3.103.59 \(\int \frac {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} (18 x^2-6 x^3)+e^{x/4} (2 x-54 x^2+36 x^3-6 x^4)+(-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} (-36 x^2+12 x^3)) \log (x)+(18 x^2-6 e^{x/4} x^2-6 x^3) \log ^2(x)+2 x^2 \log ^3(x)+(-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} (-20 x-4 x^2)+e^{x/2} (180 x-24 x^2-12 x^3)+e^{x/4} (10-543 x+251 x^2+12 x^3-12 x^4)+(-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} (60 x+12 x^2)+e^{x/4} (-360 x+48 x^2+24 x^3)) \log (x)+(180 x-24 x^2-12 x^3+e^{x/4} (-60 x-12 x^2)) \log ^2(x)+(20 x+4 x^2) \log ^3(x)) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} (90-12 x-6 x^2)+e^{x/4} (-270+126 x+6 x^2-6 x^3)+(270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} (-180+24 x+12 x^2)) \log (x)+(90+e^{x/4} (-30-6 x)-12 x-6 x^2) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ x \left (x-\frac {1}{\left (3-e^{x/4}-x+\log (x)\right )^2}\right ) \log (5+x) \]

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Rubi [F]  time = 14.28, 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 {-6 x+56 x^2-2 e^{3 x/4} x^2-54 x^3+18 x^4-2 x^5+e^{x/2} \left (18 x^2-6 x^3\right )+e^{x/4} \left (2 x-54 x^2+36 x^3-6 x^4\right )+\left (-2 x+54 x^2+6 e^{x/2} x^2-36 x^3+6 x^4+e^{x/4} \left (-36 x^2+12 x^3\right )\right ) \log (x)+\left (18 x^2-6 e^{x/4} x^2-6 x^3\right ) \log ^2(x)+2 x^2 \log ^3(x)+\left (-10+528 x-434 x^2+72 x^3+16 x^4-4 x^5+e^{3 x/4} \left (-20 x-4 x^2\right )+e^{x/2} \left (180 x-24 x^2-12 x^3\right )+e^{x/4} \left (10-543 x+251 x^2+12 x^3-12 x^4\right )+\left (-10+538 x-252 x^2-12 x^3+12 x^4+e^{x/2} \left (60 x+12 x^2\right )+e^{x/4} \left (-360 x+48 x^2+24 x^3\right )\right ) \log (x)+\left (180 x-24 x^2-12 x^3+e^{x/4} \left (-60 x-12 x^2\right )\right ) \log ^2(x)+\left (20 x+4 x^2\right ) \log ^3(x)\right ) \log (5+x)}{270+e^{3 x/4} (-10-2 x)-216 x+36 x^2+8 x^3-2 x^4+e^{x/2} \left (90-12 x-6 x^2\right )+e^{x/4} \left (-270+126 x+6 x^2-6 x^3\right )+\left (270-126 x-6 x^2+6 x^3+e^{x/2} (30+6 x)+e^{x/4} \left (-180+24 x+12 x^2\right )\right ) \log (x)+\left (90+e^{x/4} (-30-6 x)-12 x-6 x^2\right ) \log ^2(x)+(10+2 x) \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-6*x + 56*x^2 - 2*E^((3*x)/4)*x^2 - 54*x^3 + 18*x^4 - 2*x^5 + E^(x/2)*(18*x^2 - 6*x^3) + E^(x/4)*(2*x - 5
4*x^2 + 36*x^3 - 6*x^4) + (-2*x + 54*x^2 + 6*E^(x/2)*x^2 - 36*x^3 + 6*x^4 + E^(x/4)*(-36*x^2 + 12*x^3))*Log[x]
+ (18*x^2 - 6*E^(x/4)*x^2 - 6*x^3)*Log[x]^2 + 2*x^2*Log[x]^3 + (-10 + 528*x - 434*x^2 + 72*x^3 + 16*x^4 - 4*x
^5 + E^((3*x)/4)*(-20*x - 4*x^2) + E^(x/2)*(180*x - 24*x^2 - 12*x^3) + E^(x/4)*(10 - 543*x + 251*x^2 + 12*x^3
- 12*x^4) + (-10 + 538*x - 252*x^2 - 12*x^3 + 12*x^4 + E^(x/2)*(60*x + 12*x^2) + E^(x/4)*(-360*x + 48*x^2 + 24
*x^3))*Log[x] + (180*x - 24*x^2 - 12*x^3 + E^(x/4)*(-60*x - 12*x^2))*Log[x]^2 + (20*x + 4*x^2)*Log[x]^3)*Log[5
+ x])/(270 + E^((3*x)/4)*(-10 - 2*x) - 216*x + 36*x^2 + 8*x^3 - 2*x^4 + E^(x/2)*(90 - 12*x - 6*x^2) + E^(x/4)
*(-270 + 126*x + 6*x^2 - 6*x^3) + (270 - 126*x - 6*x^2 + 6*x^3 + E^(x/2)*(30 + 6*x) + E^(x/4)*(-180 + 24*x + 1
2*x^2))*Log[x] + (90 + E^(x/4)*(-30 - 6*x) - 12*x - 6*x^2)*Log[x]^2 + (10 + 2*x)*Log[x]^3),x]

[Out]

x^2*Log[5 + x] - 2*Defer[Int][Log[5 + x]/(-3 + E^(x/4) + x - Log[x])^3, x] + (7*Defer[Int][(x*Log[5 + x])/(-3
+ E^(x/4) + x - Log[x])^3, x])/2 - Defer[Int][(x^2*Log[5 + x])/(-3 + E^(x/4) + x - Log[x])^3, x]/2 - Defer[Int
][Log[5 + x]/(-3 + E^(x/4) + x - Log[x])^2, x] + Defer[Int][(x*Log[5 + x])/(-3 + E^(x/4) + x - Log[x])^2, x]/2
+ Defer[Int][(x*Log[x]*Log[5 + x])/(-3 + E^(x/4) + x - Log[x])^3, x]/2 - 4*Defer[Subst][Defer[Int][(-3 + E^x
+ 4*x - Log[4*x])^(-2), x], x, x/4] + 20*Defer[Subst][Defer[Int][1/((5 + 4*x)*(-3 + E^x + 4*x - Log[4*x])^2),
x], x, x/4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x \left (3-28 x+e^{3 x/4} x+3 e^{x/2} (-3+x) x+27 x^2-9 x^3+x^4+e^{x/4} \left (-1+27 x-18 x^2+3 x^3\right )\right )-(5+x) \left (4 e^{3 x/4} x+12 e^{x/2} (-3+x) x+e^{x/4} \left (-2+109 x-72 x^2+12 x^3\right )+2 \left (1-53 x+54 x^2-18 x^3+2 x^4\right )\right ) \log (5+x)-6 x \left (-3+e^{x/4}+x\right ) \log ^2(x) (x+2 (5+x) \log (5+x))+2 x \log ^3(x) (x+2 (5+x) \log (5+x))+2 \log (x) \left (x \left (-1+3 \left (-3+e^{x/4}\right )^2 x+6 \left (-3+e^{x/4}\right ) x^2+3 x^3\right )+(5+x) \left (-1+6 \left (-3+e^{x/4}\right )^2 x+12 \left (-3+e^{x/4}\right ) x^2+6 x^3\right ) \log (5+x)\right )}{2 (5+x) \left (3-e^{x/4}-x+\log (x)\right )^3} \, dx\\ &=\frac {1}{2} \int \frac {-2 x \left (3-28 x+e^{3 x/4} x+3 e^{x/2} (-3+x) x+27 x^2-9 x^3+x^4+e^{x/4} \left (-1+27 x-18 x^2+3 x^3\right )\right )-(5+x) \left (4 e^{3 x/4} x+12 e^{x/2} (-3+x) x+e^{x/4} \left (-2+109 x-72 x^2+12 x^3\right )+2 \left (1-53 x+54 x^2-18 x^3+2 x^4\right )\right ) \log (5+x)-6 x \left (-3+e^{x/4}+x\right ) \log ^2(x) (x+2 (5+x) \log (5+x))+2 x \log ^3(x) (x+2 (5+x) \log (5+x))+2 \log (x) \left (x \left (-1+3 \left (-3+e^{x/4}\right )^2 x+6 \left (-3+e^{x/4}\right ) x^2+3 x^3\right )+(5+x) \left (-1+6 \left (-3+e^{x/4}\right )^2 x+12 \left (-3+e^{x/4}\right ) x^2+6 x^3\right ) \log (5+x)\right )}{(5+x) \left (3-e^{x/4}-x+\log (x)\right )^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {\left (4-7 x+x^2-x \log (x)\right ) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3}+\frac {2 x (x+10 \log (5+x)+2 x \log (5+x))}{5+x}+\frac {-2 x-10 \log (5+x)+3 x \log (5+x)+x^2 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (4-7 x+x^2-x \log (x)\right ) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx\right )+\frac {1}{2} \int \frac {-2 x-10 \log (5+x)+3 x \log (5+x)+x^2 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\int \frac {x (x+10 \log (5+x)+2 x \log (5+x))}{5+x} \, dx\\ &=\frac {1}{2} \int \left (-\frac {2 x}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}-\frac {10 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}+\frac {3 x \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}+\frac {x^2 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}\right ) \, dx-\frac {1}{2} \int \left (\frac {4 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3}-\frac {7 x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3}+\frac {x^2 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3}-\frac {x \log (x) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3}\right ) \, dx+\int x \left (\frac {x}{5+x}+2 \log (5+x)\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^2 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx\right )+\frac {1}{2} \int \frac {x^2 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {x \log (x) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {3}{2} \int \frac {x \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-2 \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {7}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-5 \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-\int \frac {x}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\int \left (\frac {x^2}{5+x}+2 x \log (5+x)\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {x^2 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx\right )+\frac {1}{2} \int \frac {x \log (x) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {1}{2} \int \left (-\frac {5 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2}+\frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2}+\frac {25 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}\right ) \, dx+\frac {3}{2} \int \left (\frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2}-\frac {5 \log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}\right ) \, dx+2 \int x \log (5+x) \, dx-2 \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {7}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-5 \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\int \frac {x^2}{5+x} \, dx-\int \left (\frac {1}{\left (-3+e^{x/4}+x-\log (x)\right )^2}-\frac {5}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2}\right ) \, dx\\ &=x^2 \log (5+x)-\frac {1}{2} \int \frac {x^2 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {1}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {x \log (x) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {3}{2} \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-2 \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-\frac {5}{2} \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {7}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+5 \int \frac {1}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-5 \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-\frac {15}{2} \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {25}{2} \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-\int \frac {x^2}{5+x} \, dx+\int \left (-5+x+\frac {25}{5+x}\right ) \, dx-\int \frac {1}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx\\ &=-5 x+\frac {x^2}{2}+25 \log (5+x)+x^2 \log (5+x)-\frac {1}{2} \int \frac {x^2 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {1}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {x \log (x) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {3}{2} \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-2 \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-\frac {5}{2} \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {7}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-4 \operatorname {Subst}\left (\int \frac {1}{\left (-3+e^x+4 x-\log (4 x)\right )^2} \, dx,x,\frac {x}{4}\right )-5 \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-\frac {15}{2} \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {25}{2} \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+20 \operatorname {Subst}\left (\int \frac {1}{(5+4 x) \left (-3+e^x+4 x-\log (4 x)\right )^2} \, dx,x,\frac {x}{4}\right )-\int \left (-5+x+\frac {25}{5+x}\right ) \, dx\\ &=x^2 \log (5+x)-\frac {1}{2} \int \frac {x^2 \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {1}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {1}{2} \int \frac {x \log (x) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx+\frac {3}{2} \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-2 \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-\frac {5}{2} \int \frac {\log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {7}{2} \int \frac {x \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^3} \, dx-4 \operatorname {Subst}\left (\int \frac {1}{\left (-3+e^x+4 x-\log (4 x)\right )^2} \, dx,x,\frac {x}{4}\right )-5 \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx-\frac {15}{2} \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+\frac {25}{2} \int \frac {\log (5+x)}{(5+x) \left (-3+e^{x/4}+x-\log (x)\right )^2} \, dx+20 \operatorname {Subst}\left (\int \frac {1}{(5+4 x) \left (-3+e^x+4 x-\log (4 x)\right )^2} \, dx,x,\frac {x}{4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.41, size = 75, normalized size = 2.68 \begin {gather*} \frac {x \left (-1+\left (-3+e^{x/4}\right )^2 x+2 \left (-3+e^{x/4}\right ) x^2+x^3-2 x \left (-3+e^{x/4}+x\right ) \log (x)+x \log ^2(x)\right ) \log (5+x)}{\left (-3+e^{x/4}+x-\log (x)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*x + 56*x^2 - 2*E^((3*x)/4)*x^2 - 54*x^3 + 18*x^4 - 2*x^5 + E^(x/2)*(18*x^2 - 6*x^3) + E^(x/4)*(2
*x - 54*x^2 + 36*x^3 - 6*x^4) + (-2*x + 54*x^2 + 6*E^(x/2)*x^2 - 36*x^3 + 6*x^4 + E^(x/4)*(-36*x^2 + 12*x^3))*
Log[x] + (18*x^2 - 6*E^(x/4)*x^2 - 6*x^3)*Log[x]^2 + 2*x^2*Log[x]^3 + (-10 + 528*x - 434*x^2 + 72*x^3 + 16*x^4
- 4*x^5 + E^((3*x)/4)*(-20*x - 4*x^2) + E^(x/2)*(180*x - 24*x^2 - 12*x^3) + E^(x/4)*(10 - 543*x + 251*x^2 + 1
2*x^3 - 12*x^4) + (-10 + 538*x - 252*x^2 - 12*x^3 + 12*x^4 + E^(x/2)*(60*x + 12*x^2) + E^(x/4)*(-360*x + 48*x^
2 + 24*x^3))*Log[x] + (180*x - 24*x^2 - 12*x^3 + E^(x/4)*(-60*x - 12*x^2))*Log[x]^2 + (20*x + 4*x^2)*Log[x]^3)
*Log[5 + x])/(270 + E^((3*x)/4)*(-10 - 2*x) - 216*x + 36*x^2 + 8*x^3 - 2*x^4 + E^(x/2)*(90 - 12*x - 6*x^2) + E
^(x/4)*(-270 + 126*x + 6*x^2 - 6*x^3) + (270 - 126*x - 6*x^2 + 6*x^3 + E^(x/2)*(30 + 6*x) + E^(x/4)*(-180 + 24
*x + 12*x^2))*Log[x] + (90 + E^(x/4)*(-30 - 6*x) - 12*x - 6*x^2)*Log[x]^2 + (10 + 2*x)*Log[x]^3),x]

[Out]

(x*(-1 + (-3 + E^(x/4))^2*x + 2*(-3 + E^(x/4))*x^2 + x^3 - 2*x*(-3 + E^(x/4) + x)*Log[x] + x*Log[x]^2)*Log[5 +
x])/(-3 + E^(x/4) + x - Log[x])^2

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fricas [B]  time = 0.84, size = 112, normalized size = 4.00 \begin {gather*} \frac {{\left (x^{4} + x^{2} \log \relax (x)^{2} - 6 \, x^{3} + x^{2} e^{\left (\frac {1}{2} \, x\right )} + 9 \, x^{2} + 2 \, {\left (x^{3} - 3 \, x^{2}\right )} e^{\left (\frac {1}{4} \, x\right )} - 2 \, {\left (x^{3} + x^{2} e^{\left (\frac {1}{4} \, x\right )} - 3 \, x^{2}\right )} \log \relax (x) - x\right )} \log \left (x + 5\right )}{x^{2} + 2 \, {\left (x - 3\right )} e^{\left (\frac {1}{4} \, x\right )} - 2 \, {\left (x + e^{\left (\frac {1}{4} \, x\right )} - 3\right )} \log \relax (x) + \log \relax (x)^{2} - 6 \, x + e^{\left (\frac {1}{2} \, x\right )} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+20*x)*log(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2+180*x)*log(x)^2+((12*x^2+60*x)*exp(
1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4*x)+12*x^4-12*x^3-252*x^2+538*x-10)*log(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-
12*x^3-24*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4*x)-4*x^5+16*x^4+72*x^3-434*x^2+528
*x-10)*log(5+x)+2*x^2*log(x)^3+(-6*x^2*exp(1/4*x)-6*x^3+18*x^2)*log(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*e
xp(1/4*x)+6*x^4-36*x^3+54*x^2-2*x)*log(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*exp(1/4*x)^2+(-6*x^4+36*x^3-54*x^
2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x^2-6*x)/((2*x+10)*log(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*log(
x)^2+((6*x+30)*exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)*log(x)+(-2*x-10)*exp(1/4*x)^3+
(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x, algorithm="
fricas")

[Out]

(x^4 + x^2*log(x)^2 - 6*x^3 + x^2*e^(1/2*x) + 9*x^2 + 2*(x^3 - 3*x^2)*e^(1/4*x) - 2*(x^3 + x^2*e^(1/4*x) - 3*x
^2)*log(x) - x)*log(x + 5)/(x^2 + 2*(x - 3)*e^(1/4*x) - 2*(x + e^(1/4*x) - 3)*log(x) + log(x)^2 - 6*x + e^(1/2
*x) + 9)

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giac [B]  time = 0.86, size = 170, normalized size = 6.07 \begin {gather*} \frac {x^{4} \log \left (x + 5\right ) + 2 \, x^{3} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) - 2 \, x^{3} \log \left (x + 5\right ) \log \relax (x) - 2 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) \log \relax (x) + x^{2} \log \left (x + 5\right ) \log \relax (x)^{2} - 6 \, x^{3} \log \left (x + 5\right ) + x^{2} e^{\left (\frac {1}{2} \, x\right )} \log \left (x + 5\right ) - 6 \, x^{2} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) + 6 \, x^{2} \log \left (x + 5\right ) \log \relax (x) + 9 \, x^{2} \log \left (x + 5\right ) - x \log \left (x + 5\right )}{x^{2} + 2 \, x e^{\left (\frac {1}{4} \, x\right )} - 2 \, x \log \relax (x) - 2 \, e^{\left (\frac {1}{4} \, x\right )} \log \relax (x) + \log \relax (x)^{2} - 6 \, x + e^{\left (\frac {1}{2} \, x\right )} - 6 \, e^{\left (\frac {1}{4} \, x\right )} + 6 \, \log \relax (x) + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+20*x)*log(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2+180*x)*log(x)^2+((12*x^2+60*x)*exp(
1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4*x)+12*x^4-12*x^3-252*x^2+538*x-10)*log(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-
12*x^3-24*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4*x)-4*x^5+16*x^4+72*x^3-434*x^2+528
*x-10)*log(5+x)+2*x^2*log(x)^3+(-6*x^2*exp(1/4*x)-6*x^3+18*x^2)*log(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*e
xp(1/4*x)+6*x^4-36*x^3+54*x^2-2*x)*log(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*exp(1/4*x)^2+(-6*x^4+36*x^3-54*x^
2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x^2-6*x)/((2*x+10)*log(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*log(
x)^2+((6*x+30)*exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)*log(x)+(-2*x-10)*exp(1/4*x)^3+
(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x, algorithm="
giac")

[Out]

(x^4*log(x + 5) + 2*x^3*e^(1/4*x)*log(x + 5) - 2*x^3*log(x + 5)*log(x) - 2*x^2*e^(1/4*x)*log(x + 5)*log(x) + x
^2*log(x + 5)*log(x)^2 - 6*x^3*log(x + 5) + x^2*e^(1/2*x)*log(x + 5) - 6*x^2*e^(1/4*x)*log(x + 5) + 6*x^2*log(
x + 5)*log(x) + 9*x^2*log(x + 5) - x*log(x + 5))/(x^2 + 2*x*e^(1/4*x) - 2*x*log(x) - 2*e^(1/4*x)*log(x) + log(
x)^2 - 6*x + e^(1/2*x) - 6*e^(1/4*x) + 6*log(x) + 9)

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maple [B]  time = 0.08, size = 82, normalized size = 2.93

method	result	size
risch	\(\frac {\left (x^{3}+2 x^{2} {\mathrm e}^{\frac {x}{4}}-2 x^{2} \ln \relax (x )+x \,{\mathrm e}^{\frac {x}{2}}-2 \ln \relax (x ) {\mathrm e}^{\frac {x}{4}} x +x \ln \relax (x )^{2}-6 x^{2}-6 x \,{\mathrm e}^{\frac {x}{4}}+6 x \ln \relax (x )+9 x -1\right ) x \ln \left (5+x \right )}{\left (x +{\mathrm e}^{\frac {x}{4}}-\ln \relax (x )-3\right )^{2}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^2+20*x)*ln(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2+180*x)*ln(x)^2+((12*x^2+60*x)*exp(1/4*x)^2
+(24*x^3+48*x^2-360*x)*exp(1/4*x)+12*x^4-12*x^3-252*x^2+538*x-10)*ln(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-12*x^3-24
*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4*x)-4*x^5+16*x^4+72*x^3-434*x^2+528*x-10)*ln
(5+x)+2*x^2*ln(x)^3+(-6*x^2*exp(1/4*x)-6*x^3+18*x^2)*ln(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*exp(1/4*x)+6*
x^4-36*x^3+54*x^2-2*x)*ln(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*exp(1/4*x)^2+(-6*x^4+36*x^3-54*x^2+2*x)*exp(1/
4*x)-2*x^5+18*x^4-54*x^3+56*x^2-6*x)/((2*x+10)*ln(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*ln(x)^2+((6*x+30)*
exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)*ln(x)+(-2*x-10)*exp(1/4*x)^3+(-6*x^2-12*x+90)
*exp(1/4*x)^2+(-6*x^3+6*x^2+126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x,method=_RETURNVERBOSE)

[Out]

(x^3+2*x^2*exp(1/4*x)-2*x^2*ln(x)+x*exp(1/2*x)-2*ln(x)*exp(1/4*x)*x+x*ln(x)^2-6*x^2-6*x*exp(1/4*x)+6*x*ln(x)+9
*x-1)*x/(x+exp(1/4*x)-ln(x)-3)^2*ln(5+x)

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maxima [B]  time = 0.66, size = 121, normalized size = 4.32 \begin {gather*} \frac {x^{2} e^{\left (\frac {1}{2} \, x\right )} \log \left (x + 5\right ) + 2 \, {\left (x^{3} - x^{2} \log \relax (x) - 3 \, x^{2}\right )} e^{\left (\frac {1}{4} \, x\right )} \log \left (x + 5\right ) + {\left (x^{4} + x^{2} \log \relax (x)^{2} - 6 \, x^{3} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x^{2}\right )} \log \relax (x) - x\right )} \log \left (x + 5\right )}{x^{2} + 2 \, {\left (x - \log \relax (x) - 3\right )} e^{\left (\frac {1}{4} \, x\right )} - 2 \, {\left (x - 3\right )} \log \relax (x) + \log \relax (x)^{2} - 6 \, x + e^{\left (\frac {1}{2} \, x\right )} + 9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^2+20*x)*log(x)^3+((-12*x^2-60*x)*exp(1/4*x)-12*x^3-24*x^2+180*x)*log(x)^2+((12*x^2+60*x)*exp(
1/4*x)^2+(24*x^3+48*x^2-360*x)*exp(1/4*x)+12*x^4-12*x^3-252*x^2+538*x-10)*log(x)+(-4*x^2-20*x)*exp(1/4*x)^3+(-
12*x^3-24*x^2+180*x)*exp(1/4*x)^2+(-12*x^4+12*x^3+251*x^2-543*x+10)*exp(1/4*x)-4*x^5+16*x^4+72*x^3-434*x^2+528
*x-10)*log(5+x)+2*x^2*log(x)^3+(-6*x^2*exp(1/4*x)-6*x^3+18*x^2)*log(x)^2+(6*x^2*exp(1/4*x)^2+(12*x^3-36*x^2)*e
xp(1/4*x)+6*x^4-36*x^3+54*x^2-2*x)*log(x)-2*x^2*exp(1/4*x)^3+(-6*x^3+18*x^2)*exp(1/4*x)^2+(-6*x^4+36*x^3-54*x^
2+2*x)*exp(1/4*x)-2*x^5+18*x^4-54*x^3+56*x^2-6*x)/((2*x+10)*log(x)^3+((-6*x-30)*exp(1/4*x)-6*x^2-12*x+90)*log(
x)^2+((6*x+30)*exp(1/4*x)^2+(12*x^2+24*x-180)*exp(1/4*x)+6*x^3-6*x^2-126*x+270)*log(x)+(-2*x-10)*exp(1/4*x)^3+
(-6*x^2-12*x+90)*exp(1/4*x)^2+(-6*x^3+6*x^2+126*x-270)*exp(1/4*x)-2*x^4+8*x^3+36*x^2-216*x+270),x, algorithm="
maxima")

[Out]

(x^2*e^(1/2*x)*log(x + 5) + 2*(x^3 - x^2*log(x) - 3*x^2)*e^(1/4*x)*log(x + 5) + (x^4 + x^2*log(x)^2 - 6*x^3 +
9*x^2 - 2*(x^3 - 3*x^2)*log(x) - x)*log(x + 5))/(x^2 + 2*(x - log(x) - 3)*e^(1/4*x) - 2*(x - 3)*log(x) + log(x
)^2 - 6*x + e^(1/2*x) + 9)

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mupad [B]  time = 7.61, size = 122, normalized size = 4.36 \begin {gather*} -\frac {\ln \left (x+5\right )\,\left (x-\frac {{\mathrm {e}}^{x/2}\,\left (x^3+5\,x^2\right )}{x+5}-\frac {\left (x^3+5\,x^2\right )\,{\left (\ln \relax (x)-x+3\right )}^2}{x+5}+\frac {2\,{\mathrm {e}}^{x/4}\,\left (x^3+5\,x^2\right )\,\left (\ln \relax (x)-x+3\right )}{x+5}\right )}{{\mathrm {e}}^{x/2}-6\,x+6\,\ln \relax (x)-{\mathrm {e}}^{x/4}\,\left (2\,\ln \relax (x)-2\,x+6\right )+{\ln \relax (x)}^2-2\,x\,\ln \relax (x)+x^2+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x - log(x + 5)*(528*x - exp((3*x)/4)*(20*x + 4*x^2) + log(x)^3*(20*x + 4*x^2) - exp(x/2)*(24*x^2 - 180
*x + 12*x^3) - log(x)^2*(exp(x/4)*(60*x + 12*x^2) - 180*x + 24*x^2 + 12*x^3) + log(x)*(538*x + exp(x/2)*(60*x
+ 12*x^2) + exp(x/4)*(48*x^2 - 360*x + 24*x^3) - 252*x^2 - 12*x^3 + 12*x^4 - 10) + exp(x/4)*(251*x^2 - 543*x +
12*x^3 - 12*x^4 + 10) - 434*x^2 + 72*x^3 + 16*x^4 - 4*x^5 - 10) - exp(x/2)*(18*x^2 - 6*x^3) + 2*x^2*exp((3*x)
/4) - exp(x/4)*(2*x - 54*x^2 + 36*x^3 - 6*x^4) - 2*x^2*log(x)^3 - 56*x^2 + 54*x^3 - 18*x^4 + 2*x^5 + log(x)^2*
(6*x^2*exp(x/4) - 18*x^2 + 6*x^3) + log(x)*(2*x + exp(x/4)*(36*x^2 - 12*x^3) - 6*x^2*exp(x/2) - 54*x^2 + 36*x^
3 - 6*x^4))/(exp(x/4)*(126*x + 6*x^2 - 6*x^3 - 270) - exp(x/2)*(12*x + 6*x^2 - 90) - 216*x + log(x)*(exp(x/4)*
(24*x + 12*x^2 - 180) - 126*x + exp(x/2)*(6*x + 30) - 6*x^2 + 6*x^3 + 270) - exp((3*x)/4)*(2*x + 10) + 36*x^2
+ 8*x^3 - 2*x^4 + log(x)^3*(2*x + 10) - log(x)^2*(12*x + exp(x/4)*(6*x + 30) + 6*x^2 - 90) + 270),x)

[Out]

-(log(x + 5)*(x - (exp(x/2)*(5*x^2 + x^3))/(x + 5) - ((5*x^2 + x^3)*(log(x) - x + 3)^2)/(x + 5) + (2*exp(x/4)*
(5*x^2 + x^3)*(log(x) - x + 3))/(x + 5)))/(exp(x/2) - 6*x + 6*log(x) - exp(x/4)*(2*log(x) - 2*x + 6) + log(x)^
2 - 2*x*log(x) + x^2 + 9)

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sympy [B]  time = 1.37, size = 73, normalized size = 2.61 \begin {gather*} - \frac {x \log {\left (x + 5 \right )}}{x^{2} - 2 x \log {\relax (x )} - 6 x + \left (2 x - 2 \log {\relax (x )} - 6\right ) e^{\frac {x}{4}} + e^{\frac {x}{2}} + \log {\relax (x )}^{2} + 6 \log {\relax (x )} + 9} + \left (x^{2} - \frac {25}{3}\right ) \log {\left (x + 5 \right )} + \frac {25 \log {\left (3 x + 15 \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**2+20*x)*ln(x)**3+((-12*x**2-60*x)*exp(1/4*x)-12*x**3-24*x**2+180*x)*ln(x)**2+((12*x**2+60*x)
*exp(1/4*x)**2+(24*x**3+48*x**2-360*x)*exp(1/4*x)+12*x**4-12*x**3-252*x**2+538*x-10)*ln(x)+(-4*x**2-20*x)*exp(
1/4*x)**3+(-12*x**3-24*x**2+180*x)*exp(1/4*x)**2+(-12*x**4+12*x**3+251*x**2-543*x+10)*exp(1/4*x)-4*x**5+16*x**
4+72*x**3-434*x**2+528*x-10)*ln(5+x)+2*x**2*ln(x)**3+(-6*x**2*exp(1/4*x)-6*x**3+18*x**2)*ln(x)**2+(6*x**2*exp(
1/4*x)**2+(12*x**3-36*x**2)*exp(1/4*x)+6*x**4-36*x**3+54*x**2-2*x)*ln(x)-2*x**2*exp(1/4*x)**3+(-6*x**3+18*x**2
)*exp(1/4*x)**2+(-6*x**4+36*x**3-54*x**2+2*x)*exp(1/4*x)-2*x**5+18*x**4-54*x**3+56*x**2-6*x)/((2*x+10)*ln(x)**
3+((-6*x-30)*exp(1/4*x)-6*x**2-12*x+90)*ln(x)**2+((6*x+30)*exp(1/4*x)**2+(12*x**2+24*x-180)*exp(1/4*x)+6*x**3-
6*x**2-126*x+270)*ln(x)+(-2*x-10)*exp(1/4*x)**3+(-6*x**2-12*x+90)*exp(1/4*x)**2+(-6*x**3+6*x**2+126*x-270)*exp
(1/4*x)-2*x**4+8*x**3+36*x**2-216*x+270),x)

[Out]

-x*log(x + 5)/(x**2 - 2*x*log(x) - 6*x + (2*x - 2*log(x) - 6)*exp(x/4) + exp(x/2) + log(x)**2 + 6*log(x) + 9)
+ (x**2 - 25/3)*log(x + 5) + 25*log(3*x + 15)/3

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