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

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

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Rubi [A]  time = 0.58, antiderivative size = 319, normalized size of antiderivative = 1.00, 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 31

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

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.31, size = 293, normalized size = 0.92 \[ \frac {2^{3/5} 3^{2/5} \left (1+\sqrt {5}\right ) \log \left (2^{2/5} 3^{3/5} x^2+\left (\frac {3}{2}\right )^{4/5} \left (\sqrt {5}-1\right ) x+3\right )-2^{3/5} 3^{2/5} \left (\sqrt {5}-1\right ) \log \left (2^{2/5} 3^{3/5} x^2-\left (\frac {3}{2}\right )^{4/5} \left (1+\sqrt {5}\right ) x+3\right )+\frac {40 x^2}{2 x^5+3}-\frac {300 x^2}{\left (2 x^5+3\right )^2}-4\ 2^{3/5} 3^{2/5} \log \left (\sqrt [5]{2} 3^{4/5} x+3\right )-4 \sqrt [10]{2} 3^{2/5} \sqrt {5-\sqrt {5}} \tan ^{-1}\left (\frac {4 \sqrt [5]{2} 3^{4/5} x+3 \sqrt {5}-3}{3 \sqrt {2 \left (5+\sqrt {5}\right )}}\right )+4 \sqrt [10]{2} 3^{2/5} \sqrt {5+\sqrt {5}} \tan ^{-1}\left (\frac {4 \sqrt [5]{2} 3^{4/5} x-3 \left (1+\sqrt {5}\right )}{3 \sqrt {10-2 \sqrt {5}}}\right )}{6000} \]

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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IntegrateAlgebraic [A]  time = 0.70, size = 340, normalized size = 1.07 \[ -\frac {\sqrt [5]{5 \sqrt {5}-11} \log \left (2\ 2^{2/5} 3^{3/5} x^2-\sqrt [5]{2} 3^{4/5} \sqrt {5} x-\sqrt [5]{2} 3^{4/5} x+6\right )}{500\ 6^{3/5}}+\frac {\sqrt [5]{11+5 \sqrt {5}} \log \left (2\ 2^{2/5} 3^{3/5} x^2+\sqrt [5]{2} 3^{4/5} \sqrt {5} x-\sqrt [5]{2} 3^{4/5} x+6\right )}{500\ 6^{3/5}}+\frac {\left (4 x^5-9\right ) x^2}{300 \left (2 x^5+3\right )^2}-\frac {\log \left (\sqrt [5]{2} 3^{4/5} x+3\right )}{250\ 2^{2/5} 3^{3/5}}-\frac {\sqrt [10]{25-11 \sqrt {5}} \tan ^{-1}\left (\frac {2 \sqrt [10]{50-22 \sqrt {5}} x}{\sqrt [5]{3} 5^{3/10}}+\sqrt {1-\frac {2}{\sqrt {5}}}\right )}{50 \sqrt {2} 3^{3/5} 5^{4/5}}-\frac {\sqrt [10]{25+11 \sqrt {5}} \tan ^{-1}\left (\sqrt {1+\frac {2}{\sqrt {5}}}-\frac {2 \sqrt [10]{50+22 \sqrt {5}} x}{\sqrt [5]{3} 5^{3/10}}\right )}{50 \sqrt {2} 3^{3/5} 5^{4/5}} \]

Warning: Unable to verify antiderivative.

[In]

IntegrateAlgebraic[x^6/(3 + 2*x^5)^3,x]

[Out]

(x^2*(-9 + 4*x^5))/(300*(3 + 2*x^5)^2) - ((25 - 11*Sqrt[5])^(1/10)*ArcTan[Sqrt[1 - 2/Sqrt[5]] + (2*(50 - 22*Sq
rt[5])^(1/10)*x)/(3^(1/5)*5^(3/10))])/(50*Sqrt[2]*3^(3/5)*5^(4/5)) - ((25 + 11*Sqrt[5])^(1/10)*ArcTan[Sqrt[1 +
2/Sqrt[5]] - (2*(50 + 22*Sqrt[5])^(1/10)*x)/(3^(1/5)*5^(3/10))])/(50*Sqrt[2]*3^(3/5)*5^(4/5)) - Log[3 + 2^(1/
5)*3^(4/5)*x]/(250*2^(2/5)*3^(3/5)) - ((-11 + 5*Sqrt[5])^(1/5)*Log[6 - 2^(1/5)*3^(4/5)*x - 2^(1/5)*3^(4/5)*Sqr
t[5]*x + 2*2^(2/5)*3^(3/5)*x^2])/(500*6^(3/5)) + ((11 + 5*Sqrt[5])^(1/5)*Log[6 - 2^(1/5)*3^(4/5)*x + 2^(1/5)*3
^(4/5)*Sqrt[5]*x + 2*2^(2/5)*3^(3/5)*x^2])/(500*6^(3/5))

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fricas [C]  time = 41.32, size = 1751, normalized size = 5.49 \[ \text {result too large to display} \]

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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giac [A]  time = 1.83, size = 250, normalized size = 0.78 \[ -\frac {1}{3000} \, {\left (\sqrt {5} \left (\frac {3}{2}\right )^{\frac {2}{5}} \sqrt {2 \, \sqrt {5} + 10} - \left (\frac {3}{2}\right )^{\frac {2}{5}} \sqrt {2 \, \sqrt {5} + 10}\right )} \arctan \left (\frac {2 \, \left (\frac {3}{2}\right )^{\frac {4}{5}} {\left (\left (\frac {3}{2}\right )^{\frac {1}{5}} {\left (\sqrt {5} - 1\right )} + 4 \, x\right )}}{3 \, \sqrt {2 \, \sqrt {5} + 10}}\right ) + \frac {1}{3000} \, {\left (\sqrt {5} \left (\frac {3}{2}\right )^{\frac {2}{5}} \sqrt {-2 \, \sqrt {5} + 10} + \left (\frac {3}{2}\right )^{\frac {2}{5}} \sqrt {-2 \, \sqrt {5} + 10}\right )} \arctan \left (-\frac {2 \, \left (\frac {3}{2}\right )^{\frac {4}{5}} {\left (\left (\frac {3}{2}\right )^{\frac {1}{5}} {\left (\sqrt {5} + 1\right )} - 4 \, x\right )}}{3 \, \sqrt {-2 \, \sqrt {5} + 10}}\right ) - \frac {1}{6000} \, {\left (\left (\frac {3}{2}\right )^{\frac {2}{5}} {\left (\sqrt {5} - 5\right )} + \sqrt {5} \left (\frac {3}{2}\right )^{\frac {2}{5}} + 3 \, \left (\frac {3}{2}\right )^{\frac {2}{5}}\right )} \log \left (x^{2} - \frac {1}{2} \, x {\left (\sqrt {5} \left (\frac {3}{2}\right )^{\frac {1}{5}} + \left (\frac {3}{2}\right )^{\frac {1}{5}}\right )} + \left (\frac {3}{2}\right )^{\frac {2}{5}}\right ) + \frac {1}{6000} \, {\left (\left (\frac {3}{2}\right )^{\frac {2}{5}} {\left (\sqrt {5} + 5\right )} + \sqrt {5} \left (\frac {3}{2}\right )^{\frac {2}{5}} - 3 \, \left (\frac {3}{2}\right )^{\frac {2}{5}}\right )} \log \left (x^{2} + \frac {1}{2} \, x {\left (\sqrt {5} \left (\frac {3}{2}\right )^{\frac {1}{5}} - \left (\frac {3}{2}\right )^{\frac {1}{5}}\right )} + \left (\frac {3}{2}\right )^{\frac {2}{5}}\right ) - \frac {1}{750} \, \left (\frac {3}{2}\right )^{\frac {2}{5}} \log \left ({\left | x + \left (\frac {3}{2}\right )^{\frac {1}{5}} \right |}\right ) + \frac {4 \, x^{7} - 9 \, x^{2}}{300 \, {\left (2 \, x^{5} + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3000*(sqrt(5)*(3/2)^(2/5)*sqrt(2*sqrt(5) + 10) - (3/2)^(2/5)*sqrt(2*sqrt(5) + 10))*arctan(2/3*(3/2)^(4/5)*(
(3/2)^(1/5)*(sqrt(5) - 1) + 4*x)/sqrt(2*sqrt(5) + 10)) + 1/3000*(sqrt(5)*(3/2)^(2/5)*sqrt(-2*sqrt(5) + 10) + (
3/2)^(2/5)*sqrt(-2*sqrt(5) + 10))*arctan(-2/3*(3/2)^(4/5)*((3/2)^(1/5)*(sqrt(5) + 1) - 4*x)/sqrt(-2*sqrt(5) +
10)) - 1/6000*((3/2)^(2/5)*(sqrt(5) - 5) + sqrt(5)*(3/2)^(2/5) + 3*(3/2)^(2/5))*log(x^2 - 1/2*x*(sqrt(5)*(3/2)
^(1/5) + (3/2)^(1/5)) + (3/2)^(2/5)) + 1/6000*((3/2)^(2/5)*(sqrt(5) + 5) + sqrt(5)*(3/2)^(2/5) - 3*(3/2)^(2/5)
)*log(x^2 + 1/2*x*(sqrt(5)*(3/2)^(1/5) - (3/2)^(1/5)) + (3/2)^(2/5)) - 1/750*(3/2)^(2/5)*log(abs(x + (3/2)^(1/
5))) + 1/300*(4*x^7 - 9*x^2)/(2*x^5 + 3)^2

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maple [C]  time = 0.27, size = 47, normalized size = 0.15

method	result	size
risch	\(\frac {\frac {1}{75} x^{7}-\frac {3}{100} x^{2}}{\left (2 x^{5}+3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (2 \textit {\_Z}^{5}+3\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{500}\)	\(47\)
meijerg	\(\frac {108^{\frac {4}{5}} \left (-\frac {x^{2} 2^{\frac {2}{5}} 3^{\frac {3}{5}} \left (-\frac {28 x^{5}}{3}+21\right )}{105 \left (1+\frac {2 x^{5}}{3}\right )^{2}}+\frac {2 \,108^{\frac {1}{5}} x^{2} \left (-\frac {2^{\frac {3}{5}} 3^{\frac {2}{5}} \ln \left (1+\frac {2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}{3}\right )}{2 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {2^{\frac {3}{5}} 3^{\frac {2}{5}} \cos \left (\frac {2 \pi }{5}\right ) \ln \left (1-\frac {2 \cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}{3}+\frac {2^{\frac {2}{5}} 3^{\frac {3}{5}} \left (x^{5}\right )^{\frac {2}{5}}}{3}\right )}{2 \left (x^{5}\right )^{\frac {2}{5}}}+\frac {72^{\frac {1}{5}} \arctan \left (\frac {\sin \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}{3-\cos \left (\frac {\pi }{5}\right ) 2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}\right ) \sin \left (\frac {2 \pi }{5}\right )}{\left (x^{5}\right )^{\frac {2}{5}}}+\frac {2^{\frac {3}{5}} 3^{\frac {2}{5}} \cos \left (\frac {\pi }{5}\right ) \ln \left (1+\frac {2 \cos \left (\frac {2 \pi }{5}\right ) 2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}{3}+\frac {2^{\frac {2}{5}} 3^{\frac {3}{5}} \left (x^{5}\right )^{\frac {2}{5}}}{3}\right )}{2 \left (x^{5}\right )^{\frac {2}{5}}}-\frac {72^{\frac {1}{5}} \arctan \left (\frac {\sin \left (\frac {2 \pi }{5}\right ) 2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}{3+\cos \left (\frac {2 \pi }{5}\right ) 2^{\frac {1}{5}} 3^{\frac {4}{5}} \left (x^{5}\right )^{\frac {1}{5}}}\right ) \sin \left (\frac {\pi }{5}\right )}{\left (x^{5}\right )^{\frac {2}{5}}}\right )}{25}\right )}{6480}\)	\(277\)
default	\(\frac {\frac {1}{75} x^{7}-\frac {3}{100} x^{2}}{\left (2 x^{5}+3\right )^{2}}+\frac {48^{\frac {2}{5}} \ln \left (48^{\frac {1}{5}}+2 x \right )}{150 \left (5+\sqrt {5}\right ) \left (\sqrt {5}-5\right )}+\frac {48^{\frac {2}{5}} \ln \left (x \sqrt {5}\, 48^{\frac {1}{5}}-x 48^{\frac {1}{5}}+48^{\frac {2}{5}}+4 x^{2}\right ) \sqrt {5}}{12000}+\frac {48^{\frac {2}{5}} \ln \left (x \sqrt {5}\, 48^{\frac {1}{5}}-x 48^{\frac {1}{5}}+48^{\frac {2}{5}}+4 x^{2}\right )}{12000}-\frac {48^{\frac {3}{5}} \sqrt {5}\, \arctan \left (\frac {\sqrt {5}\, 48^{\frac {1}{5}}}{\sqrt {10 \,48^{\frac {2}{5}}+2 \sqrt {5}\, 48^{\frac {2}{5}}}}-\frac {48^{\frac {1}{5}}}{\sqrt {10 \,48^{\frac {2}{5}}+2 \sqrt {5}\, 48^{\frac {2}{5}}}}+\frac {8 x}{\sqrt {10 \,48^{\frac {2}{5}}+2 \sqrt {5}\, 48^{\frac {2}{5}}}}\right )}{1500 \sqrt {10 \,48^{\frac {2}{5}}+2 \sqrt {5}\, 48^{\frac {2}{5}}}}-\frac {48^{\frac {2}{5}} \ln \left (-x \sqrt {5}\, 48^{\frac {1}{5}}+48^{\frac {2}{5}}-x 48^{\frac {1}{5}}+4 x^{2}\right ) \sqrt {5}}{12000}+\frac {48^{\frac {2}{5}} \ln \left (-x \sqrt {5}\, 48^{\frac {1}{5}}+48^{\frac {2}{5}}-x 48^{\frac {1}{5}}+4 x^{2}\right )}{12000}+\frac {48^{\frac {3}{5}} \arctan \left (-\frac {\sqrt {5}\, 48^{\frac {1}{5}}}{\sqrt {10 \,48^{\frac {2}{5}}-2 \sqrt {5}\, 48^{\frac {2}{5}}}}-\frac {48^{\frac {1}{5}}}{\sqrt {10 \,48^{\frac {2}{5}}-2 \sqrt {5}\, 48^{\frac {2}{5}}}}+\frac {8 x}{\sqrt {10 \,48^{\frac {2}{5}}-2 \sqrt {5}\, 48^{\frac {2}{5}}}}\right ) \sqrt {5}}{1500 \sqrt {10 \,48^{\frac {2}{5}}-2 \sqrt {5}\, 48^{\frac {2}{5}}}}\)	\(354\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(1/300*x^7-3/400*x^2)/(2*x^5+3)^2+1/500*sum(1/_R^3*ln(x-_R),_R=RootOf(2*_Z^5+3))

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maxima [A]  time = 1.21, size = 335, normalized size = 1.05 \[ \frac {3^{\frac {4}{5}} 2^{\frac {4}{5}} {\left (\sqrt {5} - 5\right )} \arctan \left (\frac {3^{\frac {4}{5}} 2^{\frac {4}{5}} {\left (4 \cdot 2^{\frac {2}{5}} x + \sqrt {5} 3^{\frac {1}{5}} 2^{\frac {1}{5}} - 3^{\frac {1}{5}} 2^{\frac {1}{5}}\right )}}{6 \, \sqrt {2 \, \sqrt {5} + 10}}\right )}{750 \, {\left (\sqrt {5} 3^{\frac {2}{5}} 2^{\frac {1}{5}} - 3^{\frac {2}{5}} 2^{\frac {1}{5}}\right )} \sqrt {2 \, \sqrt {5} + 10}} + \frac {3^{\frac {4}{5}} 2^{\frac {4}{5}} {\left (\sqrt {5} + 5\right )} \arctan \left (\frac {3^{\frac {4}{5}} 2^{\frac {4}{5}} {\left (4 \cdot 2^{\frac {2}{5}} x - \sqrt {5} 3^{\frac {1}{5}} 2^{\frac {1}{5}} - 3^{\frac {1}{5}} 2^{\frac {1}{5}}\right )}}{6 \, \sqrt {-2 \, \sqrt {5} + 10}}\right )}{750 \, {\left (\sqrt {5} 3^{\frac {2}{5}} 2^{\frac {1}{5}} + 3^{\frac {2}{5}} 2^{\frac {1}{5}}\right )} \sqrt {-2 \, \sqrt {5} + 10}} - \frac {1}{1500} \cdot 3^{\frac {2}{5}} 2^{\frac {3}{5}} \log \left (2^{\frac {1}{5}} x + 3^{\frac {1}{5}}\right ) + \frac {4 \, x^{7} - 9 \, x^{2}}{300 \, {\left (4 \, x^{10} + 12 \, x^{5} + 9\right )}} - \frac {\log \left (2 \cdot 2^{\frac {2}{5}} x^{2} - x {\left (\sqrt {5} 3^{\frac {1}{5}} 2^{\frac {1}{5}} + 3^{\frac {1}{5}} 2^{\frac {1}{5}}\right )} + 2 \cdot 3^{\frac {2}{5}}\right )}{250 \, {\left (\sqrt {5} 3^{\frac {3}{5}} 2^{\frac {2}{5}} + 3^{\frac {3}{5}} 2^{\frac {2}{5}}\right )}} + \frac {\log \left (2 \cdot 2^{\frac {2}{5}} x^{2} + x {\left (\sqrt {5} 3^{\frac {1}{5}} 2^{\frac {1}{5}} - 3^{\frac {1}{5}} 2^{\frac {1}{5}}\right )} + 2 \cdot 3^{\frac {2}{5}}\right )}{250 \, {\left (\sqrt {5} 3^{\frac {3}{5}} 2^{\frac {2}{5}} - 3^{\frac {3}{5}} 2^{\frac {2}{5}}\right )}} \]

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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mupad [B]  time = 1.60, size = 285, normalized size = 0.89 \[ \frac {3^{2/5}\,\ln \left (x-\frac {3^{1/5}\,{\left (2\,2^{1/10}\,\sqrt {-\sqrt {5}-5}-2^{3/5}\,\left (\sqrt {5}-1\right )\right )}^3}{256}\right )\,\left (2\,2^{1/10}\,\sqrt {-\sqrt {5}-5}-2^{3/5}\,\left (\sqrt {5}-1\right )\right )}{6000}-\frac {\frac {3\,x^2}{400}-\frac {x^7}{300}}{x^{10}+3\,x^5+\frac {9}{4}}-\frac {3^{2/5}\,\ln \left (x+\frac {3^{1/5}\,{\left (2\,2^{1/10}\,\sqrt {-\sqrt {5}-5}+2^{3/5}\,\left (\sqrt {5}-1\right )\right )}^3}{256}\right )\,\left (2\,2^{1/10}\,\sqrt {-\sqrt {5}-5}+2^{3/5}\,\left (\sqrt {5}-1\right )\right )}{6000}-\frac {{72}^{1/5}\,\ln \left (x+\frac {{72}^{3/5}}{12}\right )}{1500}+\frac {3^{2/5}\,\ln \left (x-\frac {3^{1/5}\,{\left (2^{3/5}\,\left (\sqrt {5}+1\right )-2\,2^{1/10}\,\sqrt {\sqrt {5}-5}\right )}^3}{256}\right )\,\left (2^{3/5}\,\left (\sqrt {5}+1\right )-2\,2^{1/10}\,\sqrt {\sqrt {5}-5}\right )}{6000}+\frac {3^{2/5}\,\ln \left (x-\frac {3^{1/5}\,{\left (2^{3/5}\,\left (\sqrt {5}+1\right )+2\,2^{1/10}\,\sqrt {\sqrt {5}-5}\right )}^3}{256}\right )\,\left (2^{3/5}\,\left (\sqrt {5}+1\right )+2\,2^{1/10}\,\sqrt {\sqrt {5}-5}\right )}{6000} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(2*x^5 + 3)^3,x)

[Out]

(3^(2/5)*log(x - (3^(1/5)*(2*2^(1/10)*(- 5^(1/2) - 5)^(1/2) - 2^(3/5)*(5^(1/2) - 1))^3)/256)*(2*2^(1/10)*(- 5^
(1/2) - 5)^(1/2) - 2^(3/5)*(5^(1/2) - 1)))/6000 - ((3*x^2)/400 - x^7/300)/(3*x^5 + x^10 + 9/4) - (3^(2/5)*log(
x + (3^(1/5)*(2*2^(1/10)*(- 5^(1/2) - 5)^(1/2) + 2^(3/5)*(5^(1/2) - 1))^3)/256)*(2*2^(1/10)*(- 5^(1/2) - 5)^(1
/2) + 2^(3/5)*(5^(1/2) - 1)))/6000 - (72^(1/5)*log(x + 72^(3/5)/12))/1500 + (3^(2/5)*log(x - (3^(1/5)*(2^(3/5)
*(5^(1/2) + 1) - 2*2^(1/10)*(5^(1/2) - 5)^(1/2))^3)/256)*(2^(3/5)*(5^(1/2) + 1) - 2*2^(1/10)*(5^(1/2) - 5)^(1/
2)))/6000 + (3^(2/5)*log(x - (3^(1/5)*(2^(3/5)*(5^(1/2) + 1) + 2*2^(1/10)*(5^(1/2) - 5)^(1/2))^3)/256)*(2^(3/5
)*(5^(1/2) + 1) + 2*2^(1/10)*(5^(1/2) - 5)^(1/2)))/6000

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sympy [A]  time = 0.30, size = 37, normalized size = 0.12 \[ \frac {4 x^{7} - 9 x^{2}}{1200 x^{10} + 3600 x^{5} + 2700} + \operatorname {RootSum} {\left (105468750000000 t^{5} + 1, \left (t \mapsto t \log {\left (- 281250000 t^{3} + x \right )} \right )\right )} \]

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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