\(\int \frac {b+a^3 x^2}{(-b+a^3 x^2) \sqrt [3]{-b x^2+a^3 x^3}} \, dx\) [2380]

Optimal result

Integrand size = 41, antiderivative size = 190 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b x^2+a^3 x^3}}\right )}{a}-\frac {\log \left (-a x+\sqrt [3]{-b x^2+a^3 x^3}\right )}{a}+\frac {\log \left (a^2 x^2+a x \sqrt [3]{-b x^2+a^3 x^3}+\left (-b x^2+a^3 x^3\right )^{2/3}\right )}{2 a}+\text {RootSum}\left [a^6-a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(788\) vs. \(2(190)=380\).

Time = 0.58 (sec) , antiderivative size = 788, normalized size of antiderivative = 4.15, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {2081, 6857, 61, 926, 93} \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} a \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{a \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \log (x) \sqrt [3]{a^3 x-b}}{2 a \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{a \sqrt [3]{x}}-1\right )}{2 a \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}-\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt {a} \sqrt [3]{x} \sqrt [3]{a^{3/2}+\sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-a^{3/2} x\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (a^{3/2} x+\sqrt {b}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}-\sqrt [3]{x}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}} \]

[In]

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

[Out]

-((Sqrt[3]*x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*a*x^(1/3))])/(a*(-(b*
x^2) + a^3*x^3)^(1/3))) + (Sqrt[3]*x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[
3]*Sqrt[a]*(a^(3/2) - Sqrt[b])^(1/3)*x^(1/3))])/(Sqrt[a]*(a^(3/2) - Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3))
+ (Sqrt[3]*x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*Sqrt[a]*(a^(3/2) + S
qrt[b])^(1/3)*x^(1/3))])/(Sqrt[a]*(a^(3/2) + Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x
)^(1/3)*Log[x])/(2*a*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[Sqrt[b] - a^(3/2)*x])/(2*Sq
rt[a]*(a^(3/2) - Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[Sqrt[b] + a^(3/2
)*x])/(2*Sqrt[a]*(a^(3/2) + Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) + (3*x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^
(1/3) + (-b + a^3*x)^(1/3)/(Sqrt[a]*(a^(3/2) - Sqrt[b])^(1/3))])/(2*Sqrt[a]*(a^(3/2) - Sqrt[b])^(1/3)*(-(b*x^2
) + a^3*x^3)^(1/3)) + (3*x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^(1/3) + (-b + a^3*x)^(1/3)/(Sqrt[a]*(a^(3/2) + Sqrt
[b])^(1/3))])/(2*Sqrt[a]*(a^(3/2) + Sqrt[b])^(1/3)*(-(b*x^2) + a^3*x^3)^(1/3)) - (3*x^(2/3)*(-b + a^3*x)^(1/3)
*Log[-1 + (-b + a^3*x)^(1/3)/(a*x^(1/3))])/(2*a*(-(b*x^2) + a^3*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 93

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 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

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*} \text {integral}& = \frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {b+a^3 x^2}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^2\right )} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (\frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x}}+\frac {2 b}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^2\right )}\right ) \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = \frac {\left (x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x}} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}+\frac {\left (2 b x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (-b+a^3 x^2\right )} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = -\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}+\frac {\left (2 b x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \left (-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{2 \sqrt {b} x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}}\right ) \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = -\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}-a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}}-\frac {\left (\sqrt {b} x^{2/3} \sqrt [3]{-b+a^3 x}\right ) \int \frac {1}{x^{2/3} \left (\sqrt {b}+a^{3/2} x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{\sqrt [3]{-b x^2+a^3 x^3}} \\ & = -\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} a \sqrt [3]{x}}\right )}{a \sqrt [3]{-b x^2+a^3 x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{x}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{-b+a^3 x} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{-b+a^3 x}}{\sqrt {3} \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{x}}\right )}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log (x)}{2 a \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}-a^{3/2} x\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (\sqrt {b}+a^{3/2} x\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}-\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}+\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}}}\right )}{2 \sqrt {a} \sqrt [3]{a^{3/2}+\sqrt {b}} \sqrt [3]{-b x^2+a^3 x^3}}-\frac {3 x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-1+\frac {\sqrt [3]{-b+a^3 x}}{a \sqrt [3]{x}}\right )}{2 a \sqrt [3]{-b x^2+a^3 x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.19 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{-b+a^3 x}}\right )-2 \log \left (a \left (a \sqrt [3]{x}-\sqrt [3]{-b+a^3 x}\right )\right )+\log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{-b+a^3 x}+\left (-b+a^3 x\right )^{2/3}\right )+2 a \text {RootSum}\left [a^6-a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{2 a \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 0.20 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}-a^{3} b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) a -2 \ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )+\ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2 a}\)	\(175\)

[In]

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

[Out]

1/2*(-2*3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x-b))^(1/3))*3^(1/2)/a/x)+2*sum(ln((-_R*x+(x^2*(a^3*x-b))^(1/3))/x
)/_R,_R=RootOf(_Z^6-2*_Z^3*a^3+a^6-a^3*b))*a-2*ln((-a*x+(x^2*(a^3*x-b))^(1/3))/x)+ln((a^2*x^2+a*(x^2*(a^3*x-b)
)^(1/3)*x+(x^2*(a^3*x-b))^(2/3))/x^2))/a

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.28 (sec) , antiderivative size = 1225, normalized size of antiderivative = 6.45 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/2*((sqrt(-3)*a - a)*(((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) + 1)/(a^3 - b))^(1/3)*log(2*((sqrt(-3)*a^3
*x + a^3*x - (sqrt(-3)*(a^6 - a^3*b)*x + (a^6 - a^3*b)*x)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)))*(((a^3 - b)*sqrt(
b/(a^9 - 2*a^6*b + a^3*b^2)) + 1)/(a^3 - b))^(2/3) + 2*(a^3*x^3 - b*x^2)^(1/3))/x) - (sqrt(-3)*a + a)*(((a^3 -
b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) + 1)/(a^3 - b))^(1/3)*log(-2*((sqrt(-3)*a^3*x - a^3*x - (sqrt(-3)*(a^6 -
a^3*b)*x - (a^6 - a^3*b)*x)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)))*(((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2))
+ 1)/(a^3 - b))^(2/3) - 2*(a^3*x^3 - b*x^2)^(1/3))/x) + 2*a*(((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) + 1)
/(a^3 - b))^(1/3)*log(-2*((a^3*x - (a^6 - a^3*b)*x*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)))*(((a^3 - b)*sqrt(b/(a^9
- 2*a^6*b + a^3*b^2)) + 1)/(a^3 - b))^(2/3) - (a^3*x^3 - b*x^2)^(1/3))/x) + (sqrt(-3)*a - a)*(-((a^3 - b)*sqrt
(b/(a^9 - 2*a^6*b + a^3*b^2)) - 1)/(a^3 - b))^(1/3)*log(2*((sqrt(-3)*a^3*x + a^3*x + (sqrt(-3)*(a^6 - a^3*b)*x
+ (a^6 - a^3*b)*x)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)))*(-((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) - 1)/(a^
3 - b))^(2/3) + 2*(a^3*x^3 - b*x^2)^(1/3))/x) - (sqrt(-3)*a + a)*(-((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)
) - 1)/(a^3 - b))^(1/3)*log(-2*((sqrt(-3)*a^3*x - a^3*x + (sqrt(-3)*(a^6 - a^3*b)*x - (a^6 - a^3*b)*x)*sqrt(b/
(a^9 - 2*a^6*b + a^3*b^2)))*(-((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) - 1)/(a^3 - b))^(2/3) - 2*(a^3*x^3
- b*x^2)^(1/3))/x) + 2*a*(-((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) - 1)/(a^3 - b))^(1/3)*log(-2*((a^3*x +
(a^6 - a^3*b)*x*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)))*(-((a^3 - b)*sqrt(b/(a^9 - 2*a^6*b + a^3*b^2)) - 1)/(a^3 -
b))^(2/3) - (a^3*x^3 - b*x^2)^(1/3))/x) - 2*sqrt(3)*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 - b*x^2)^(1/
3))/(a*x)) - 2*log(-(a*x - (a^3*x^3 - b*x^2)^(1/3))/x) + log((a^2*x^2 + (a^3*x^3 - b*x^2)^(1/3)*a*x + (a^3*x^3
- b*x^2)^(2/3))/x^2))/a

Sympy [N/A]

Not integrable

Time = 2.84 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.16 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {a^{3} x^{2} + b}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{2} - b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {a^{3} x^{2} + b}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} - b\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {a^{3} x^{2} + b}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} - b\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.22 \[ \int \frac {b+a^3 x^2}{\left (-b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int -\frac {a^3\,x^2+b}{\left (b-a^3\,x^2\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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