\(\int \frac {(-b+x^3) (b+x^3) (-c+x^3)}{\sqrt [3]{a x^2+x^3}} \, dx\) [2927]

Optimal result

Integrand size = 33, antiderivative size = 339 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {\left (a x^2+x^3\right )^{2/3} \left (38038000 a^8-77157360 a^2 b^2+49533120 a^5 c-28528500 a^7 x+57868020 a b^2 x-37149840 a^4 c x+24453000 a^6 x^2-49601160 b^2 x^2+31842720 a^3 c x^2-22007700 a^5 x^3-28658448 a^2 c x^3+20314800 a^4 x^4+26453952 a c x^4-19045125 a^3 x^5-24800580 c x^5+18042750 a^2 x^6-17222625 a x^7+16533720 x^8\right )}{148803480 x}+\frac {\left (-135850 \sqrt {3} a^9+275562 \sqrt {3} a^3 b^2-176904 \sqrt {3} a^6 c+1594323 \sqrt {3} b^2 c\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a x^2+x^3}}\right )}{1594323}+\frac {\left (135850 a^9-275562 a^3 b^2+176904 a^6 c-1594323 b^2 c\right ) \log \left (-x+\sqrt [3]{a x^2+x^3}\right )}{1594323}+\frac {\left (-135850 a^9+275562 a^3 b^2-176904 a^6 c+1594323 b^2 c\right ) \log \left (x^2+x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}\right )}{3188646} \]

[Out]

1/148803480*(a*x^2+x^3)^(2/3)*(38038000*a^8-28528500*a^7*x+24453000*a^6*x^2-22007700*a^5*x^3+20314800*a^4*x^4-
19045125*a^3*x^5+18042750*a^2*x^6-17222625*a*x^7+16533720*x^8+49533120*a^5*c-37149840*a^4*c*x+31842720*a^3*c*x
^2-28658448*a^2*c*x^3+26453952*a*c*x^4-24800580*c*x^5-77157360*a^2*b^2+57868020*a*b^2*x-49601160*b^2*x^2)/x+1/
1594323*(-135850*3^(1/2)*a^9+275562*3^(1/2)*a^3*b^2-176904*3^(1/2)*a^6*c+1594323*3^(1/2)*b^2*c)*arctan(3^(1/2)
*x/(x+2*(a*x^2+x^3)^(1/3)))+1/1594323*(135850*a^9+176904*a^6*c-275562*a^3*b^2-1594323*b^2*c)*ln(-x+(a*x^2+x^3)
^(1/3))+1/3188646*(-135850*a^9-176904*a^6*c+275562*a^3*b^2+1594323*b^2*c)*ln(x^2+x*(a*x^2+x^3)^(1/3)+(a*x^2+x^
3)^(2/3))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(993\) vs. \(2(339)=678\).

Time = 0.68 (sec) , antiderivative size = 993, normalized size of antiderivative = 2.93, number of steps used = 28, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2078, 2036, 61, 2049} \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {135850 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) a^9}{531441 \sqrt {3} \sqrt [3]{x^3+a x^2}}+\frac {67925 x^{2/3} \sqrt [3]{a+x} \log (x) a^9}{1594323 \sqrt [3]{x^3+a x^2}}+\frac {67925 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right ) a^9}{531441 \sqrt [3]{x^3+a x^2}}+\frac {135850 \left (x^3+a x^2\right )^{2/3} a^8}{531441 x}-\frac {67925 \left (x^3+a x^2\right )^{2/3} a^7}{354294}+\frac {728 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) a^6}{2187 \sqrt {3} \sqrt [3]{x^3+a x^2}}+\frac {364 c x^{2/3} \sqrt [3]{a+x} \log (x) a^6}{6561 \sqrt [3]{x^3+a x^2}}+\frac {364 c x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right ) a^6}{2187 \sqrt [3]{x^3+a x^2}}+\frac {67925 x \left (x^3+a x^2\right )^{2/3} a^6}{413343}-\frac {13585 x^2 \left (x^3+a x^2\right )^{2/3} a^5}{91854}+\frac {728 c \left (x^3+a x^2\right )^{2/3} a^5}{2187 x}+\frac {2090 x^3 \left (x^3+a x^2\right )^{2/3} a^4}{15309}-\frac {182}{729} c \left (x^3+a x^2\right )^{2/3} a^4-\frac {14 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right ) a^3}{27 \sqrt {3} \sqrt [3]{x^3+a x^2}}-\frac {7 b^2 x^{2/3} \sqrt [3]{a+x} \log (x) a^3}{81 \sqrt [3]{x^3+a x^2}}-\frac {7 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right ) a^3}{27 \sqrt [3]{x^3+a x^2}}-\frac {5225 x^4 \left (x^3+a x^2\right )^{2/3} a^3}{40824}+\frac {52}{243} c x \left (x^3+a x^2\right )^{2/3} a^3+\frac {275 x^5 \left (x^3+a x^2\right )^{2/3} a^2}{2268}-\frac {26}{135} c x^2 \left (x^3+a x^2\right )^{2/3} a^2-\frac {14 b^2 \left (x^3+a x^2\right )^{2/3} a^2}{27 x}-\frac {25}{216} x^6 \left (x^3+a x^2\right )^{2/3} a+\frac {8}{45} c x^3 \left (x^3+a x^2\right )^{2/3} a+\frac {7}{18} b^2 \left (x^3+a x^2\right )^{2/3} a-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{x^3+a x^2}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{x^3+a x^2}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{x^3+a x^2}}+\frac {1}{9} x^7 \left (x^3+a x^2\right )^{2/3}-\frac {1}{6} c x^4 \left (x^3+a x^2\right )^{2/3}-\frac {1}{3} b^2 x \left (x^3+a x^2\right )^{2/3} \]

[In]

Int[((-b + x^3)*(b + x^3)*(-c + x^3))/(a*x^2 + x^3)^(1/3),x]

[Out]

(-67925*a^7*(a*x^2 + x^3)^(2/3))/354294 + (7*a*b^2*(a*x^2 + x^3)^(2/3))/18 - (182*a^4*c*(a*x^2 + x^3)^(2/3))/7
29 + (135850*a^8*(a*x^2 + x^3)^(2/3))/(531441*x) - (14*a^2*b^2*(a*x^2 + x^3)^(2/3))/(27*x) + (728*a^5*c*(a*x^2
+ x^3)^(2/3))/(2187*x) + (67925*a^6*x*(a*x^2 + x^3)^(2/3))/413343 - (b^2*x*(a*x^2 + x^3)^(2/3))/3 + (52*a^3*c
*x*(a*x^2 + x^3)^(2/3))/243 - (13585*a^5*x^2*(a*x^2 + x^3)^(2/3))/91854 - (26*a^2*c*x^2*(a*x^2 + x^3)^(2/3))/1
35 + (2090*a^4*x^3*(a*x^2 + x^3)^(2/3))/15309 + (8*a*c*x^3*(a*x^2 + x^3)^(2/3))/45 - (5225*a^3*x^4*(a*x^2 + x^
3)^(2/3))/40824 - (c*x^4*(a*x^2 + x^3)^(2/3))/6 + (275*a^2*x^5*(a*x^2 + x^3)^(2/3))/2268 - (25*a*x^6*(a*x^2 +
x^3)^(2/3))/216 + (x^7*(a*x^2 + x^3)^(2/3))/9 + (135850*a^9*x^(2/3)*(a + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x
)^(1/3))/(Sqrt[3]*x^(1/3))])/(531441*Sqrt[3]*(a*x^2 + x^3)^(1/3)) - (14*a^3*b^2*x^(2/3)*(a + x)^(1/3)*ArcTan[1
/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(27*Sqrt[3]*(a*x^2 + x^3)^(1/3)) + (728*a^6*c*x^(2/3)*(a + x)
^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(2187*Sqrt[3]*(a*x^2 + x^3)^(1/3)) - (Sqrt[3]*
b^2*c*x^(2/3)*(a + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(a*x^2 + x^3)^(1/3) + (67
925*a^9*x^(2/3)*(a + x)^(1/3)*Log[x])/(1594323*(a*x^2 + x^3)^(1/3)) - (7*a^3*b^2*x^(2/3)*(a + x)^(1/3)*Log[x])
/(81*(a*x^2 + x^3)^(1/3)) + (364*a^6*c*x^(2/3)*(a + x)^(1/3)*Log[x])/(6561*(a*x^2 + x^3)^(1/3)) - (b^2*c*x^(2/
3)*(a + x)^(1/3)*Log[x])/(2*(a*x^2 + x^3)^(1/3)) + (67925*a^9*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(
1/3)])/(531441*(a*x^2 + x^3)^(1/3)) - (7*a^3*b^2*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)])/(27*(a
*x^2 + x^3)^(1/3)) + (364*a^6*c*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)])/(2187*(a*x^2 + x^3)^(1/
3)) - (3*b^2*c*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)])/(2*(a*x^2 + x^3)^(1/3))

Rule 61

Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[d/b, 3]}, Simp[(-Sqrt
[3])*(q/d)*ArcTan[2*q*((a + b*x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3))) + 1/Sqrt[3]], x] + (-Simp[3*(q/(2*d))*Log[q*
((a + b*x)^(1/3)/(c + d*x)^(1/3)) - 1], x] - Simp[(q/(2*d))*Log[c + d*x], x])] /; FreeQ[{a, b, c, d}, x] && Ne
Q[b*c - a*d, 0] && PosQ[d/b]

Rule 2036

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[(a*x^j + b*x^n)^FracPart[p]/(x^(j*FracPart[p
])*(a + b*x^(n - j))^FracPart[p]), Int[x^(j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, j, n, p}, x] &&  !I
ntegerQ[p] && NeQ[n, j] && PosQ[n - j]

Rule 2049

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

Rule 2078

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[Pq*(a*x^j + b*x^n)^p, x]
, x] /; FreeQ[{a, b, j, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IntegerQ[p] && NeQ[n, j]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^2 c}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 x^3}{\sqrt [3]{a x^2+x^3}}-\frac {c x^6}{\sqrt [3]{a x^2+x^3}}+\frac {x^9}{\sqrt [3]{a x^2+x^3}}\right ) \, dx \\ & = -\left (b^2 \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx\right )-c \int \frac {x^6}{\sqrt [3]{a x^2+x^3}} \, dx+\left (b^2 c\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx+\int \frac {x^9}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = -\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {1}{27} (25 a) \int \frac {x^8}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {1}{9} \left (7 a b^2\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {1}{9} (8 a c) \int \frac {x^5}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {\left (b^2 c x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{\sqrt [3]{a x^2+x^3}} \\ & = \frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {1}{324} \left (275 a^2\right ) \int \frac {x^7}{\sqrt [3]{a x^2+x^3}} \, dx-\frac {1}{27} \left (14 a^2 b^2\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx-\frac {1}{135} \left (104 a^2 c\right ) \int \frac {x^4}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = \frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (5225 a^3\right ) \int \frac {x^6}{\sqrt [3]{a x^2+x^3}} \, dx}{6804}+\frac {1}{81} \left (14 a^3 b^2\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {1}{81} \left (52 a^3 c\right ) \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx \\ & = \frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (10450 a^4\right ) \int \frac {x^5}{\sqrt [3]{a x^2+x^3}} \, dx}{15309}-\frac {1}{729} \left (364 a^4 c\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx+\frac {\left (14 a^3 b^2 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{81 \sqrt [3]{a x^2+x^3}} \\ & = \frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (27170 a^5\right ) \int \frac {x^4}{\sqrt [3]{a x^2+x^3}} \, dx}{45927}+\frac {\left (728 a^5 c\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx}{2187} \\ & = \frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (67925 a^6\right ) \int \frac {x^3}{\sqrt [3]{a x^2+x^3}} \, dx}{137781}-\frac {\left (728 a^6 c\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx}{6561} \\ & = \frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (67925 a^7\right ) \int \frac {x^2}{\sqrt [3]{a x^2+x^3}} \, dx}{177147}-\frac {\left (728 a^6 c x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{6561 \sqrt [3]{a x^2+x^3}} \\ & = -\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}+\frac {\left (135850 a^8\right ) \int \frac {x}{\sqrt [3]{a x^2+x^3}} \, dx}{531441} \\ & = -\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^8 \left (a x^2+x^3\right )^{2/3}}{531441 x}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (135850 a^9\right ) \int \frac {1}{\sqrt [3]{a x^2+x^3}} \, dx}{1594323} \\ & = -\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^8 \left (a x^2+x^3\right )^{2/3}}{531441 x}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {\left (135850 a^9 x^{2/3} \sqrt [3]{a+x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{a+x}} \, dx}{1594323 \sqrt [3]{a x^2+x^3}} \\ & = -\frac {67925 a^7 \left (a x^2+x^3\right )^{2/3}}{354294}+\frac {7}{18} a b^2 \left (a x^2+x^3\right )^{2/3}-\frac {182}{729} a^4 c \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^8 \left (a x^2+x^3\right )^{2/3}}{531441 x}-\frac {14 a^2 b^2 \left (a x^2+x^3\right )^{2/3}}{27 x}+\frac {728 a^5 c \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {67925 a^6 x \left (a x^2+x^3\right )^{2/3}}{413343}-\frac {1}{3} b^2 x \left (a x^2+x^3\right )^{2/3}+\frac {52}{243} a^3 c x \left (a x^2+x^3\right )^{2/3}-\frac {13585 a^5 x^2 \left (a x^2+x^3\right )^{2/3}}{91854}-\frac {26}{135} a^2 c x^2 \left (a x^2+x^3\right )^{2/3}+\frac {2090 a^4 x^3 \left (a x^2+x^3\right )^{2/3}}{15309}+\frac {8}{45} a c x^3 \left (a x^2+x^3\right )^{2/3}-\frac {5225 a^3 x^4 \left (a x^2+x^3\right )^{2/3}}{40824}-\frac {1}{6} c x^4 \left (a x^2+x^3\right )^{2/3}+\frac {275 a^2 x^5 \left (a x^2+x^3\right )^{2/3}}{2268}-\frac {25}{216} a x^6 \left (a x^2+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (a x^2+x^3\right )^{2/3}+\frac {135850 a^9 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{531441 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {14 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{27 \sqrt {3} \sqrt [3]{a x^2+x^3}}+\frac {728 a^6 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {\sqrt {3} b^2 c x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {67925 a^9 x^{2/3} \sqrt [3]{a+x} \log (x)}{1594323 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{81 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {b^2 c x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {67925 a^9 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{531441 \sqrt [3]{a x^2+x^3}}-\frac {7 a^3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{27 \sqrt [3]{a x^2+x^3}}+\frac {364 a^6 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {3 b^2 c x^{2/3} \sqrt [3]{a+x} \log \left (-1+\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}\right )}{2 \sqrt [3]{a x^2+x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.81 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.18 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {x^{2/3} \left (3 \sqrt [3]{x} (a+x) \left (38038000 a^8-28528500 a^7 x+24453000 a^6 x^2+21060 a^5 \left (2352 c-1045 x^3\right )+19440 a^4 \left (-1911 c x+1045 x^4\right )+3645 a^3 \left (8736 c x^2-5225 x^5\right )-13122 a^2 \left (5880 b^2+2184 c x^3-1375 x^6\right )+137781 a \left (420 b^2 x+192 c x^4-125 x^7\right )-8266860 \left (6 b^2 x^2+3 c x^5-2 x^8\right )\right )+280 \sqrt {3} \left (182468 a^9-91854 a^3 b^2+58968 a^6 c+1594323 b^2 c\right ) \sqrt [3]{a+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{a+x}}\right )+11760 \sqrt {3} a^3 \left (7579 a^6-8748 b^2+5616 a^3 c\right ) \sqrt [3]{a+x} \arctan \left (\frac {1+\frac {2 \sqrt [3]{a+x}}{\sqrt [3]{x}}}{\sqrt {3}}\right )+280 \left (135850 a^9-275562 a^3 b^2+176904 a^6 c-1594323 b^2 c\right ) \sqrt [3]{a+x} \log \left (-\sqrt [3]{x}+\sqrt [3]{a+x}\right )+140 \left (-135850 a^9+275562 a^3 b^2-176904 a^6 c+1594323 b^2 c\right ) \sqrt [3]{a+x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{a+x}+(a+x)^{2/3}\right )\right )}{446410440 \sqrt [3]{x^2 (a+x)}} \]

[In]

Integrate[((-b + x^3)*(b + x^3)*(-c + x^3))/(a*x^2 + x^3)^(1/3),x]

[Out]

(x^(2/3)*(3*x^(1/3)*(a + x)*(38038000*a^8 - 28528500*a^7*x + 24453000*a^6*x^2 + 21060*a^5*(2352*c - 1045*x^3)
+ 19440*a^4*(-1911*c*x + 1045*x^4) + 3645*a^3*(8736*c*x^2 - 5225*x^5) - 13122*a^2*(5880*b^2 + 2184*c*x^3 - 137
5*x^6) + 137781*a*(420*b^2*x + 192*c*x^4 - 125*x^7) - 8266860*(6*b^2*x^2 + 3*c*x^5 - 2*x^8)) + 280*Sqrt[3]*(18
2468*a^9 - 91854*a^3*b^2 + 58968*a^6*c + 1594323*b^2*c)*(a + x)^(1/3)*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(a
+ x)^(1/3))] + 11760*Sqrt[3]*a^3*(7579*a^6 - 8748*b^2 + 5616*a^3*c)*(a + x)^(1/3)*ArcTan[(1 + (2*(a + x)^(1/3
))/x^(1/3))/Sqrt[3]] + 280*(135850*a^9 - 275562*a^3*b^2 + 176904*a^6*c - 1594323*b^2*c)*(a + x)^(1/3)*Log[-x^(
1/3) + (a + x)^(1/3)] + 140*(-135850*a^9 + 275562*a^3*b^2 - 176904*a^6*c + 1594323*b^2*c)*(a + x)^(1/3)*Log[x^
(2/3) + x^(1/3)*(a + x)^(1/3) + (a + x)^(2/3)]))/(446410440*(x^2*(a + x))^(1/3))

Maple [A] (verified)

Time = 7.01 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {67925 \left (x \left (\frac {6804}{5225} a^{6} c -\frac {137781}{67925} a^{3} b^{2}-\frac {1594323}{135850} b^{2} c +a^{9}\right ) \ln \left (\frac {\left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )-2 \left (\frac {6804}{5225} a^{6} c -\frac {137781}{67925} a^{3} b^{2}-\frac {1594323}{135850} b^{2} c +a^{9}\right ) \sqrt {3}\, x \arctan \left (\frac {\left (2 \left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )-2 x \left (\frac {6804}{5225} a^{6} c -\frac {137781}{67925} a^{3} b^{2}-\frac {1594323}{135850} b^{2} c +a^{9}\right ) \ln \left (\frac {\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}-x}{x}\right )-6 \left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}} \left (\frac {59049 x^{8}}{135850}-\frac {19683 a \,x^{7}}{43472}+\frac {6561 a^{2} x^{6}}{13832}+\left (-\frac {729 a^{3}}{1456}-\frac {177147 c}{271700}\right ) x^{5}+\frac {243 a \left (a^{3}+\frac {6804 c}{5225}\right ) x^{4}}{455}-\frac {81 a^{2} \left (a^{3}+\frac {6804 c}{5225}\right ) x^{3}}{140}+\left (\frac {9}{14} a^{6}+\frac {4374}{5225} a^{3} c -\frac {177147}{135850} b^{2}\right ) x^{2}-\frac {3 \left (a^{6}+\frac {6804}{5225} a^{3} c -\frac {137781}{67925} b^{2}\right ) a x}{4}+a^{2} \left (a^{6}+\frac {6804}{5225} a^{3} c -\frac {137781}{67925} b^{2}\right )\right )\right ) a^{9} x^{17}}{1594323 {\left (\left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}+x \left (x +\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}\right )\right )}^{9} {\left (-\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}+x \right )}^{9}}\)	\(333\)

[In]

int((x^3-b)*(x^3+b)*(x^3-c)/(a*x^2+x^3)^(1/3),x,method=_RETURNVERBOSE)

[Out]

67925/1594323*(x*(6804/5225*a^6*c-137781/67925*a^3*b^2-1594323/135850*b^2*c+a^9)*ln(((x^2*(a+x))^(2/3)+(x^2*(a
+x))^(1/3)*x+x^2)/x^2)-2*(6804/5225*a^6*c-137781/67925*a^3*b^2-1594323/135850*b^2*c+a^9)*3^(1/2)*x*arctan(1/3*
(2*(x^2*(a+x))^(1/3)+x)*3^(1/2)/x)-2*x*(6804/5225*a^6*c-137781/67925*a^3*b^2-1594323/135850*b^2*c+a^9)*ln(((x^
2*(a+x))^(1/3)-x)/x)-6*(x^2*(a+x))^(2/3)*(59049/135850*x^8-19683/43472*a*x^7+6561/13832*a^2*x^6+(-729/1456*a^3
-177147/271700*c)*x^5+243/455*a*(a^3+6804/5225*c)*x^4-81/140*a^2*(a^3+6804/5225*c)*x^3+(9/14*a^6+4374/5225*a^3
*c-177147/135850*b^2)*x^2-3/4*(a^6+6804/5225*a^3*c-137781/67925*b^2)*a*x+a^2*(a^6+6804/5225*a^3*c-137781/67925
*b^2)))*a^9*x^17/((x^2*(a+x))^(2/3)+x*(x+(x^2*(a+x))^(1/3)))^9/(-(x^2*(a+x))^(1/3)+x)^9

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {280 \, \sqrt {3} {\left (135850 \, a^{9} - 275562 \, a^{3} b^{2} + 243 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} c\right )} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 280 \, {\left (135850 \, a^{9} - 275562 \, a^{3} b^{2} + 243 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} c\right )} x \log \left (-\frac {x - {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - 140 \, {\left (135850 \, a^{9} - 275562 \, a^{3} b^{2} + 243 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} c\right )} x \log \left (\frac {x^{2} + {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}} x + {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (38038000 \, a^{8} + 18042750 \, a^{2} x^{6} - 17222625 \, a x^{7} + 16533720 \, x^{8} + 49533120 \, a^{5} c - 3645 \, {\left (5225 \, a^{3} + 6804 \, c\right )} x^{5} + 3888 \, {\left (5225 \, a^{4} + 6804 \, a c\right )} x^{4} - 77157360 \, a^{2} b^{2} - 4212 \, {\left (5225 \, a^{5} + 6804 \, a^{2} c\right )} x^{3} + 360 \, {\left (67925 \, a^{6} + 88452 \, a^{3} c - 137781 \, b^{2}\right )} x^{2} - 420 \, {\left (67925 \, a^{7} + 88452 \, a^{4} c - 137781 \, a b^{2}\right )} x\right )} {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{446410440 \, x} \]

[In]

integrate((x^3-b)*(x^3+b)*(x^3-c)/(a*x^2+x^3)^(1/3),x, algorithm="fricas")

[Out]

1/446410440*(280*sqrt(3)*(135850*a^9 - 275562*a^3*b^2 + 243*(728*a^6 - 6561*b^2)*c)*x*arctan(1/3*(sqrt(3)*x +
2*sqrt(3)*(a*x^2 + x^3)^(1/3))/x) + 280*(135850*a^9 - 275562*a^3*b^2 + 243*(728*a^6 - 6561*b^2)*c)*x*log(-(x -
(a*x^2 + x^3)^(1/3))/x) - 140*(135850*a^9 - 275562*a^3*b^2 + 243*(728*a^6 - 6561*b^2)*c)*x*log((x^2 + (a*x^2
+ x^3)^(1/3)*x + (a*x^2 + x^3)^(2/3))/x^2) + 3*(38038000*a^8 + 18042750*a^2*x^6 - 17222625*a*x^7 + 16533720*x^
8 + 49533120*a^5*c - 3645*(5225*a^3 + 6804*c)*x^5 + 3888*(5225*a^4 + 6804*a*c)*x^4 - 77157360*a^2*b^2 - 4212*(
5225*a^5 + 6804*a^2*c)*x^3 + 360*(67925*a^6 + 88452*a^3*c - 137781*b^2)*x^2 - 420*(67925*a^7 + 88452*a^4*c - 1
37781*a*b^2)*x)*(a*x^2 + x^3)^(2/3))/x

Sympy [F]

\[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int \frac {\left (- b + x^{3}\right ) \left (b + x^{3}\right ) \left (- c + x^{3}\right )}{\sqrt [3]{x^{2} \left (a + x\right )}}\, dx \]

[In]

integrate((x**3-b)*(x**3+b)*(x**3-c)/(a*x**2+x**3)**(1/3),x)

[Out]

Integral((-b + x**3)*(b + x**3)*(-c + x**3)/(x**2*(a + x))**(1/3), x)

Maxima [F]

\[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int { \frac {{\left (x^{3} + b\right )} {\left (x^{3} - b\right )} {\left (x^{3} - c\right )}}{{\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^3-b)*(x^3+b)*(x^3-c)/(a*x^2+x^3)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^3 + b)*(x^3 - b)*(x^3 - c)/(a*x^2 + x^3)^(1/3), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.69 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {280 \, \sqrt {3} {\left (135850 \, a^{10} + 176904 \, a^{7} c - 275562 \, a^{4} b^{2} - 1594323 \, a b^{2} c\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 140 \, {\left (135850 \, a^{10} + 176904 \, a^{7} c - 275562 \, a^{4} b^{2} - 1594323 \, a b^{2} c\right )} \log \left ({\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 280 \, {\left (135850 \, a^{10} + 176904 \, a^{7} c - 275562 \, a^{4} b^{2} - 1594323 \, a b^{2} c\right )} \log \left ({\left | {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) + \frac {3 \, {\left (38038000 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {26}{3}} - 332832500 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {23}{3}} + 1289216500 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {20}{3}} + 49533120 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {26}{3}} - 2897952200 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} - 433414800 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {23}{3}} + 4158305800 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} + 1678818960 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {20}{3}} - 77157360 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {26}{3}} - 3938066825 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} - 3773716128 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} + 675126900 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {23}{3}} + 2448101425 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 5414949792 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} - 2615083380 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {20}{3}} - 952462700 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 5128154388 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} + 5833647540 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} + 204186220 \, a^{10} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + 3164424732 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} - 8170413300 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} - 1170879948 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} + 7338216060 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} + 198438660 \, a^{7} c {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} - 4119651900 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 1319941980 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 184626540 \, a^{4} b^{2} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )} x^{9}}{a^{9}}}{446410440 \, a} \]

[In]

integrate((x^3-b)*(x^3+b)*(x^3-c)/(a*x^2+x^3)^(1/3),x, algorithm="giac")

[Out]

1/446410440*(280*sqrt(3)*(135850*a^10 + 176904*a^7*c - 275562*a^4*b^2 - 1594323*a*b^2*c)*arctan(1/3*sqrt(3)*(2
*(a/x + 1)^(1/3) + 1)) - 140*(135850*a^10 + 176904*a^7*c - 275562*a^4*b^2 - 1594323*a*b^2*c)*log((a/x + 1)^(2/
3) + (a/x + 1)^(1/3) + 1) + 280*(135850*a^10 + 176904*a^7*c - 275562*a^4*b^2 - 1594323*a*b^2*c)*log(abs((a/x +
1)^(1/3) - 1)) + 3*(38038000*a^10*(a/x + 1)^(26/3) - 332832500*a^10*(a/x + 1)^(23/3) + 1289216500*a^10*(a/x +
1)^(20/3) + 49533120*a^7*c*(a/x + 1)^(26/3) - 2897952200*a^10*(a/x + 1)^(17/3) - 433414800*a^7*c*(a/x + 1)^(2
3/3) + 4158305800*a^10*(a/x + 1)^(14/3) + 1678818960*a^7*c*(a/x + 1)^(20/3) - 77157360*a^4*b^2*(a/x + 1)^(26/3
) - 3938066825*a^10*(a/x + 1)^(11/3) - 3773716128*a^7*c*(a/x + 1)^(17/3) + 675126900*a^4*b^2*(a/x + 1)^(23/3)
+ 2448101425*a^10*(a/x + 1)^(8/3) + 5414949792*a^7*c*(a/x + 1)^(14/3) - 2615083380*a^4*b^2*(a/x + 1)^(20/3) -
952462700*a^10*(a/x + 1)^(5/3) - 5128154388*a^7*c*(a/x + 1)^(11/3) + 5833647540*a^4*b^2*(a/x + 1)^(17/3) + 204
186220*a^10*(a/x + 1)^(2/3) + 3164424732*a^7*c*(a/x + 1)^(8/3) - 8170413300*a^4*b^2*(a/x + 1)^(14/3) - 1170879
948*a^7*c*(a/x + 1)^(5/3) + 7338216060*a^4*b^2*(a/x + 1)^(11/3) + 198438660*a^7*c*(a/x + 1)^(2/3) - 4119651900
*a^4*b^2*(a/x + 1)^(8/3) + 1319941980*a^4*b^2*(a/x + 1)^(5/3) - 184626540*a^4*b^2*(a/x + 1)^(2/3))*x^9/a^9)/a

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right ) \left (-c+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int \frac {\left (x^3+b\right )\,\left (b-x^3\right )\,\left (c-x^3\right )}{{\left (x^3+a\,x^2\right )}^{1/3}} \,d x \]

[In]

int(((b + x^3)*(b - x^3)*(c - x^3))/(a*x^2 + x^3)^(1/3),x)

[Out]

int(((b + x^3)*(b - x^3)*(c - x^3))/(a*x^2 + x^3)^(1/3), x)