3.29.76 \(\int \frac {-b x+a x^3}{(-b+2 a x^3) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\) [2876]

Optimal. Leaf size=310 \[ \frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{b^2 x^2+a^3 x^3}}\right )}{2 a}-\frac {\log \left (-a x+\sqrt [3]{b^2 x^2+a^3 x^3}\right )}{2 a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{b^2 x^2+a^3 x^3}+\left (b^2 x^2+a^3 x^3\right )^{2/3}\right )}{4 a}+\frac {1}{6} \text {RootSum}\left [a^9+2 a b^5-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\& ,\frac {-a^3 \log (x)-2 b^2 \log (x)+a^3 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+2 b^2 \log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}-\text {$\#$1}^4}\& \right ] \]

[Out]

Unintegrable

________________________________________________________________________________________

Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(1782\) vs. \(2(310)=620\).
time = 2.07, antiderivative size = 1782, normalized size of antiderivative = 5.75, number of steps used = 16, number of rules used = 7, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1607, 2081, 6857, 129, 494, 245, 384} \begin {gather*} \frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{x} a}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \text {ArcTan}\left (\frac {\frac {2 \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}{\sqrt [3]{x a^3+b^2}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{-2} \sqrt [3]{a} x+\sqrt [3]{b}\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (a \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/3} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{x a^3+b^2} \log \left (\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}-\sqrt [3]{x a^3+b^2}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{a^3 x^3+b^2 x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a^(1/3) - (-2)^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqr
t[3]])/(2*Sqrt[3]*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3
)*ArcTan[(1 + (2*a*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((-
1)^(1/3)*((-1)^(2/3)*a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a*x^(1/3))/(b^2 + a
^3*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) - (-2)^(2/3)*b^(1/3))*x^(2/3)
*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))
/Sqrt[3]])/(2*Sqrt[3]*a^(4/9)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) - 2^
(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*x^(1/3))/(
b^2 + a^3*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(4/9)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))
+ ((-1)^(1/3)*((-1)^(2/3)*a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*ArcTan[(1 + (2*a^(1/9)*(a^(8/
3) + (-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*x^(1/3))/(b^2 + a^3*x)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*a^(4/9)*(a^(8/3) + (
-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) - (-2)^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a
^3*x)^(1/3)*Log[b^(1/3) + (-2)^(1/3)*a^(1/3)*x])/(12*a^(4/9)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a
^3*x^3)^(1/3)) - ((a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) - 2^(1/3)*a^(1/3)*x])/(1
2*a^(4/9)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) + (-1)^(1/3)*2^(2/3)*b^(1/3
))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[b^(1/3) - (-1)^(2/3)*2^(1/3)*a^(1/3)*x])/(12*a^(4/9)*(a^(8/3) + (-1)^(2/3)*
2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) - (-2)^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3
)*Log[a*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(4*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(1/3) - 2^(2/3)*b^(1/3))*x
^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(4*a^(4/3)*(b^2*x^2 + a^3*x^3)^(1/3)) - ((a^(
1/3) + (-1)^(1/3)*2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(4*a^(4/3
)*(b^2*x^2 + a^3*x^3)^(1/3)) + ((a^(1/3) - (-2)^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a^(1/9)*(a^(8/3
) - (-2)^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(4*a^(4/9)*(a^(8/3) - (-2)^(1/3)*b^(5/3))^(1/3)*
(b^2*x^2 + a^3*x^3)^(1/3)) + ((a^(1/3) - 2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a^(1/9)*(a^(8/3) + 2
^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(4*a^(4/9)*(a^(8/3) + 2^(1/3)*b^(5/3))^(1/3)*(b^2*x^2 +
a^3*x^3)^(1/3)) + ((a^(1/3) + (-1)^(1/3)*2^(2/3)*b^(1/3))*x^(2/3)*(b^2 + a^3*x)^(1/3)*Log[a^(1/9)*(a^(8/3) + (
-1)^(2/3)*2^(1/3)*b^(5/3))^(1/3)*x^(1/3) - (b^2 + a^3*x)^(1/3)])/(4*a^(4/9)*(a^(8/3) + (-1)^(2/3)*2^(1/3)*b^(5
/3))^(1/3)*(b^2*x^2 + a^3*x^3)^(1/3))

Rule 129

Int[((e_.)*(x_))^(p_)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> With[{k = Denominator[p]
}, Dist[k/e, Subst[Int[x^(k*(p + 1) - 1)*(a + b*(x^k/e))^m*(c + d*(x^k/e))^n, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] && FractionQ[p] && IntegerQ[m]

Rule 245

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + 2*Rt[b, 3]*(x/(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 384

Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Simp[
ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q
), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 494

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[a*(e^n/b), Int[(e*x)^(m - n)*((c + d*x^n)^q/(a + b*x^n)), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2081

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 6857

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 {-b x+a x^3}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx &=\int \frac {x \left (-b+a x^2\right )}{\left (-b+2 a x^3\right ) \sqrt [3]{b^2 x^2+a^3 x^3}} \, dx\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x} \left (-b+a x^2\right )}{\sqrt [3]{b^2+a^3 x} \left (-b+2 a x^3\right )} \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \left (-\frac {\left (-\frac {\sqrt [3]{-1} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\left (\frac {\sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}-\frac {\left (\frac {(-1)^{2/3} \sqrt [3]{a} b}{2^{2/3}}-b^{4/3}\right ) \sqrt [3]{x}}{3 b \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}}\right ) \, dx}{\sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {\sqrt [3]{x}}{\left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{-\frac {1}{2}} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\left (\sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{-\frac {1}{2}} \left ((-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a}-2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left ((-1)^{2/3} \left (\sqrt [3]{-2} \sqrt [3]{a}+2 \sqrt [3]{b}\right ) \sqrt [3]{b} x^{2/3} \sqrt [3]{b^2+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right ) \sqrt [3]{b^2+a^3 x}} \, dx}{6 \sqrt [3]{2} \sqrt [3]{a} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ &=-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{b^2+a^3 x}}{\sqrt {3} \sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{x}}\right )}{2 \sqrt {3} a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{12 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{12 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log (x)}{12 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}+\sqrt [3]{-2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-\sqrt [3]{2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (\sqrt [3]{b}-(-1)^{2/3} \sqrt [3]{2} \sqrt [3]{a} x\right )}{12 a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}}}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}-\sqrt [3]{-2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}}}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+\sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}+\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{b^2+a^3 x}}{\sqrt [9]{a} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}}}\right )}{4 a^{4/9} \sqrt [3]{a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}-\frac {\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) x^{2/3} \sqrt [3]{b^2+a^3 x} \log \left (-1+\frac {\sqrt [3]{b^2+a^3 x}}{a \sqrt [3]{x}}\right )}{4 a^{4/3} \sqrt [3]{b^2 x^2+a^3 x^3}}\\ \end {align*}

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Mathematica [A]
time = 10.41, size = 268, normalized size = 0.86 \begin {gather*} \frac {3 \sqrt [3]{a} x \sqrt [3]{1+\frac {a^3 x}{b^2}} \, _2F_1\left (\frac {1}{3},\frac {1}{3};\frac {4}{3};-\frac {a^3 x}{b^2}\right )-x \left (\left (\sqrt [3]{a}-(-2)^{2/3} \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}-\sqrt [3]{-2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}+\sqrt [3]{-1} 2^{2/3} \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {\sqrt [3]{a} \left (a^{8/3}+(-1)^{2/3} \sqrt [3]{2} b^{5/3}\right ) x}{b^2+a^3 x}\right )+\left (\sqrt [3]{a}-2^{2/3} \sqrt [3]{b}\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {a^3 x+\sqrt [3]{2} \sqrt [3]{a} b^{5/3} x}{b^2+a^3 x}\right )\right )}{2 \sqrt [3]{a} \sqrt [3]{x^2 \left (b^2+a^3 x\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^(1/3)*x*(1 + (a^3*x)/b^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, -((a^3*x)/b^2)] - x*((a^(1/3) - (-2)^(2/
3)*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3) - (-2)^(1/3)*b^(5/3))*x)/(b^2 + a^3*x)] + (a^(1/3
) + (-1)^(1/3)*2^(2/3)*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^(1/3)*(a^(8/3) + (-1)^(2/3)*2^(1/3)*b^(5/3))
*x)/(b^2 + a^3*x)] + (a^(1/3) - 2^(2/3)*b^(1/3))*Hypergeometric2F1[1/3, 1, 4/3, (a^3*x + 2^(1/3)*a^(1/3)*b^(5/
3)*x)/(b^2 + a^3*x)]))/(2*a^(1/3)*(x^2*(b^2 + a^3*x))^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 102.55, size = 146381, normalized size = 472.20 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(3)*(1/6)^(1/3)*(1/18)^(1/3)*a*((6*a^6 - 24*a^5*b + 36*a^3*b^2 + 72*b^4 + (a^9 + 2*a*b^5)*(2*(1/2)
^(2/3)*(-I*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^9 + 2*a*b^5)^2 - (a^4 + 28*a^3*b + 16*a^2*b
^2 + 6*a*b^2 + 48*b^3)/(a^10 + 2*a^2*b^5))/(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^9 + 2*a*b^5)^3 - 3*(a^
6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^10 + 2*a^2*b^5)*(a^9 +
2*a*b^5)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 4
8*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)*b/((a^8 + 2*b^5)^2*a^4))
^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^9 + 2*a*b^5)^3 - 3*(a^6 - 4*
a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^10 + 2*a^2*b^5)*(a^9 + 2*a*b^
5)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^
6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)*b/((a^8 + 2*b^5)^2*a^4))^(1/3)
- 2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^9 + 2*a*b^5)) + 3*sqrt(1/3)*(a^9 + 2*a*b^5)*sqrt(-(4*a^12 + 544*a
^11*b + 1344*a^8*b^3 - 720*a^6*b^4 - 832*a^3*b^6 + 64*(8*a^2 + 3*a)*b^7 - 192*b^8 + 32*(36*a^5 + a^4)*b^5 + (a
^18 + 4*a^10*b^5 + 4*a^2*b^10)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^9 +
2*a*b^5)^2 - (a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/(a^10 + 2*a^2*b^5))/(2*(a^6 - 4*a^5*b + 6*a^3*b^
2 + 12*b^4)^3/(a^9 + 2*a*b^5)^3 - 3*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^
2 + 48*b^3)/((a^10 + 2*a^2*b^5)*(a^9 + 2*a*b^5)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2
*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a
^6 - a^5)*b^3)*b/((a^8 + 2*b^5)^2*a^4))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12
*b^4)^3/(a^9 + 2*a*b^5)^3 - 3*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48
*b^3)/((a^10 + 2*a^2*b^5)*(a^9 + 2*a*b^5)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a
^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a
^5)*b^3)*b/((a^8 + 2*b^5)^2*a^4))^(1/3) - 2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^9 + 2*a*b^5))^2 + 16*(4*a^
10 - 3*a^9)*b^2 + 4*(a^15 - 4*a^14*b + 6*a^12*b^2 + 12*a^9*b^4 + 2*a^7*b^5 - 8*a^6*b^6 + 12*a^4*b^7 + 24*a*b^9
)*(2*(1/2)^(2/3)*(-I*sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^9 + 2*a*b^5)^2 - (a^4 + 28*a^3*b
+ 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/(a^10 + 2*a^2*b^5))/(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^9 + 2*a*b^5)
^3 - 3*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^10 + 2*a^2*b^
5)*(a^9 + 2*a*b^5)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a^10 + 216*a^9*b + 216*a
^7*b^2 - 48*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)*b/((a^8 + 2*b^
5)^2*a^4))^(1/3) + (1/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^9 + 2*a*b^5)^3 - 3
*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^10 + 2*a^2*b^5)*(a^
9 + 2*a*b^5)) + (a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2
- 48*a*b^6 + 64*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)*b/((a^8 + 2*b^5)^2*a
^4))^(1/3) - 2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^9 + 2*a*b^5)))/(a^18 + 4*a^10*b^5 + 4*a^2*b^10)))/(a^9
+ 2*a*b^5))^(1/3)*arctan(-1/24*(3*sqrt(1/2)*(1/6)^(1/3)*(1/18)^(1/3)*(sqrt(3)*(12*a^17 - a^15*b + 16*a^14*b^2
- 8*a^12*b^3 + 8*a^11*b^4 + 24*a^9*b^5 - 2*a^7*b^6 + 32*a^6*b^7 - 16*a^4*b^8 + 16*a^3*b^9)*(2*(1/2)^(2/3)*(-I*
sqrt(3) + 1)*((a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^2/(a^9 + 2*a*b^5)^2 - (a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^
2 + 48*b^3)/(a^10 + 2*a^2*b^5))/(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^9 + 2*a*b^5)^3 - 3*(a^6 - 4*a^5*b
+ 6*a^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^10 + 2*a^2*b^5)*(a^9 + 2*a*b^5)) +
(a^3 - 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 6
4*b^7 + 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)*b/((a^8 + 2*b^5)^2*a^4))^(1/3) + (1
/2)^(1/3)*(I*sqrt(3) + 1)*(2*(a^6 - 4*a^5*b + 6*a^3*b^2 + 12*b^4)^3/(a^9 + 2*a*b^5)^3 - 3*(a^6 - 4*a^5*b + 6*a
^3*b^2 + 12*b^4)*(a^4 + 28*a^3*b + 16*a^2*b^2 + 6*a*b^2 + 48*b^3)/((a^10 + 2*a^2*b^5)*(a^9 + 2*a*b^5)) + (a^3
- 12*a^2*b + 48*a*b^2 - 64*b^3)/(a^12 + 2*a^4*b^5) + 2*(54*a^10 + 216*a^9*b + 216*a^7*b^2 - 48*a*b^6 + 64*b^7
+ 96*(3*a^3 - a^2)*b^5 + (72*a^4 - a^3)*b^4 + 36*(12*a^6 - a^5)*b^3)*b/((a^8 + 2*b^5)^2*a^4))^(1/3) - 2*(a^6 -
4*a^5*b + 6*a^3*b^2 + 12*b^4)/(a^9 + 2*a*b^5))^2*x + 4*sqrt(3)*(21*a^14 + 140*a^11*b^2 + 420*a^8*b^4 - 140*a^
3*b^7 + 112*a^2*b^8 + (336*a^5 + a^4)*b^6 - 12*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \left (a x^{2} - b\right )}{\sqrt [3]{x^{2} \left (a^{3} x + b^{2}\right )} \left (2 a x^{3} - b\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
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((a*x^3-b*x)/(2*a*x^3-b)/(a^3*x^3+b^2*x^2)^(1/3),x, algorithm="giac")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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