\(\int \frac {1}{(b+a^3 x^3) \sqrt [3]{-b x^2+a^3 x^3}} \, dx\) [1062]

Optimal result

Integrand size = 30, antiderivative size = 80 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {\text {RootSum}\left [a^9+a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(945\) vs. \(2(80)=160\).

Time = 1.23 (sec) , antiderivative size = 945, normalized size of antiderivative = 11.81, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2081, 6857, 93} \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (-a x-\sqrt [3]{b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt [3]{-1} a x-\sqrt [3]{b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (-(-1)^{2/3} a x-\sqrt [3]{b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}}-\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}}}-\sqrt [3]{x}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} b \sqrt [3]{a^3 x^3-b x^2}} \]

[In]

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

[Out]

-((x^(2/3)*(-b + a^3*x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 + b^(2/3))^(1/3)
*x^(1/3))])/(Sqrt[3]*a^(1/3)*(a^2 + b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*x^3)^(1/3))) - (x^(2/3)*(-b + a^3*x)^(1/3
)*ArcTan[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 - (-1)^(1/3)*b^(2/3))^(1/3)*x^(1/3))])/(Sqrt
[3]*a^(1/3)*(a^2 - (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*ArcTa
n[1/Sqrt[3] + (2*(-b + a^3*x)^(1/3))/(Sqrt[3]*a^(1/3)*(a^2 + (-1)^(2/3)*b^(2/3))^(1/3)*x^(1/3))])/(Sqrt[3]*a^(
1/3)*(a^2 + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*x^3)^(1/3)) + (x^(2/3)*(-b + a^3*x)^(1/3)*Log[-b^(1/3)
- a*x])/(6*a^(1/3)*(a^2 + b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*x^3)^(1/3)) + (x^(2/3)*(-b + a^3*x)^(1/3)*Log[-b^(
1/3) + (-1)^(1/3)*a*x])/(6*a^(1/3)*(a^2 - (-1)^(1/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*x^3)^(1/3)) + (x^(2/3)*(
-b + a^3*x)^(1/3)*Log[-b^(1/3) - (-1)^(2/3)*a*x])/(6*a^(1/3)*(a^2 + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a^
3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^(1/3) + (-b + a^3*x)^(1/3)/(a^(1/3)*(a^2 + b^(2/3))^(1/3))]
)/(2*a^(1/3)*(a^2 + b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^(1/3) +
(-b + a^3*x)^(1/3)/(a^(1/3)*(a^2 - (-1)^(1/3)*b^(2/3))^(1/3))])/(2*a^(1/3)*(a^2 - (-1)^(1/3)*b^(2/3))^(1/3)*b*
(-(b*x^2) + a^3*x^3)^(1/3)) - (x^(2/3)*(-b + a^3*x)^(1/3)*Log[-x^(1/3) + (-b + a^3*x)^(1/3)/(a^(1/3)*(a^2 + (-
1)^(2/3)*b^(2/3))^(1/3))])/(2*a^(1/3)*(a^2 + (-1)^(2/3)*b^(2/3))^(1/3)*b*(-(b*x^2) + a^3*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 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 {1}{x^{2/3} \sqrt [3]{-b+a^3 x} \left (b+a^3 x^3\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}{3 b^{2/3} x^{2/3} \left (-\sqrt [3]{b}-a x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{3 b^{2/3} x^{2/3} \left (-\sqrt [3]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{-b+a^3 x}}-\frac {1}{3 b^{2/3} x^{2/3} \left (-\sqrt [3]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{-b+a^3 x}}\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} \left (-\sqrt [3]{b}-a x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{3 b^{2/3} \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} \left (-\sqrt [3]{b}+\sqrt [3]{-1} a x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{3 b^{2/3} \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} \left (-\sqrt [3]{b}-(-1)^{2/3} a x\right ) \sqrt [3]{-b+a^3 x}} \, dx}{3 b^{2/3} \sqrt [3]{-b x^2+a^3 x^3}} \\ & = -\frac {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 [3]{a} \sqrt [3]{a^2+b^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {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 [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {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 [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}\right )}{\sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{b}-a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{b}+\sqrt [3]{-1} a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}+\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{b}-(-1)^{2/3} a x\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2-\sqrt [3]{-1} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}}-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \log \left (-\sqrt [3]{x}+\frac {\sqrt [3]{-b+a^3 x}}{\sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}}}\right )}{2 \sqrt [3]{a} \sqrt [3]{a^2+(-1)^{2/3} b^{2/3}} b \sqrt [3]{-b x^2+a^3 x^3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \text {RootSum}\left [a^9+a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 4.25 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.98 (sec) , antiderivative size = 22203, normalized size of antiderivative = 277.54 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [N/A]

Not integrable

Time = 1.76 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{3} + b\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} + b\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} + b\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {1}{\left (a^3\,x^3+b\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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