\(\int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx\) [2857]

Optimal result

Integrand size = 30, antiderivative size = 299 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\sqrt {3} \arctan \left (\frac {\sqrt {3} a x}{a x+2 \sqrt [3]{-b^2 x^2+a^3 x^3}}\right )-\log \left (-a x+\sqrt [3]{-b^2 x^2+a^3 x^3}\right )+\frac {1}{2} \log \left (a^2 x^2+a x \sqrt [3]{-b^2 x^2+a^3 x^3}+\left (-b^2 x^2+a^3 x^3\right )^{2/3}\right )+\frac {1}{2} \text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-a^5 \log (x)-b^3 \log (x)+a^5 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+b^3 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^3-a^2 \log \left (\sqrt [3]{-b^2 x^2+a^3 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{a^3 \text {$\#$1}^2-\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(866\) vs. \(2(299)=598\).

Time = 0.92 (sec) , antiderivative size = 866, normalized size of antiderivative = 2.90, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2081, 920, 61, 6857, 93} \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\frac {\sqrt {3} \sqrt [3]{a^3 x^3-b^2 x^2} \arctan \left (\frac {2 \sqrt [3]{x} a}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \arctan \left (\frac {2 \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {\sqrt {3} \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \arctan \left (\frac {2 \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{a^3 x-b^2}}+\frac {1}{\sqrt {3}}\right )}{2 a x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {b}-\sqrt {-a} x\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt {-a} x+\sqrt {b}\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {\sqrt [3]{a^3 x^3-b^2 x^2} \log \left (a^3 x-b^2\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}-\frac {3 \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\frac {a \sqrt [3]{x}}{\sqrt [3]{a^3 x-b^2}}-1\right )}{2 x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [3]{a^3-\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}}+\frac {3 \sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{a^3 x^3-b^2 x^2} \log \left (\sqrt [3]{a^3+\sqrt {-a} b^{3/2}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b^2}\right )}{4 a x^{2/3} \sqrt [3]{a^3 x-b^2}} \]

[In]

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

[Out]

-((Sqrt[3]*(-(b^2*x^2) + a^3*x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*a*x^(1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/(x^(
2/3)*(-b^2 + a^3*x)^(1/3))) + (Sqrt[3]*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*ArcTan[1/Sq
rt[3] + (2*(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/(2*a*x^(2/3)*(-b^2 + a^3*x
)^(1/3)) + (Sqrt[3]*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a^3 + S
qrt[-a]*b^(3/2))^(1/3)*x^(1/3))/(Sqrt[3]*(-b^2 + a^3*x)^(1/3))])/(2*a*x^(2/3)*(-b^2 + a^3*x)^(1/3)) - ((a^3 -
Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*Log[Sqrt[b] - Sqrt[-a]*x])/(4*a*x^(2/3)*(-b^2 + a^3*x)^(1
/3)) - ((a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*Log[Sqrt[b] + Sqrt[-a]*x])/(4*a*x^(2/3)*(-
b^2 + a^3*x)^(1/3)) - ((-(b^2*x^2) + a^3*x^3)^(1/3)*Log[-b^2 + a^3*x])/(2*x^(2/3)*(-b^2 + a^3*x)^(1/3)) - (3*(
-(b^2*x^2) + a^3*x^3)^(1/3)*Log[-1 + (a*x^(1/3))/(-b^2 + a^3*x)^(1/3)])/(2*x^(2/3)*(-b^2 + a^3*x)^(1/3)) + (3*
(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3)^(1/3)*Log[(a^3 - Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3) - (-b^2
+ a^3*x)^(1/3)])/(4*a*x^(2/3)*(-b^2 + a^3*x)^(1/3)) + (3*(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*(-(b^2*x^2) + a^3*x^3
)^(1/3)*Log[(a^3 + Sqrt[-a]*b^(3/2))^(1/3)*x^(1/3) - (-b^2 + a^3*x)^(1/3)])/(4*a*x^(2/3)*(-b^2 + a^3*x)^(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 920

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

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\frac {x^{4/3} \left (-b^2+a^3 x\right )^{2/3} \left (6 \sqrt {3} \arctan \left (\frac {\sqrt {3} a \sqrt [3]{x}}{a \sqrt [3]{x}+2 \sqrt [3]{-b^2+a^3 x}}\right )+6 \log \left (-a \sqrt [3]{x}+\sqrt [3]{-b^2+a^3 x}\right )-3 \log \left (a^2 x^{2/3}+a \sqrt [3]{x} \sqrt [3]{-b^2+a^3 x}+\left (-b^2+a^3 x\right )^{2/3}\right )+\text {RootSum}\left [a^6+a b^3-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-a^5 \log (x)-b^3 \log (x)+3 a^5 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )+3 b^3 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )+a^2 \log (x) \text {$\#$1}^3-3 a^2 \log \left (\sqrt [3]{-b^2+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a^3 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]\right )}{6 \left (x^2 \left (-b^2+a^3 x\right )\right )^{2/3}} \]

[In]

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

[Out]

-1/6*(x^(4/3)*(-b^2 + a^3*x)^(2/3)*(6*Sqrt[3]*ArcTan[(Sqrt[3]*a*x^(1/3))/(a*x^(1/3) + 2*(-b^2 + a^3*x)^(1/3))]
+ 6*Log[-(a*x^(1/3)) + (-b^2 + a^3*x)^(1/3)] - 3*Log[a^2*x^(2/3) + a*x^(1/3)*(-b^2 + a^3*x)^(1/3) + (-b^2 + a
^3*x)^(2/3)] + RootSum[a^6 + a*b^3 - 2*a^3*#1^3 + #1^6 & , (-(a^5*Log[x]) - b^3*Log[x] + 3*a^5*Log[(-b^2 + a^3
*x)^(1/3) - x^(1/3)*#1] + 3*b^3*Log[(-b^2 + a^3*x)^(1/3) - x^(1/3)*#1] + a^2*Log[x]*#1^3 - 3*a^2*Log[(-b^2 + a
^3*x)^(1/3) - x^(1/3)*#1]*#1^3)/(-(a^3*#1^2) + #1^5) & ]))/(x^2*(-b^2 + a^3*x))^(2/3)

Maple [N/A] (verified)

Time = 0.68 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.69

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+a \,b^{3}\right )}{\sum }\frac {\left (-\textit {\_R}^{3} a^{2}+a^{5}+b^{3}\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )}{-\textit {\_R}^{5}+\textit {\_R}^{2} a^{3}}\right )}{2}-\ln \left (\frac {-a x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}}{x}\right )+\frac {\ln \left (\frac {a^{2} x^{2}+a \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}} x +\left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a x +2 \left (x^{2} \left (a^{3} x -b^{2}\right )\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a x}\right )\)	\(206\)

[In]

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

[Out]

1/2*sum((-_R^3*a^2+a^5+b^3)*ln((-_R*x+(x^2*(a^3*x-b^2))^(1/3))/x)/(-_R^5+_R^2*a^3),_R=RootOf(_Z^6-2*_Z^3*a^3+a
^6+a*b^3))-ln((-a*x+(x^2*(a^3*x-b^2))^(1/3))/x)+1/2*ln((a^2*x^2+a*(x^2*(a^3*x-b^2))^(1/3)*x+(x^2*(a^3*x-b^2))^
(2/3))/x^2)+3^(1/2)*arctan(1/3*(a*x+2*(x^2*(a^3*x-b^2))^(1/3))*3^(1/2)/a/x)

Fricas [C] (verification not implemented)

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

Time = 0.28 (sec) , antiderivative size = 554, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=-\frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {-3} a x + a x\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (\sqrt {-3} a x - a x\right )} {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{4} \, {\left (\sqrt {-3} + 1\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (\frac {{\left (\sqrt {-3} a x + a x\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} + 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{4} \, {\left (\sqrt {-3} - 1\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {{\left (\sqrt {-3} a x - a x\right )} {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - 2 \, {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \sqrt {3} \arctan \left (\frac {\sqrt {3} a x + 2 \, \sqrt {3} {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{3 \, a x}\right ) + \frac {1}{2} \, {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {a x {\left (\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} \log \left (-\frac {a x {\left (-\sqrt {-\frac {b^{3}}{a^{5}}} + 1\right )}^{\frac {1}{3}} - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) - \log \left (-\frac {a x - {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {a^{2} x^{2} + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}} a x + {\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]

[In]

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

[Out]

-1/4*(sqrt(-3) + 1)*(sqrt(-b^3/a^5) + 1)^(1/3)*log(((sqrt(-3)*a*x + a*x)*(sqrt(-b^3/a^5) + 1)^(1/3) + 2*(a^3*x
^3 - b^2*x^2)^(1/3))/x) + 1/4*(sqrt(-3) - 1)*(sqrt(-b^3/a^5) + 1)^(1/3)*log(-((sqrt(-3)*a*x - a*x)*(sqrt(-b^3/
a^5) + 1)^(1/3) - 2*(a^3*x^3 - b^2*x^2)^(1/3))/x) - 1/4*(sqrt(-3) + 1)*(-sqrt(-b^3/a^5) + 1)^(1/3)*log(((sqrt(
-3)*a*x + a*x)*(-sqrt(-b^3/a^5) + 1)^(1/3) + 2*(a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/4*(sqrt(-3) - 1)*(-sqrt(-b^3/
a^5) + 1)^(1/3)*log(-((sqrt(-3)*a*x - a*x)*(-sqrt(-b^3/a^5) + 1)^(1/3) - 2*(a^3*x^3 - b^2*x^2)^(1/3))/x) + sqr
t(3)*arctan(1/3*(sqrt(3)*a*x + 2*sqrt(3)*(a^3*x^3 - b^2*x^2)^(1/3))/(a*x)) + 1/2*(sqrt(-b^3/a^5) + 1)^(1/3)*lo
g(-(a*x*(sqrt(-b^3/a^5) + 1)^(1/3) - (a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/2*(-sqrt(-b^3/a^5) + 1)^(1/3)*log(-(a*x
*(-sqrt(-b^3/a^5) + 1)^(1/3) - (a^3*x^3 - b^2*x^2)^(1/3))/x) - log(-(a*x - (a^3*x^3 - b^2*x^2)^(1/3))/x) + 1/2
*log((a^2*x^2 + (a^3*x^3 - b^2*x^2)^(1/3)*a*x + (a^3*x^3 - b^2*x^2)^(2/3))/x^2)

Sympy [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.07 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int \frac {\sqrt [3]{x^{2} \left (a^{3} x - b^{2}\right )}}{a x^{2} + b}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int { \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} + b} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int { \frac {{\left (a^{3} x^{3} - b^{2} x^{2}\right )}^{\frac {1}{3}}}{a x^{2} + b} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 6.81 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.10 \[ \int \frac {\sqrt [3]{-b^2 x^2+a^3 x^3}}{b+a x^2} \, dx=\int \frac {{\left (a^3\,x^3-b^2\,x^2\right )}^{1/3}}{a\,x^2+b} \,d x \]

[In]

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

[Out]

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