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

Optimal. Leaf size=339 \[ \frac {\left (135850 a^9+176904 a^6 c-275562 a^3 b^2-1594323 b^2 c\right ) \log \left (\sqrt [3]{a x^2+x^3}-x\right )}{1594323}+\frac {\left (-135850 a^9-176904 a^6 c+275562 a^3 b^2+1594323 b^2 c\right ) \log \left (x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}+x^2\right )}{3188646}+\frac {\left (-135850 \sqrt {3} a^9-176904 \sqrt {3} a^6 c+275562 \sqrt {3} a^3 b^2+1594323 \sqrt {3} b^2 c\right ) \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{a x^2+x^3}+x}\right )}{1594323}+\frac {\left (a x^2+x^3\right )^{2/3} \left (38038000 a^8-28528500 a^7 x+24453000 a^6 x^2+49533120 a^5 c-22007700 a^5 x^3-37149840 a^4 c x+20314800 a^4 x^4+31842720 a^3 c x^2-19045125 a^3 x^5-77157360 a^2 b^2-28658448 a^2 c x^3+18042750 a^2 x^6+57868020 a b^2 x+26453952 a c x^4-17222625 a x^7-49601160 b^2 x^2-24800580 c x^5+16533720 x^8\right )}{148803480 x} \]

________________________________________________________________________________________

Rubi [B]  time = 1.07, antiderivative size = 993, normalized size of antiderivative = 2.93, number of steps used = 28, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2053, 2011, 59, 2024} \begin {gather*} \frac {135850 x^{2/3} \sqrt [3]{a+x} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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 59

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

Rule 2011

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 2024

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 2053

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*} \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 \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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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} \tan ^{-1}\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 [C]  time = 0.28, size = 319, normalized size = 0.94 \begin {gather*} \frac {3 x \sqrt [3]{\frac {a+x}{a}} \left (a^9 \, _2F_1\left (-\frac {26}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-9 a^9 \, _2F_1\left (-\frac {23}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+36 a^9 \, _2F_1\left (-\frac {20}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+a^3 \left (84 a^6+20 a^3 c-b^2\right ) \, _2F_1\left (-\frac {8}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-3 a^3 \left (12 a^6+5 a^3 c-b^2\right ) \, _2F_1\left (-\frac {5}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+3 a^3 \left (3 a^6+2 a^3 c-b^2\right ) \, _2F_1\left (-\frac {2}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-\left (a^6-b^2\right ) \left (a^3+c\right ) \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-a^6 \left (84 a^3+c\right ) \, _2F_1\left (-\frac {17}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )+6 a^6 \left (21 a^3+c\right ) \, _2F_1\left (-\frac {14}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )-3 a^6 \left (42 a^3+5 c\right ) \, _2F_1\left (-\frac {11}{3},\frac {1}{3};\frac {4}{3};-\frac {x}{a}\right )\right )}{\sqrt [3]{x^2 (a+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*x*((a + x)/a)^(1/3)*(a^9*Hypergeometric2F1[-26/3, 1/3, 4/3, -(x/a)] - 9*a^9*Hypergeometric2F1[-23/3, 1/3, 4
/3, -(x/a)] + 36*a^9*Hypergeometric2F1[-20/3, 1/3, 4/3, -(x/a)] - a^6*(84*a^3 + c)*Hypergeometric2F1[-17/3, 1/
3, 4/3, -(x/a)] + 6*a^6*(21*a^3 + c)*Hypergeometric2F1[-14/3, 1/3, 4/3, -(x/a)] - 3*a^6*(42*a^3 + 5*c)*Hyperge
ometric2F1[-11/3, 1/3, 4/3, -(x/a)] + a^3*(84*a^6 - b^2 + 20*a^3*c)*Hypergeometric2F1[-8/3, 1/3, 4/3, -(x/a)]
- 3*a^3*(12*a^6 - b^2 + 5*a^3*c)*Hypergeometric2F1[-5/3, 1/3, 4/3, -(x/a)] + 3*a^3*(3*a^6 - b^2 + 2*a^3*c)*Hyp
ergeometric2F1[-2/3, 1/3, 4/3, -(x/a)] - (a^6 - b^2)*(a^3 + c)*Hypergeometric2F1[1/3, 1/3, 4/3, -(x/a)]))/(x^2
*(a + x))^(1/3)

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 4.41, size = 339, normalized size = 1.00 \begin {gather*} \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 ) \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a*x^2 + x^3)^(2/3)*(38038000*a^8 - 77157360*a^2*b^2 + 49533120*a^5*c - 28528500*a^7*x + 57868020*a*b^2*x - 3
7149840*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))/(148803480*x) + ((-135850*Sqrt[3]*a^9 + 275562*Sqrt[3]*a^3*b^2 - 176904*Sqrt[3]*a^6*c +
1594323*Sqrt[3]*b^2*c)*ArcTan[(Sqrt[3]*x)/(x + 2*(a*x^2 + x^3)^(1/3))])/1594323 + ((135850*a^9 - 275562*a^3*b^
2 + 176904*a^6*c - 1594323*b^2*c)*Log[-x + (a*x^2 + x^3)^(1/3)])/1594323 + ((-135850*a^9 + 275562*a^3*b^2 - 17
6904*a^6*c + 1594323*b^2*c)*Log[x^2 + x*(a*x^2 + x^3)^(1/3) + (a*x^2 + x^3)^(2/3)])/3188646

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 325, normalized size = 0.96 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

giac [A]  time = 0.49, size = 572, normalized size = 1.69 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-b \right ) \left (x^{3}+b \right ) \left (x^{3}-c \right )}{\left (a \,x^{2}+x^{3}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________