3.22.41 \(\int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} (-b^2+a^2 x^6)} \, dx\)

Optimal. Leaf size=156 \[ \frac {\text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a-3 \text {$\#$1}^3 a^2+a^3-a b^2\& ,\frac {\log \left (\sqrt [3]{a x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 a b}-\frac {\text {RootSum}\left [-\text {$\#$1}^9+3 \text {$\#$1}^6 a-3 \text {$\#$1}^3 a^2+a^3+a b^2\& ,\frac {\log \left (\sqrt [3]{a x^3-b x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ]}{6 a b} \]

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Rubi [B]  time = 6.02, antiderivative size = 2641, normalized size of antiderivative = 16.93, number of steps used = 67, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2056, 6725, 105, 50, 59, 91}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[x^3/((-(b*x^2) + a*x^3)^(1/3)*(-b^2 + a^2*x^6)),x]

[Out]

-1/3*(x*(b - a*x))/(a^(5/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) + ((-1)^(1/3)*x*(b - a*x))/(3*a^(5/3)*b^(4/3)*(-
(b*x^2) + a*x^3)^(1/3)) - ((-1)^(2/3)*x*(b - a*x))/(3*a^(5/3)*b^(4/3)*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b
+ a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/3)*x^(1/3))])/(3*Sqrt[3]*a^2*b^(1/3)*(-(b*
x^2) + a*x^3)^(1/3)) + ((-1)^(1/3)*x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*a
^(1/3)*x^(1/3))])/(3*Sqrt[3]*a^2*b^(1/3)*(-(b*x^2) + a*x^3)^(1/3)) - ((-1)^(2/3)*x^(2/3)*(-b + a*x)^(1/3)*ArcT
an[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/3)*x^(1/3))])/(3*Sqrt[3]*a^2*b^(1/3)*(-(b*x^2) + a*x^3)^(1/3
)) + (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(2/3) - b^(2/3))^(1
/3)*x^(1/3))])/(2*Sqrt[3]*a^(10/9)*(a^(2/3) - b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)
^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(2/3) + b^(2/3))^(1/3)*x^(1/3))])/(2*Sqrt[3
]*a^(10/9)*(a^(2/3) + b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3]
+ (2*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(10/9)*(a
^(2/3) - (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] +
(2*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(10/9)*(a^(
2/3) + (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2
*(-b + a*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(10/9)*(a^(2/
3) - (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(
-b + a*x)^(1/3))/(Sqrt[3]*a^(1/9)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/3)*x^(1/3))])/(2*Sqrt[3]*a^(10/9)*(a^(2/3)
+ (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[x])/(18*a^2*b^(1/3)*(
-(b*x^2) + a*x^3)^(1/3)) + ((-1)^(1/3)*x^(2/3)*(-b + a*x)^(1/3)*Log[x])/(18*a^2*b^(1/3)*(-(b*x^2) + a*x^3)^(1/
3)) - ((-1)^(2/3)*x^(2/3)*(-b + a*x)^(1/3)*Log[x])/(18*a^2*b^(1/3)*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b +
a*x)^(1/3)*Log[-b^(1/3) - a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) + b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^
(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) - a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) - b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(
1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[-b^(1/3) + (-1)^(1/3)*a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) - (-1)^(1/3)*b^(
2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) + (-1)^(1/3)*a^(1/3)*x])/(12*a
^(10/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[-b^(1
/3) - (-1)^(2/3)*a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (
x^(2/3)*(-b + a*x)^(1/3)*Log[b^(1/3) - (-1)^(2/3)*a^(1/3)*x])/(12*a^(10/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(1/3
)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a^(1/9)*(a^(2/3) -
b^(2/3))^(1/3))])/(4*a^(10/9)*(a^(2/3) - b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3
)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a^(1/9)*(a^(2/3) + b^(2/3))^(1/3))])/(4*a^(10/9)*(a^(2/3) + b^(2/3))^(1/3)*
b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a^(1/9)*(a^(2/3) - (-
1)^(1/3)*b^(2/3))^(1/3))])/(4*a^(10/9)*(a^(2/3) - (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(
2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a^(1/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1/3))])/(4*a^(1
0/9)*(a^(2/3) + (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) + (x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3)
+ (-b + a*x)^(1/3)/(a^(1/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3))^(1/3))])/(4*a^(10/9)*(a^(2/3) - (-1)^(2/3)*b^(2/3)
)^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)) - (x^(2/3)*(-b + a*x)^(1/3)*Log[-x^(1/3) + (-b + a*x)^(1/3)/(a^(1/9)*(a^(2
/3) + (-1)^(2/3)*b^(2/3))^(1/3))])/(4*a^(10/9)*(a^(2/3) + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a*x^3)^(1/3)
) - (x^(2/3)*(-b + a*x)^(1/3)*Log[-1 + (-b + a*x)^(1/3)/(a^(1/3)*x^(1/3))])/(6*a^2*b^(1/3)*(-(b*x^2) + a*x^3)^
(1/3)) + ((-1)^(1/3)*x^(2/3)*(-b + a*x)^(1/3)*Log[-1 + (-b + a*x)^(1/3)/(a^(1/3)*x^(1/3))])/(6*a^2*b^(1/3)*(-(
b*x^2) + a*x^3)^(1/3)) - ((-1)^(2/3)*x^(2/3)*(-b + a*x)^(1/3)*Log[-1 + (-b + a*x)^(1/3)/(a^(1/3)*x^(1/3))])/(6
*a^2*b^(1/3)*(-(b*x^2) + a*x^3)^(1/3))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 91

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

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt [3]{-b x^2+a x^3} \left (-b^2+a^2 x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-b+a x} \left (-b^2+a^2 x^6\right )} \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {x^{7/3}}{2 b \sqrt [3]{-b+a x} \left (b-a x^3\right )}-\frac {x^{7/3}}{2 b \sqrt [3]{-b+a x} \left (b+a x^3\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-b+a x} \left (b-a x^3\right )} \, dx}{2 b \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\sqrt [3]{-b+a x} \left (b+a x^3\right )} \, dx}{2 b \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (-\frac {x^{7/3}}{3 b^{2/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}-\frac {x^{7/3}}{3 b^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}-\frac {x^{7/3}}{3 b^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 b \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \left (\frac {x^{7/3}}{3 b^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}+\frac {x^{7/3}}{3 b^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}+\frac {x^{7/3}}{3 b^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}}\right ) \, dx}{2 b \sqrt [3]{-b x^2+a x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{7/3}}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 b^{5/3} \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{4/3}}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 \sqrt [3]{a} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{4/3}}{\left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 \sqrt [3]{a} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{4/3}}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 \sqrt [3]{a} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{4/3}}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 \sqrt [3]{a} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{4/3}}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 \sqrt [3]{a} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {x^{4/3}}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 \sqrt [3]{a} b^{4/3} \sqrt [3]{-b x^2+a x^3}}\\ &=2 \frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-2 \frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+2 \frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b \sqrt [3]{-b x^2+a x^3}}-\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b \sqrt [3]{-b x^2+a x^3}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b \sqrt [3]{-b x^2+a x^3}}+\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b \sqrt [3]{-b x^2+a x^3}}-\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a^{2/3} b \sqrt [3]{-b x^2+a x^3}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a b^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a b^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a b^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a b^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a b^{2/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right ) \sqrt [3]{-b+a x}} \, dx}{6 a b^{2/3} \sqrt [3]{-b x^2+a x^3}}+2 \left (-\frac {x (b-a x)}{6 a^{5/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x}} \, dx}{18 a^{5/3} \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}\right )-2 \left (-\frac {\sqrt [3]{-1} x (b-a x)}{6 a^{5/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left (\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x}} \, dx}{18 a^{5/3} \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}\right )+2 \left (-\frac {(-1)^{2/3} x (b-a x)}{6 a^{5/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}+\frac {\left ((-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a x}} \, dx}{18 a^{5/3} \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}\right )\\ &=\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{2/3}-b^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{2/3}+b^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{b}-\sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [3]{b}+\sqrt [3]{-1} \sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{a} x\right )}{12 a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [9]{a} \sqrt [3]{a^{2/3}-b^{2/3}}}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}-b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [9]{a} \sqrt [3]{a^{2/3}+b^{2/3}}}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}+b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [9]{a} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}}}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [9]{a} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}+\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [9]{a} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}-(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a x}}{\sqrt [9]{a} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}}}\right )}{4 a^{10/9} \sqrt [3]{a^{2/3}+(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a x^3}}+2 \left (-\frac {x (b-a x)}{6 a^{5/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{x}}\right )}{6 \sqrt {3} a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log (x)}{36 a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a x} \log \left (-1+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a} \sqrt [3]{x}}\right )}{12 a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}\right )-2 \left (-\frac {\sqrt [3]{-1} x (b-a x)}{6 a^{5/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{x}}\right )}{6 \sqrt {3} a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x} \log (x)}{36 a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}-\frac {\sqrt [3]{-1} x^{2/3} \sqrt [3]{-b+a x} \log \left (-1+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a} \sqrt [3]{x}}\right )}{12 a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}\right )+2 \left (-\frac {(-1)^{2/3} x (b-a x)}{6 a^{5/3} b^{4/3} \sqrt [3]{-b x^2+a x^3}}-\frac {(-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a x}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{x}}\right )}{6 \sqrt {3} a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}-\frac {(-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x} \log (x)}{36 a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}-\frac {(-1)^{2/3} x^{2/3} \sqrt [3]{-b+a x} \log \left (-1+\frac {\sqrt [3]{-b+a x}}{\sqrt [3]{a} \sqrt [3]{x}}\right )}{12 a^2 \sqrt [3]{b} \sqrt [3]{-b x^2+a x^3}}\right )\\ \end {align*}

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Mathematica [C]  time = 0.81, size = 295, normalized size = 1.89 \begin {gather*} \frac {x \left (-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a x-\sqrt [3]{a} b^{2/3} x}{a x-b}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a x+\sqrt [3]{a} b^{2/3} x}{a x-b}\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{a} b^{2/3} x-2 a x}{2 (b-a x)}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{a} b^{2/3} x-2 a x}{2 (b-a x)}\right )+\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{a} b^{2/3} x-2 a x}{2 (b-a x)}\right )-\, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {i \left (i+\sqrt {3}\right ) \sqrt [3]{a} b^{2/3} x-2 a x}{2 (b-a x)}\right )\right )}{2 a b \sqrt [3]{x^2 (a x-b)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3/((-(b*x^2) + a*x^3)^(1/3)*(-b^2 + a^2*x^6)),x]

[Out]

(x*(-Hypergeometric2F1[1/3, 1, 4/3, (a*x - a^(1/3)*b^(2/3)*x)/(-b + a*x)] + Hypergeometric2F1[1/3, 1, 4/3, (a*
x + a^(1/3)*b^(2/3)*x)/(-b + a*x)] - Hypergeometric2F1[1/3, 1, 4/3, (-2*a*x + (-1 - I*Sqrt[3])*a^(1/3)*b^(2/3)
*x)/(2*(b - a*x))] + Hypergeometric2F1[1/3, 1, 4/3, (-2*a*x + (1 - I*Sqrt[3])*a^(1/3)*b^(2/3)*x)/(2*(b - a*x))
] + Hypergeometric2F1[1/3, 1, 4/3, (-2*a*x + (1 + I*Sqrt[3])*a^(1/3)*b^(2/3)*x)/(2*(b - a*x))] - Hypergeometri
c2F1[1/3, 1, 4/3, (-2*a*x + I*(I + Sqrt[3])*a^(1/3)*b^(2/3)*x)/(2*(b - a*x))]))/(2*a*b*(x^2*(-b + a*x))^(1/3))

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IntegrateAlgebraic [A]  time = 0.54, size = 156, normalized size = 1.00 \begin {gather*} \frac {\text {RootSum}\left [a^3-a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 a b}-\frac {\text {RootSum}\left [a^3+a b^2-3 a^2 \text {$\#$1}^3+3 a \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 a b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/((-(b*x^2) + a*x^3)^(1/3)*(-b^2 + a^2*x^6)),x]

[Out]

RootSum[a^3 - a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*#1])/#1 & ]
/(6*a*b) - RootSum[a^3 + a*b^2 - 3*a^2*#1^3 + 3*a*#1^6 - #1^9 & , (-Log[x] + Log[(-(b*x^2) + a*x^3)^(1/3) - x*
#1])/#1 & ]/(6*a*b)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (a^{2} x^{6} - b^{2}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x, algorithm="giac")

[Out]

integrate(x^3/((a^2*x^6 - b^2)*(a*x^3 - b*x^2)^(1/3)), x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (a \,x^{3}-b \,x^{2}\right )^{\frac {1}{3}} \left (a^{2} x^{6}-b^{2}\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x)

[Out]

int(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (a^{2} x^{6} - b^{2}\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a*x^3-b*x^2)^(1/3)/(a^2*x^6-b^2),x, algorithm="maxima")

[Out]

integrate(x^3/((a^2*x^6 - b^2)*(a*x^3 - b*x^2)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {x^3}{\left (b^2-a^2\,x^6\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^3/((b^2 - a^2*x^6)*(a*x^3 - b*x^2)^(1/3)),x)

[Out]

-int(x^3/((b^2 - a^2*x^6)*(a*x^3 - b*x^2)^(1/3)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a*x**3-b*x**2)**(1/3)/(a**2*x**6-b**2),x)

[Out]

Timed out

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