\(\int \frac {1}{\sqrt [3]{-b x^2+a x^3} (-b+a x^4)} \, dx\) [1098]

Optimal result

Integrand size = 28, antiderivative size = 82 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\frac {\text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^6-4 a \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(1208\) vs. \(2(82)=164\).

Time = 1.43 (sec) , antiderivative size = 1208, normalized size of antiderivative = 14.73, number of steps used = 11, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2081, 6857, 926, 93} \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {\sqrt {3} x^{2/3} \sqrt [3]{a x-b} \arctan \left (\frac {2 \sqrt [3]{a x-b}}{\sqrt {3} \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{4 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt {-\sqrt {a}} x\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt {-\sqrt {a}} x+\sqrt [4]{b}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{b}-\sqrt [4]{a} x\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a x-b} \log \left (\sqrt [4]{a} x+\sqrt [4]{b}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}-b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [12]{a} \sqrt [3]{a^{3/4}+b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a-\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}}+\frac {3 x^{2/3} \sqrt [3]{a x-b} \log \left (\frac {\sqrt [3]{a x-b}}{\sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}}}-\sqrt [3]{x}\right )}{8 \sqrt [3]{a+\sqrt {-\sqrt {a}} b^{3/4}} b \sqrt [3]{a x^3-b x^2}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a x} \text {RootSum}\left [a^4-a b^3-4 a^3 \text {$\#$1}^3+6 a^2 \text {$\#$1}^6-4 a \text {$\#$1}^9+\text {$\#$1}^{12}\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{4 b \sqrt [3]{x^2 (-b+a x)}} \]

[In]

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

[Out]

(x^(2/3)*(-b + a*x)^(1/3)*RootSum[a^4 - a*b^3 - 4*a^3*#1^3 + 6*a^2*#1^6 - 4*a*#1^9 + #1^12 & , (-Log[x^(1/3)]
+ Log[(-b + a*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(4*b*(x^2*(-b + a*x))^(1/3))

Maple [N/A] (verified)

Time = 1.51 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{12}-4 a \,\textit {\_Z}^{9}+6 a^{2} \textit {\_Z}^{6}-4 a^{3} \textit {\_Z}^{3}+a^{4}-a \,b^{3}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{4 b}\)	\(72\)

[In]

int(1/(a*x^3-b*x^2)^(1/3)/(a*x^4-b),x,method=_RETURNVERBOSE)

[Out]

1/4*sum(ln((-_R*x+(x^2*(a*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^12-4*_Z^9*a+6*_Z^6*a^2-4*_Z^3*a^3+a^4-a*b^3))/b

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/(a*x^3-b*x^2)^(1/3)/(a*x^4-b),x, algorithm="fricas")

[Out]

Timed out

Sympy [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a x - b\right )} \left (a x^{4} - b\right )}\, dx \]

[In]

integrate(1/(a*x**3-b*x**2)**(1/3)/(a*x**4-b),x)

[Out]

Integral(1/((x**2*(a*x - b))**(1/3)*(a*x**4 - b)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} - b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(a*x^3-b*x^2)^(1/3)/(a*x^4-b),x, algorithm="maxima")

[Out]

integrate(1/((a*x^4 - b)*(a*x^3 - b*x^2)^(1/3)), x)

Giac [N/A]

Not integrable

Time = 8.31 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=\int { \frac {1}{{\left (a x^{4} - b\right )} {\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate(1/(a*x^3-b*x^2)^(1/3)/(a*x^4-b),x, algorithm="giac")

[Out]

integrate(1/((a*x^4 - b)*(a*x^3 - b*x^2)^(1/3)), x)

Mupad [N/A]

Not integrable

Time = 5.67 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.35 \[ \int \frac {1}{\sqrt [3]{-b x^2+a x^3} \left (-b+a x^4\right )} \, dx=-\int \frac {1}{\left (b-a\,x^4\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \]

[In]

int(-1/((b - a*x^4)*(a*x^3 - b*x^2)^(1/3)),x)

[Out]

-int(1/((b - a*x^4)*(a*x^3 - b*x^2)^(1/3)), x)