3.177 \(\int \frac{x^6}{(3+2 x^5)^3} \, dx\)

Optimal. Leaf size=319 \[ \frac{x^2}{150 \left (2 x^5+3\right )}-\frac{x^2}{20 \left (2 x^5+3\right )^2}+\frac{\left (1+\sqrt{5}\right ) \log \left (2^{2/5} x^2-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right ) x}{2^{4/5}}+3^{2/5}\right )}{1000\ 2^{2/5} 3^{3/5}}+\frac{\left (1-\sqrt{5}\right ) \log \left (2^{2/5} x^2-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right ) x}{2^{4/5}}+3^{2/5}\right )}{1000\ 2^{2/5} 3^{3/5}}-\frac{\log \left (\sqrt [5]{2} x+\sqrt [5]{3}\right )}{250\ 2^{2/5} 3^{3/5}}-\frac{\sqrt{5+\sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{2\ 2^{7/10} x}{\sqrt [5]{3} \sqrt{5-\sqrt{5}}}\right )}{250\ 2^{9/10} 3^{3/5}}-\frac{\sqrt{5-\sqrt{5}} \tan ^{-1}\left (\frac{2\ 2^{7/10} x}{\sqrt [5]{3} \sqrt{5+\sqrt{5}}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{250\ 2^{9/10} 3^{3/5}} \]

[Out]

-x^2/(20*(3 + 2*x^5)^2) + x^2/(150*(3 + 2*x^5)) - (Sqrt[5 + Sqrt[5]]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (2*2^(7/
10)*x)/(3^(1/5)*Sqrt[5 - Sqrt[5]])])/(250*2^(9/10)*3^(3/5)) - (Sqrt[5 - Sqrt[5]]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5
] + (2*2^(7/10)*x)/(3^(1/5)*Sqrt[5 + Sqrt[5]])])/(250*2^(9/10)*3^(3/5)) - Log[3^(1/5) + 2^(1/5)*x]/(250*2^(2/5
)*3^(3/5)) + ((1 + Sqrt[5])*Log[3^(2/5) - (3^(1/5)*(1 - Sqrt[5])*x)/2^(4/5) + 2^(2/5)*x^2])/(1000*2^(2/5)*3^(3
/5)) + ((1 - Sqrt[5])*Log[3^(2/5) - (3^(1/5)*(1 + Sqrt[5])*x)/2^(4/5) + 2^(2/5)*x^2])/(1000*2^(2/5)*3^(3/5))

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Rubi [A]  time = 0.583666, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {288, 290, 293, 634, 618, 204, 628, 31} \[ \frac{x^2}{150 \left (2 x^5+3\right )}-\frac{x^2}{20 \left (2 x^5+3\right )^2}+\frac{\left (1+\sqrt{5}\right ) \log \left (2^{2/5} x^2-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right ) x}{2^{4/5}}+3^{2/5}\right )}{1000\ 2^{2/5} 3^{3/5}}+\frac{\left (1-\sqrt{5}\right ) \log \left (2^{2/5} x^2-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right ) x}{2^{4/5}}+3^{2/5}\right )}{1000\ 2^{2/5} 3^{3/5}}-\frac{\log \left (\sqrt [5]{2} x+\sqrt [5]{3}\right )}{250\ 2^{2/5} 3^{3/5}}-\frac{\sqrt{5+\sqrt{5}} \tan ^{-1}\left (\sqrt{\frac{1}{5} \left (5+2 \sqrt{5}\right )}-\frac{2\ 2^{7/10} x}{\sqrt [5]{3} \sqrt{5-\sqrt{5}}}\right )}{250\ 2^{9/10} 3^{3/5}}-\frac{\sqrt{5-\sqrt{5}} \tan ^{-1}\left (\frac{2\ 2^{7/10} x}{\sqrt [5]{3} \sqrt{5+\sqrt{5}}}+\sqrt{\frac{1}{5} \left (5-2 \sqrt{5}\right )}\right )}{250\ 2^{9/10} 3^{3/5}} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(3 + 2*x^5)^3,x]

[Out]

-x^2/(20*(3 + 2*x^5)^2) + x^2/(150*(3 + 2*x^5)) - (Sqrt[5 + Sqrt[5]]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (2*2^(7/
10)*x)/(3^(1/5)*Sqrt[5 - Sqrt[5]])])/(250*2^(9/10)*3^(3/5)) - (Sqrt[5 - Sqrt[5]]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5
] + (2*2^(7/10)*x)/(3^(1/5)*Sqrt[5 + Sqrt[5]])])/(250*2^(9/10)*3^(3/5)) - Log[3^(1/5) + 2^(1/5)*x]/(250*2^(2/5
)*3^(3/5)) + ((1 + Sqrt[5])*Log[3^(2/5) - (3^(1/5)*(1 - Sqrt[5])*x)/2^(4/5) + 2^(2/5)*x^2])/(1000*2^(2/5)*3^(3
/5)) + ((1 - Sqrt[5])*Log[3^(2/5) - (3^(1/5)*(1 + Sqrt[5])*x)/2^(4/5) + 2^(2/5)*x^2])/(1000*2^(2/5)*3^(3/5))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 293

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x]; -(((-r)^(m + 1)*Int[1/(r + s*x), x])/(a*n*s^m)) + Dist[(2*r^(m + 1))/(a*n*s
^m), Sum[u, {k, 1, (n - 1)/2}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 1)/2, 0] && IGtQ[m, 0] && LtQ[m, n -
1] && PosQ[a/b]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{x^6}{\left (3+2 x^5\right )^3} \, dx &=-\frac{x^2}{20 \left (3+2 x^5\right )^2}+\frac{1}{10} \int \frac{x}{\left (3+2 x^5\right )^2} \, dx\\ &=-\frac{x^2}{20 \left (3+2 x^5\right )^2}+\frac{x^2}{150 \left (3+2 x^5\right )}+\frac{1}{50} \int \frac{x}{3+2 x^5} \, dx\\ &=-\frac{x^2}{20 \left (3+2 x^5\right )^2}+\frac{x^2}{150 \left (3+2 x^5\right )}-\frac{\int \frac{1}{\sqrt [5]{3}+\sqrt [5]{2} x} \, dx}{250 \sqrt [5]{2} 3^{3/5}}+\frac{\int \frac{\frac{1}{4} \sqrt [5]{3} \left (1-\sqrt{5}\right )-\frac{\left (-1-\sqrt{5}\right ) x}{2\ 2^{4/5}}}{3^{2/5}-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right ) x}{2^{4/5}}+2^{2/5} x^2} \, dx}{125 \sqrt [5]{2} 3^{3/5}}+\frac{\int \frac{\frac{1}{4} \sqrt [5]{3} \left (1+\sqrt{5}\right )-\frac{\left (-1+\sqrt{5}\right ) x}{2\ 2^{4/5}}}{3^{2/5}-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right ) x}{2^{4/5}}+2^{2/5} x^2} \, dx}{125 \sqrt [5]{2} 3^{3/5}}\\ &=-\frac{x^2}{20 \left (3+2 x^5\right )^2}+\frac{x^2}{150 \left (3+2 x^5\right )}-\frac{\log \left (\sqrt [5]{3}+\sqrt [5]{2} x\right )}{250\ 2^{2/5} 3^{3/5}}-\frac{\int \frac{1}{3^{2/5}-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right ) x}{2^{4/5}}+2^{2/5} x^2} \, dx}{100 \sqrt [5]{2} 3^{2/5} \sqrt{5}}+\frac{\int \frac{1}{3^{2/5}-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right ) x}{2^{4/5}}+2^{2/5} x^2} \, dx}{100 \sqrt [5]{2} 3^{2/5} \sqrt{5}}+\frac{\left (1-\sqrt{5}\right ) \int \frac{-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right )}{2^{4/5}}+2\ 2^{2/5} x}{3^{2/5}-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right ) x}{2^{4/5}}+2^{2/5} x^2} \, dx}{1000\ 2^{2/5} 3^{3/5}}+\frac{\left (1+\sqrt{5}\right ) \int \frac{-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right )}{2^{4/5}}+2\ 2^{2/5} x}{3^{2/5}-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right ) x}{2^{4/5}}+2^{2/5} x^2} \, dx}{1000\ 2^{2/5} 3^{3/5}}\\ &=-\frac{x^2}{20 \left (3+2 x^5\right )^2}+\frac{x^2}{150 \left (3+2 x^5\right )}-\frac{\log \left (\sqrt [5]{3}+\sqrt [5]{2} x\right )}{250\ 2^{2/5} 3^{3/5}}+\frac{\left (1-\sqrt{5}\right ) \log \left (2\ 3^{2/5}-\sqrt [5]{6} x-\sqrt{5} \sqrt [5]{6} x+2\ 2^{2/5} x^2\right )}{1000\ 2^{2/5} 3^{3/5}}+\frac{\left (1+\sqrt{5}\right ) \log \left (2\ 3^{2/5}-\sqrt [5]{6} x+\sqrt{5} \sqrt [5]{6} x+2\ 2^{2/5} x^2\right )}{1000\ 2^{2/5} 3^{3/5}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{3^{2/5} \left (5-\sqrt{5}\right )}{2^{3/5}}-x^2} \, dx,x,-\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right )}{2^{4/5}}+2\ 2^{2/5} x\right )}{50 \sqrt [5]{2} 3^{2/5} \sqrt{5}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{3^{2/5} \left (5+\sqrt{5}\right )}{2^{3/5}}-x^2} \, dx,x,-\frac{\sqrt [5]{3} \left (1-\sqrt{5}\right )}{2^{4/5}}+2\ 2^{2/5} x\right )}{50 \sqrt [5]{2} 3^{2/5} \sqrt{5}}\\ &=-\frac{x^2}{20 \left (3+2 x^5\right )^2}+\frac{x^2}{150 \left (3+2 x^5\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt [5]{3} \left (1+\sqrt{5}\right )-4 \sqrt [5]{2} x}{\sqrt [5]{3} \sqrt{2 \left (5-\sqrt{5}\right )}}\right )}{25\ 2^{9/10} 3^{3/5} \sqrt{5 \left (5-\sqrt{5}\right )}}-\frac{\tan ^{-1}\left (\frac{\sqrt [5]{3} \sqrt{3-\sqrt{5}}+2\ 2^{7/10} x}{\sqrt [5]{3} \sqrt{5+\sqrt{5}}}\right )}{25\ 2^{9/10} 3^{3/5} \sqrt{5 \left (5+\sqrt{5}\right )}}-\frac{\log \left (\sqrt [5]{3}+\sqrt [5]{2} x\right )}{250\ 2^{2/5} 3^{3/5}}+\frac{\left (1-\sqrt{5}\right ) \log \left (2\ 3^{2/5}-\sqrt [5]{6} x-\sqrt{5} \sqrt [5]{6} x+2\ 2^{2/5} x^2\right )}{1000\ 2^{2/5} 3^{3/5}}+\frac{\left (1+\sqrt{5}\right ) \log \left (2\ 3^{2/5}-\sqrt [5]{6} x+\sqrt{5} \sqrt [5]{6} x+2\ 2^{2/5} x^2\right )}{1000\ 2^{2/5} 3^{3/5}}\\ \end{align*}

Mathematica [A]  time = 0.303986, size = 293, normalized size = 0.92 \[ \frac{\frac{40 x^2}{2 x^5+3}-\frac{300 x^2}{\left (2 x^5+3\right )^2}+2^{3/5} 3^{2/5} \left (1+\sqrt{5}\right ) \log \left (2^{2/5} 3^{3/5} x^2+\left (\frac{3}{2}\right )^{4/5} \left (\sqrt{5}-1\right ) x+3\right )-2^{3/5} 3^{2/5} \left (\sqrt{5}-1\right ) \log \left (2^{2/5} 3^{3/5} x^2-\left (\frac{3}{2}\right )^{4/5} \left (1+\sqrt{5}\right ) x+3\right )-4\ 2^{3/5} 3^{2/5} \log \left (\sqrt [5]{2} 3^{4/5} x+3\right )-4 \sqrt [10]{2} 3^{2/5} \sqrt{5-\sqrt{5}} \tan ^{-1}\left (\frac{4 \sqrt [5]{2} 3^{4/5} x+3 \sqrt{5}-3}{3 \sqrt{2 \left (5+\sqrt{5}\right )}}\right )+4 \sqrt [10]{2} 3^{2/5} \sqrt{5+\sqrt{5}} \tan ^{-1}\left (\frac{4 \sqrt [5]{2} 3^{4/5} x-3 \left (1+\sqrt{5}\right )}{3 \sqrt{10-2 \sqrt{5}}}\right )}{6000} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(3 + 2*x^5)^3,x]

[Out]

((-300*x^2)/(3 + 2*x^5)^2 + (40*x^2)/(3 + 2*x^5) - 4*2^(1/10)*3^(2/5)*Sqrt[5 - Sqrt[5]]*ArcTan[(-3 + 3*Sqrt[5]
+ 4*2^(1/5)*3^(4/5)*x)/(3*Sqrt[2*(5 + Sqrt[5])])] + 4*2^(1/10)*3^(2/5)*Sqrt[5 + Sqrt[5]]*ArcTan[(-3*(1 + Sqrt
[5]) + 4*2^(1/5)*3^(4/5)*x)/(3*Sqrt[10 - 2*Sqrt[5]])] - 4*2^(3/5)*3^(2/5)*Log[3 + 2^(1/5)*3^(4/5)*x] + 2^(3/5)
*3^(2/5)*(1 + Sqrt[5])*Log[3 + (3/2)^(4/5)*(-1 + Sqrt[5])*x + 2^(2/5)*3^(3/5)*x^2] - 2^(3/5)*3^(2/5)*(-1 + Sqr
t[5])*Log[3 - (3/2)^(4/5)*(1 + Sqrt[5])*x + 2^(2/5)*3^(3/5)*x^2])/6000

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Maple [A]  time = 0.101, size = 354, normalized size = 1.1 \begin{align*} 4\,{\frac{1}{ \left ( 2\,{x}^{5}+3 \right ) ^{2}} \left ({\frac{{x}^{7}}{300}}-{\frac{3\,{x}^{2}}{400}} \right ) }+{\frac{{48}^{{\frac{2}{5}}}\ln \left ( \sqrt [5]{48}+2\,x \right ) }{ \left ( 150\,\sqrt{5}-750 \right ) \left ( 5+\sqrt{5} \right ) }}+{\frac{{48}^{{\frac{2}{5}}}\ln \left ( -x\sqrt{5}\sqrt [5]{48}+{48}^{{\frac{2}{5}}}-x\sqrt [5]{48}+4\,{x}^{2} \right ) }{12000}}-{\frac{{48}^{{\frac{2}{5}}}\ln \left ( -x\sqrt{5}\sqrt [5]{48}+{48}^{{\frac{2}{5}}}-x\sqrt [5]{48}+4\,{x}^{2} \right ) \sqrt{5}}{12000}}+{\frac{\sqrt{5}{48}^{{\frac{3}{5}}}}{1500\,\sqrt{10\,{48}^{2/5}-2\,\sqrt{5}{48}^{2/5}}}\arctan \left ( -{\frac{\sqrt{5}\sqrt [5]{48}}{\sqrt{10\,{48}^{2/5}-2\,\sqrt{5}{48}^{2/5}}}}-{\frac{\sqrt [5]{48}}{\sqrt{10\,{48}^{2/5}-2\,\sqrt{5}{48}^{2/5}}}}+8\,{\frac{x}{\sqrt{10\,{48}^{2/5}-2\,\sqrt{5}{48}^{2/5}}}} \right ) }+{\frac{{48}^{{\frac{2}{5}}}\ln \left ( x\sqrt{5}\sqrt [5]{48}-x\sqrt [5]{48}+{48}^{{\frac{2}{5}}}+4\,{x}^{2} \right ) \sqrt{5}}{12000}}+{\frac{{48}^{{\frac{2}{5}}}\ln \left ( x\sqrt{5}\sqrt [5]{48}-x\sqrt [5]{48}+{48}^{{\frac{2}{5}}}+4\,{x}^{2} \right ) }{12000}}-{\frac{\sqrt{5}{48}^{{\frac{3}{5}}}}{1500\,\sqrt{10\,{48}^{2/5}+2\,\sqrt{5}{48}^{2/5}}}\arctan \left ({\frac{\sqrt{5}\sqrt [5]{48}}{\sqrt{10\,{48}^{2/5}+2\,\sqrt{5}{48}^{2/5}}}}-{\frac{\sqrt [5]{48}}{\sqrt{10\,{48}^{2/5}+2\,\sqrt{5}{48}^{2/5}}}}+8\,{\frac{x}{\sqrt{10\,{48}^{2/5}+2\,\sqrt{5}{48}^{2/5}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(2*x^5+3)^3,x)

[Out]

4*(1/300*x^7-3/400*x^2)/(2*x^5+3)^2+1/150*48^(2/5)/(5^(1/2)-5)/(5+5^(1/2))*ln(48^(1/5)+2*x)+1/12000*48^(2/5)*l
n(-x*5^(1/2)*48^(1/5)+48^(2/5)-x*48^(1/5)+4*x^2)-1/12000*48^(2/5)*ln(-x*5^(1/2)*48^(1/5)+48^(2/5)-x*48^(1/5)+4
*x^2)*5^(1/2)+1/1500*48^(3/5)/(10*48^(2/5)-2*5^(1/2)*48^(2/5))^(1/2)*arctan(-1/(10*48^(2/5)-2*5^(1/2)*48^(2/5)
)^(1/2)*5^(1/2)*48^(1/5)-1/(10*48^(2/5)-2*5^(1/2)*48^(2/5))^(1/2)*48^(1/5)+8*x/(10*48^(2/5)-2*5^(1/2)*48^(2/5)
)^(1/2))*5^(1/2)+1/12000*48^(2/5)*ln(x*5^(1/2)*48^(1/5)-x*48^(1/5)+48^(2/5)+4*x^2)*5^(1/2)+1/12000*48^(2/5)*ln
(x*5^(1/2)*48^(1/5)-x*48^(1/5)+48^(2/5)+4*x^2)-1/1500*48^(3/5)*5^(1/2)/(10*48^(2/5)+2*5^(1/2)*48^(2/5))^(1/2)*
arctan(1/(10*48^(2/5)+2*5^(1/2)*48^(2/5))^(1/2)*5^(1/2)*48^(1/5)-1/(10*48^(2/5)+2*5^(1/2)*48^(2/5))^(1/2)*48^(
1/5)+8*x/(10*48^(2/5)+2*5^(1/2)*48^(2/5))^(1/2))

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Maxima [A]  time = 1.44039, size = 452, normalized size = 1.42 \begin{align*} \frac{3^{\frac{4}{5}} 2^{\frac{4}{5}}{\left (\sqrt{5} - 5\right )} \arctan \left (\frac{3^{\frac{4}{5}} 2^{\frac{4}{5}}{\left (4 \cdot 2^{\frac{2}{5}} x + \sqrt{5} 3^{\frac{1}{5}} 2^{\frac{1}{5}} - 3^{\frac{1}{5}} 2^{\frac{1}{5}}\right )}}{6 \, \sqrt{2 \, \sqrt{5} + 10}}\right )}{750 \,{\left (\sqrt{5} 3^{\frac{2}{5}} 2^{\frac{1}{5}} - 3^{\frac{2}{5}} 2^{\frac{1}{5}}\right )} \sqrt{2 \, \sqrt{5} + 10}} + \frac{3^{\frac{4}{5}} 2^{\frac{4}{5}}{\left (\sqrt{5} + 5\right )} \arctan \left (\frac{3^{\frac{4}{5}} 2^{\frac{4}{5}}{\left (4 \cdot 2^{\frac{2}{5}} x - \sqrt{5} 3^{\frac{1}{5}} 2^{\frac{1}{5}} - 3^{\frac{1}{5}} 2^{\frac{1}{5}}\right )}}{6 \, \sqrt{-2 \, \sqrt{5} + 10}}\right )}{750 \,{\left (\sqrt{5} 3^{\frac{2}{5}} 2^{\frac{1}{5}} + 3^{\frac{2}{5}} 2^{\frac{1}{5}}\right )} \sqrt{-2 \, \sqrt{5} + 10}} - \frac{1}{1500} \cdot 3^{\frac{2}{5}} 2^{\frac{3}{5}} \log \left (2^{\frac{1}{5}} x + 3^{\frac{1}{5}}\right ) + \frac{4 \, x^{7} - 9 \, x^{2}}{300 \,{\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} - \frac{\log \left (2 \cdot 2^{\frac{2}{5}} x^{2} - x{\left (\sqrt{5} 3^{\frac{1}{5}} 2^{\frac{1}{5}} + 3^{\frac{1}{5}} 2^{\frac{1}{5}}\right )} + 2 \cdot 3^{\frac{2}{5}}\right )}{250 \,{\left (\sqrt{5} 3^{\frac{3}{5}} 2^{\frac{2}{5}} + 3^{\frac{3}{5}} 2^{\frac{2}{5}}\right )}} + \frac{\log \left (2 \cdot 2^{\frac{2}{5}} x^{2} + x{\left (\sqrt{5} 3^{\frac{1}{5}} 2^{\frac{1}{5}} - 3^{\frac{1}{5}} 2^{\frac{1}{5}}\right )} + 2 \cdot 3^{\frac{2}{5}}\right )}{250 \,{\left (\sqrt{5} 3^{\frac{3}{5}} 2^{\frac{2}{5}} - 3^{\frac{3}{5}} 2^{\frac{2}{5}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(2*x^5+3)^3,x, algorithm="maxima")

[Out]

1/750*3^(4/5)*2^(4/5)*(sqrt(5) - 5)*arctan(1/6*3^(4/5)*2^(4/5)*(4*2^(2/5)*x + sqrt(5)*3^(1/5)*2^(1/5) - 3^(1/5
)*2^(1/5))/sqrt(2*sqrt(5) + 10))/((sqrt(5)*3^(2/5)*2^(1/5) - 3^(2/5)*2^(1/5))*sqrt(2*sqrt(5) + 10)) + 1/750*3^
(4/5)*2^(4/5)*(sqrt(5) + 5)*arctan(1/6*3^(4/5)*2^(4/5)*(4*2^(2/5)*x - sqrt(5)*3^(1/5)*2^(1/5) - 3^(1/5)*2^(1/5
))/sqrt(-2*sqrt(5) + 10))/((sqrt(5)*3^(2/5)*2^(1/5) + 3^(2/5)*2^(1/5))*sqrt(-2*sqrt(5) + 10)) - 1/1500*3^(2/5)
*2^(3/5)*log(2^(1/5)*x + 3^(1/5)) + 1/300*(4*x^7 - 9*x^2)/(4*x^10 + 12*x^5 + 9) - 1/250*log(2*2^(2/5)*x^2 - x*
(sqrt(5)*3^(1/5)*2^(1/5) + 3^(1/5)*2^(1/5)) + 2*3^(2/5))/(sqrt(5)*3^(3/5)*2^(2/5) + 3^(3/5)*2^(2/5)) + 1/250*l
og(2*2^(2/5)*x^2 + x*(sqrt(5)*3^(1/5)*2^(1/5) - 3^(1/5)*2^(1/5)) + 2*3^(2/5))/(sqrt(5)*3^(3/5)*2^(2/5) - 3^(3/
5)*2^(2/5))

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Fricas [C]  time = 145.437, size = 6294, normalized size = 19.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(2*x^5+3)^3,x, algorithm="fricas")

[Out]

1/216000*(2880*x^7 - 2*108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)*log(
-1/6912*108^(3/5)*(-1)^(2/5)*(108^(4/5)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^
(1/5))*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 1/64*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10
) + 1)^3 - 1/6912*108^(4/5)*(-1)^(1/5)*(108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 4*108
^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) + 16*108^(3/5)*(-1)^(2/5))*(sqrt(5) + I*sqrt(-2*sqrt
(5) + 10) + 1) + 1/16*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 1/4*108^(2/5)*(-1)^(3/5
)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) + 108^(2/5)*(-1)^(3/5) + 6*x) - 2*108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*
x^5 + 9)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)*log(1/384*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) +
10) + 1)^3 + x) + 8*108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9)*log(-108^(2/5)*(-1)^(3/5) + 6*x) - 6480*x^2 +
(108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) + 108^(4/5)*(-1)^(1/5)*(4*
x^10 + 12*x^5 + 9)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9) - 24
*sqrt(3)*(4*x^10 + 12*x^5 + 9)*sqrt(-1/864*108^(4/5)*(-1)^(1/5)*(108^(4/5)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqr
t(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5))*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 3/16*108^(3/5)*(-1)^(2/5)*
(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 3/16*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^
2 + 1/2*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 3*108^(3/5)*(-1)^(2/5)))*log(1/768*108^
(3/5)*(-1)^(2/5)*(108^(4/5)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5))*(sqrt
(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2 + 1/768*108^(4/5)*(-1)^(1/5)*(108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*s
qrt(5) + 10) + 1)^2 - 4*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) + 16*108^(3/5)*(-1)^(2/5)
)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 9/16*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^
2 + 9/4*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) + 1/3456*(108^(4/5)*(-1)^(1/5)*(108^(4/5)
*sqrt(3)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*sqrt(3)*(-1)^(1/5))*(sqrt(5) + I*sqr
t(-2*sqrt(5) + 10) + 1) - 432*108^(3/5)*sqrt(3)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1))*sqrt(-1/86
4*108^(4/5)*(-1)^(1/5)*(108^(4/5)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5))
*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 3/16*108^(3/5)*(-1)^(2/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2
- 3/16*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^2 + 1/2*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I
*sqrt(-2*sqrt(5) + 10) + 1) - 3*108^(3/5)*(-1)^(2/5)) + 108*x) + (108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9)*(
sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) + 108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9)*(sqrt(5) - I*sqrt(-2*sqrt(
5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5)*(4*x^10 + 12*x^5 + 9) + 24*sqrt(3)*(4*x^10 + 12*x^5 + 9)*sqrt(-1/864*10
8^(4/5)*(-1)^(1/5)*(108^(4/5)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5))*(sq
rt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 3/16*108^(3/5)*(-1)^(2/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 3
/16*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^2 + 1/2*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqr
t(-2*sqrt(5) + 10) + 1) - 3*108^(3/5)*(-1)^(2/5)))*log(1/768*108^(3/5)*(-1)^(2/5)*(108^(4/5)*(-1)^(1/5)*(sqrt(
5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5))*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2 + 1/768*
108^(4/5)*(-1)^(1/5)*(108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 4*108^(3/5)*(-1)^(2/5)*
(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) + 16*108^(3/5)*(-1)^(2/5))*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 9
/16*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1)^2 + 9/4*108^(2/5)*(-1)^(3/5)*(sqrt(5) - I*sqr
t(-2*sqrt(5) + 10) + 1) - 1/3456*(108^(4/5)*(-1)^(1/5)*(108^(4/5)*sqrt(3)*(-1)^(1/5)*(sqrt(5) - I*sqrt(-2*sqrt
(5) + 10) + 1) - 4*108^(4/5)*sqrt(3)*(-1)^(1/5))*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 432*108^(3/5)*sqrt(
3)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1))*sqrt(-1/864*108^(4/5)*(-1)^(1/5)*(108^(4/5)*(-1)^(1/5)*
(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 4*108^(4/5)*(-1)^(1/5))*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1) - 3/
16*108^(3/5)*(-1)^(2/5)*(sqrt(5) + I*sqrt(-2*sqrt(5) + 10) + 1)^2 - 3/16*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqr
t(-2*sqrt(5) + 10) + 1)^2 + 1/2*108^(3/5)*(-1)^(2/5)*(sqrt(5) - I*sqrt(-2*sqrt(5) + 10) + 1) - 3*108^(3/5)*(-1
)^(2/5)) + 108*x))/(4*x^10 + 12*x^5 + 9)

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Sympy [A]  time = 0.270478, size = 37, normalized size = 0.12 \begin{align*} \frac{4 x^{7} - 9 x^{2}}{1200 x^{10} + 3600 x^{5} + 2700} + \operatorname{RootSum}{\left (105468750000000 t^{5} + 1, \left ( t \mapsto t \log{\left (- 281250000 t^{3} + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6/(2*x**5+3)**3,x)

[Out]

(4*x**7 - 9*x**2)/(1200*x**10 + 3600*x**5 + 2700) + RootSum(105468750000000*_t**5 + 1, Lambda(_t, _t*log(-2812
50000*_t**3 + x)))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6/(2*x^5+3)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError