\(\int \frac {1}{(b+a^3 x^2) \sqrt [3]{-b x^2+a^3 x^3}} \, dx\) [876]

Optimal result

Integrand size = 30, antiderivative size = 67 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {\text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b} \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(651\) vs. \(2(67)=134\).

Time = 0.17 (sec) , antiderivative size = 651, normalized size of antiderivative = 9.72, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2081, 926, 93} \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {\sqrt {3} x^{2/3} \sqrt [3]{a^3 x-b} \arctan \left (\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt {3} \sqrt [3]{x} \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3}}+\frac {1}{\sqrt {3}}\right )}{2 b \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3} \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {b}-\sqrt {-a^3} x\right )}{4 b \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}+\frac {x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\sqrt {-a^3} x+\sqrt {b}\right )}{4 b \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3} \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{a^3-\sqrt {-a^3} \sqrt {b}} \sqrt [3]{a^3 x^3-b x^2}}-\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \log \left (\frac {\sqrt [3]{a^3 x-b}}{\sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3}}-\sqrt [3]{x}\right )}{4 b \sqrt [3]{\sqrt {-a^3} \sqrt {b}+a^3} \sqrt [3]{a^3 x^3-b x^2}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.57 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \text {RootSum}\left [a^6+a^3 b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{2 b \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \]

[In]

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

[Out]

-1/2*(x^(2/3)*(-b + a^3*x)^(1/3)*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 & ])/(b*(x^2*(-b + a^3*x))^(1/3))

Maple [N/A] (verified)

Time = 1.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+a^{3} b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{2 b}\)	\(59\)

[In]

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

[Out]

-1/2*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))/b

Fricas [C] (verification not implemented)

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

Time = 0.30 (sec) , antiderivative size = 1372, normalized size of antiderivative = 20.48 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*(sqrt(-3) - 1)*(-((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) + 1)/(a^3*b^3 + b^4))^(1/3)*log
(1/2*((sqrt(-3)*a^3*b^2*x + a^3*b^2*x - (sqrt(-3)*(a^6*b^5 + a^3*b^6)*x + (a^6*b^5 + a^3*b^6)*x)*sqrt(-1/(a^9*
b^5 + 2*a^6*b^6 + a^3*b^7)))*(-((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) + 1)/(a^3*b^3 + b^4))
^(2/3) + 2*(a^3*x^3 - b*x^2)^(1/3))/x) - 1/4*(sqrt(-3) + 1)*(-((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 +
a^3*b^7)) + 1)/(a^3*b^3 + b^4))^(1/3)*log(-1/2*((sqrt(-3)*a^3*b^2*x - a^3*b^2*x - (sqrt(-3)*(a^6*b^5 + a^3*b^6
)*x - (a^6*b^5 + a^3*b^6)*x)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)))*(-((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2
*a^6*b^6 + a^3*b^7)) + 1)/(a^3*b^3 + b^4))^(2/3) - 2*(a^3*x^3 - b*x^2)^(1/3))/x) + 1/4*(sqrt(-3) - 1)*(((a^3*b
^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) - 1)/(a^3*b^3 + b^4))^(1/3)*log(1/2*((sqrt(-3)*a^3*b^2*x +
a^3*b^2*x + (sqrt(-3)*(a^6*b^5 + a^3*b^6)*x + (a^6*b^5 + a^3*b^6)*x)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)))
*(((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) - 1)/(a^3*b^3 + b^4))^(2/3) + 2*(a^3*x^3 - b*x^2)^
(1/3))/x) - 1/4*(sqrt(-3) + 1)*(((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) - 1)/(a^3*b^3 + b^4)
)^(1/3)*log(-1/2*((sqrt(-3)*a^3*b^2*x - a^3*b^2*x + (sqrt(-3)*(a^6*b^5 + a^3*b^6)*x - (a^6*b^5 + a^3*b^6)*x)*s
qrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)))*(((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) - 1)/(a^3*
b^3 + b^4))^(2/3) - 2*(a^3*x^3 - b*x^2)^(1/3))/x) + 1/2*(-((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*
b^7)) + 1)/(a^3*b^3 + b^4))^(1/3)*log(-((a^3*b^2*x - (a^6*b^5 + a^3*b^6)*x*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*
b^7)))*(-((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) + 1)/(a^3*b^3 + b^4))^(2/3) - (a^3*x^3 - b*
x^2)^(1/3))/x) + 1/2*(((a^3*b^3 + b^4)*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)) - 1)/(a^3*b^3 + b^4))^(1/3)*lo
g(-((a^3*b^2*x + (a^6*b^5 + a^3*b^6)*x*sqrt(-1/(a^9*b^5 + 2*a^6*b^6 + a^3*b^7)))*(((a^3*b^3 + b^4)*sqrt(-1/(a^
9*b^5 + 2*a^6*b^6 + a^3*b^7)) - 1)/(a^3*b^3 + b^4))^(2/3) - (a^3*x^3 - b*x^2)^(1/3))/x)

Sympy [N/A]

Not integrable

Time = 0.86 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.36 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{2} + b\right )}\, dx \]

[In]

integrate(1/(a**3*x**2+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**2 + b)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} + b\right )}} \,d x } \]

[In]

integrate(1/(a^3*x^2+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^2 + b)), x)

Giac [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{2} + b\right )}} \,d x } \]

[In]

integrate(1/(a^3*x^2+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^2 + b)), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\left (b+a^3 x^2\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {1}{\left (a^3\,x^2+b\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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