2\left (\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3};5 x^3\right ) (1+2 \log (x))\right ) \] Input:

Integrate[E^(5*x^3)*(32 - 1920*x^2 + 480*x^3) + E^(10*x^3)*(-128 + 32*x + 
7680*x^2 - 3840*x^3 + 480*x^4) + (64*x + E^(5*x^3)*(-256*x + 64*x^2))*Log[ 
x] + (64*x + E^(5*x^3)*(-256*x + 96*x^2 - 1920*x^4 + 480*x^5))*Log[x]^2 + 
64*x^3*Log[x]^3 + 64*x^3*Log[x]^4,x]
 

Output:

32*(1/2 - 4*E^(5*x^3) + 8*E^(10*x^3) + E^(5*x^3)*x - 4*E^(10*x^3)*x + (E^( 
10*x^3)*x^2)/2 - 2*x^2*HypergeometricPFQ[{2/3, 2/3, 2/3}, {5/3, 5/3, 5/3}, 
 5*x^3] - (24*x^5*HypergeometricPFQ[{5/3, 5/3, 5/3}, {8/3, 8/3, 8/3}, 5*x^ 
3])/25 + (8*(-x^3)^(1/3)*Gamma[2/3, 0, -5*x^3]*Log[x])/(3*5^(2/3)*x) + (24 
*x^5*HypergeometricPFQ[{5/3, 5/3}, {8/3, 8/3}, 5*x^3]*Log[x])/5 + x^2*Log[ 
x]^2 + E^(5*x^3)*x^3*Log[x]^2 + (8*(-x^3)^(1/3)*Gamma[2/3, 0, -5*x^3]*Log[ 
x]^2)/(3*5^(2/3)*x) + (4*x^2*Gamma[5/3, 0, -5*x^3]*Log[x]^2)/(5^(2/3)*(-x^ 
3)^(2/3)) + (x^4*Log[x]^4)/2 + 2*x^2*HypergeometricPFQ[{2/3, 2/3}, {5/3, 5 
/3}, 5*x^3]*(1 + 2*Log[x]))
 

Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \left (64 x^3 \log ^4(x)+64 x^3 \log ^3(x)+e^{5 x^3} \left (480 x^3-1920 x^2+32\right )+\left (e^{5 x^3} \left (64 x^2-256 x\right )+64 x\right ) \log (x)+e^{10 x^3} \left (480 x^4-3840 x^3+7680 x^2+32 x-128\right )+\left (e^{5 x^3} \left (480 x^5-1920 x^4+96 x^2-256 x\right )+64 x\right ) \log ^2(x)\right ) \, dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -256 \int e^{5 x^3} x \log ^2(x)dx+480 \int e^{5 x^3} x^5 \log ^2(x)dx-1920 \int e^{5 x^3} x^4 \log ^2(x)dx+96 \int e^{5 x^3} x^2 \log ^2(x)dx+64 x^2 \, _2F_2\left (\frac {2}{3},\frac {2}{3};\frac {5}{3},\frac {5}{3};5 x^3\right )-\frac {64 \operatorname {ExpIntegralEi}\left (5 x^3\right )}{45}+16 x^4 \log ^4(x)-128 e^{5 x^3}+256 e^{10 x^3}+\frac {64}{15} e^{5 x^3} \log (x)+\frac {64\ 2^{2/3} x \Gamma \left (\frac {1}{3},-10 x^3\right )}{3 \sqrt [3]{5} \sqrt [3]{-x^3}}-\frac {32 x \Gamma \left (\frac {1}{3},-5 x^3\right )}{3 \sqrt [3]{5} \sqrt [3]{-x^3}}+32 x^2 \log ^2(x)-\frac {8 \sqrt [3]{2} x^5 \Gamma \left (\frac {5}{3},-10 x^3\right )}{5^{2/3} \left (-x^3\right )^{5/3}}+\frac {64\ 2^{2/3} x^4 \Gamma \left (\frac {4}{3},-10 x^3\right )}{\sqrt [3]{5} \left (-x^3\right )^{4/3}}-\frac {32 x^4 \Gamma \left (\frac {4}{3},-5 x^3\right )}{\sqrt [3]{5} \left (-x^3\right )^{4/3}}-\frac {16 \sqrt [3]{2} x^2 \Gamma \left (\frac {2}{3},-10 x^3\right )}{3\ 5^{2/3} \left (-x^3\right )^{2/3}}-\frac {256 x^2 \operatorname {Gamma}\left (\frac {2}{3}\right ) \log (x)}{3\ 5^{2/3} \left (-x^3\right )^{2/3}}+\frac {256 x^2 \log (x) \Gamma \left (\frac {2}{3},-5 x^3\right )}{3\ 5^{2/3} \left (-x^3\right )^{2/3}}\)

Input:

Int[E^(5*x^3)*(32 - 1920*x^2 + 480*x^3) + E^(10*x^3)*(-128 + 32*x + 7680*x 
^2 - 3840*x^3 + 480*x^4) + (64*x + E^(5*x^3)*(-256*x + 64*x^2))*Log[x] + ( 
64*x + E^(5*x^3)*(-256*x + 96*x^2 - 1920*x^4 + 480*x^5))*Log[x]^2 + 64*x^3 
*Log[x]^3 + 64*x^3*Log[x]^4,x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(74\) vs. \(2(24)=48\).

Time = 0.06 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.00

\[16 x^{4} \ln \left (x \right )^{4}+\left (16 x^{2}-128 x +256\right ) {\mathrm e}^{10 x^{3}}+\left (-128+32 x \right ) {\mathrm e}^{5 x^{3}}+\left (32 \,{\mathrm e}^{5 x^{3}} x^{3}-128 x^{2} {\mathrm e}^{5 x^{3}}+32 x^{2}\right ) \ln \left (x \right )^{2}\]

Input:

int(64*x^3*ln(x)^4+64*x^3*ln(x)^3+((480*x^5-1920*x^4+96*x^2-256*x)*exp(5*x 
^3)+64*x)*ln(x)^2+((64*x^2-256*x)*exp(5*x^3)+64*x)*ln(x)+(480*x^4-3840*x^3 
+7680*x^2+32*x-128)*exp(5*x^3)^2+(480*x^3-1920*x^2+32)*exp(5*x^3),x)
 

Output:

16*x^4*ln(x)^4+(16*x^2-128*x+256)*exp(5*x^3)^2+(-128+32*x)*exp(5*x^3)+(32* 
exp(5*x^3)*x^3-128*x^2*exp(5*x^3)+32*x^2)*ln(x)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=16 \, x^{4} \log \left (x\right )^{4} + 32 \, {\left (x^{2} + {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (5 \, x^{3}\right )}\right )} \log \left (x\right )^{2} + 16 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (10 \, x^{3}\right )} + 32 \, {\left (x - 4\right )} e^{\left (5 \, x^{3}\right )} \] Input:

integrate(64*x^3*log(x)^4+64*x^3*log(x)^3+((480*x^5-1920*x^4+96*x^2-256*x) 
*exp(5*x^3)+64*x)*log(x)^2+((64*x^2-256*x)*exp(5*x^3)+64*x)*log(x)+(480*x^ 
4-3840*x^3+7680*x^2+32*x-128)*exp(5*x^3)^2+(480*x^3-1920*x^2+32)*exp(5*x^3 
),x, algorithm="fricas")
 

Output:

16*x^4*log(x)^4 + 32*(x^2 + (x^3 - 4*x^2)*e^(5*x^3))*log(x)^2 + 16*(x^2 - 
8*x + 16)*e^(10*x^3) + 32*(x - 4)*e^(5*x^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (22) = 44\).

Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.72 \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=16 x^{4} \log {\left (x \right )}^{4} + 32 x^{2} \log {\left (x \right )}^{2} + \left (16 x^{2} - 128 x + 256\right ) e^{10 x^{3}} + \left (32 x^{3} \log {\left (x \right )}^{2} - 128 x^{2} \log {\left (x \right )}^{2} + 32 x - 128\right ) e^{5 x^{3}} \] Input:

integrate(64*x**3*ln(x)**4+64*x**3*ln(x)**3+((480*x**5-1920*x**4+96*x**2-2 
56*x)*exp(5*x**3)+64*x)*ln(x)**2+((64*x**2-256*x)*exp(5*x**3)+64*x)*ln(x)+ 
(480*x**4-3840*x**3+7680*x**2+32*x-128)*exp(5*x**3)**2+(480*x**3-1920*x**2 
+32)*exp(5*x**3),x)
 

Output:

16*x**4*log(x)**4 + 32*x**2*log(x)**2 + (16*x**2 - 128*x + 256)*exp(10*x** 
3) + (32*x**3*log(x)**2 - 128*x**2*log(x)**2 + 32*x - 128)*exp(5*x**3)
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.07 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.24 \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=16 \, x^{4} \log \left (x\right )^{4} - \frac {32 \cdot 5^{\frac {2}{3}} x^{4} \Gamma \left (\frac {4}{3}, -5 \, x^{3}\right )}{5 \, \left (-x^{3}\right )^{\frac {4}{3}}} + 32 \, x^{2} \log \left (x\right )^{2} + 32 \, {\left (x^{3} - 4 \, x^{2}\right )} e^{\left (5 \, x^{3}\right )} \log \left (x\right )^{2} - \frac {32 \cdot 5^{\frac {2}{3}} x \Gamma \left (\frac {1}{3}, -5 \, x^{3}\right )}{15 \, \left (-x^{3}\right )^{\frac {1}{3}}} + 16 \, {\left (x^{2} - 8 \, x + 16\right )} e^{\left (10 \, x^{3}\right )} - 128 \, e^{\left (5 \, x^{3}\right )} \] Input:

integrate(64*x^3*log(x)^4+64*x^3*log(x)^3+((480*x^5-1920*x^4+96*x^2-256*x) 
*exp(5*x^3)+64*x)*log(x)^2+((64*x^2-256*x)*exp(5*x^3)+64*x)*log(x)+(480*x^ 
4-3840*x^3+7680*x^2+32*x-128)*exp(5*x^3)^2+(480*x^3-1920*x^2+32)*exp(5*x^3 
),x, algorithm="maxima")
 

Output:

16*x^4*log(x)^4 - 32/5*5^(2/3)*x^4*gamma(4/3, -5*x^3)/(-x^3)^(4/3) + 32*x^ 
2*log(x)^2 + 32*(x^3 - 4*x^2)*e^(5*x^3)*log(x)^2 - 32/15*5^(2/3)*x*gamma(1 
/3, -5*x^3)/(-x^3)^(1/3) + 16*(x^2 - 8*x + 16)*e^(10*x^3) - 128*e^(5*x^3)
 

Giac [F]

\[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=\int { 64 \, x^{3} \log \left (x\right )^{4} + 64 \, x^{3} \log \left (x\right )^{3} + 32 \, {\left ({\left (15 \, x^{5} - 60 \, x^{4} + 3 \, x^{2} - 8 \, x\right )} e^{\left (5 \, x^{3}\right )} + 2 \, x\right )} \log \left (x\right )^{2} + 32 \, {\left (15 \, x^{4} - 120 \, x^{3} + 240 \, x^{2} + x - 4\right )} e^{\left (10 \, x^{3}\right )} + 32 \, {\left (15 \, x^{3} - 60 \, x^{2} + 1\right )} e^{\left (5 \, x^{3}\right )} + 64 \, {\left ({\left (x^{2} - 4 \, x\right )} e^{\left (5 \, x^{3}\right )} + x\right )} \log \left (x\right ) \,d x } \] Input:

integrate(64*x^3*log(x)^4+64*x^3*log(x)^3+((480*x^5-1920*x^4+96*x^2-256*x) 
*exp(5*x^3)+64*x)*log(x)^2+((64*x^2-256*x)*exp(5*x^3)+64*x)*log(x)+(480*x^ 
4-3840*x^3+7680*x^2+32*x-128)*exp(5*x^3)^2+(480*x^3-1920*x^2+32)*exp(5*x^3 
),x, algorithm="giac")
 

Output:

integrate(64*x^3*log(x)^4 + 64*x^3*log(x)^3 + 32*((15*x^5 - 60*x^4 + 3*x^2 
 - 8*x)*e^(5*x^3) + 2*x)*log(x)^2 + 32*(15*x^4 - 120*x^3 + 240*x^2 + x - 4 
)*e^(10*x^3) + 32*(15*x^3 - 60*x^2 + 1)*e^(5*x^3) + 64*((x^2 - 4*x)*e^(5*x 
^3) + x)*log(x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=\int 64\,x^3\,{\ln \left (x\right )}^3+64\,x^3\,{\ln \left (x\right )}^4+{\ln \left (x\right )}^2\,\left (64\,x-{\mathrm {e}}^{5\,x^3}\,\left (-480\,x^5+1920\,x^4-96\,x^2+256\,x\right )\right )+{\mathrm {e}}^{5\,x^3}\,\left (480\,x^3-1920\,x^2+32\right )+{\mathrm {e}}^{10\,x^3}\,\left (480\,x^4-3840\,x^3+7680\,x^2+32\,x-128\right )+\ln \left (x\right )\,\left (64\,x-{\mathrm {e}}^{5\,x^3}\,\left (256\,x-64\,x^2\right )\right ) \,d x \] Input:

int(64*x^3*log(x)^3 + 64*x^3*log(x)^4 + log(x)^2*(64*x - exp(5*x^3)*(256*x 
 - 96*x^2 + 1920*x^4 - 480*x^5)) + exp(5*x^3)*(480*x^3 - 1920*x^2 + 32) + 
exp(10*x^3)*(32*x + 7680*x^2 - 3840*x^3 + 480*x^4 - 128) + log(x)*(64*x - 
exp(5*x^3)*(256*x - 64*x^2)),x)
 

Output:

int(64*x^3*log(x)^3 + 64*x^3*log(x)^4 + log(x)^2*(64*x - exp(5*x^3)*(256*x 
 - 96*x^2 + 1920*x^4 - 480*x^5)) + exp(5*x^3)*(480*x^3 - 1920*x^2 + 32) + 
exp(10*x^3)*(32*x + 7680*x^2 - 3840*x^3 + 480*x^4 - 128) + log(x)*(64*x - 
exp(5*x^3)*(256*x - 64*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 4.04 \[ \int \left (e^{5 x^3} \left (32-1920 x^2+480 x^3\right )+e^{10 x^3} \left (-128+32 x+7680 x^2-3840 x^3+480 x^4\right )+\left (64 x+e^{5 x^3} \left (-256 x+64 x^2\right )\right ) \log (x)+\left (64 x+e^{5 x^3} \left (-256 x+96 x^2-1920 x^4+480 x^5\right )\right ) \log ^2(x)+64 x^3 \log ^3(x)+64 x^3 \log ^4(x)\right ) \, dx=16 e^{10 x^{3}} x^{2}-128 e^{10 x^{3}} x +256 e^{10 x^{3}}+32 e^{5 x^{3}} \mathrm {log}\left (x \right )^{2} x^{3}-128 e^{5 x^{3}} \mathrm {log}\left (x \right )^{2} x^{2}+32 e^{5 x^{3}} x -128 e^{5 x^{3}}+16 \mathrm {log}\left (x \right )^{4} x^{4}+32 \mathrm {log}\left (x \right )^{2} x^{2} \] Input:

int(64*x^3*log(x)^4+64*x^3*log(x)^3+((480*x^5-1920*x^4+96*x^2-256*x)*exp(5 
*x^3)+64*x)*log(x)^2+((64*x^2-256*x)*exp(5*x^3)+64*x)*log(x)+(480*x^4-3840 
*x^3+7680*x^2+32*x-128)*exp(5*x^3)^2+(480*x^3-1920*x^2+32)*exp(5*x^3),x)
 

Output:

16*(e**(10*x**3)*x**2 - 8*e**(10*x**3)*x + 16*e**(10*x**3) + 2*e**(5*x**3) 
*log(x)**2*x**3 - 8*e**(5*x**3)*log(x)**2*x**2 + 2*e**(5*x**3)*x - 8*e**(5 
*x**3) + log(x)**4*x**4 + 2*log(x)**2*x**2)