\(\int \frac {-b+a x^6}{(b+a x^6) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\) [861]

Optimal result

Integrand size = 39, antiderivative size = 65 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\frac {1}{6} \text {RootSum}\left [a^6+4 a b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b+a^3 x^3+a x^6}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [F]

\[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \]

[In]

Int[(-b + a*x^6)/((b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]

[Out]

(x*(1 + (2*Sqrt[a]*x^3)/(a^(5/2) - Sqrt[a^5 + 4*b]))^(1/3)*(1 + (2*Sqrt[a]*x^3)/(a^(5/2) + Sqrt[a^5 + 4*b]))^(
1/3)*AppellF1[1/3, 1/3, 1/3, 4/3, (-2*a*x^3)/(a^3 - Sqrt[a^6 + 4*a*b]), (-2*a*x^3)/(a^3 + Sqrt[a^6 + 4*a*b])])
/(-b + a^3*x^3 + a*x^6)^(1/3) - Sqrt[b]*Defer[Int][1/((Sqrt[b] - Sqrt[-a]*x^3)*(-b + a^3*x^3 + a*x^6)^(1/3)),
x] - Sqrt[b]*Defer[Int][1/((Sqrt[b] + Sqrt[-a]*x^3)*(-b + a^3*x^3 + a*x^6)^(1/3)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}}-\frac {2 b}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}\right ) \, dx \\ & = -\left ((2 b) \int \frac {1}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx\right )+\int \frac {1}{\sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \\ & = -\left ((2 b) \int \left (\frac {1}{2 \sqrt {b} \left (\sqrt {b}-\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}+\frac {1}{2 \sqrt {b} \left (\sqrt {b}+\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}}\right ) \, dx\right )+\frac {\left (\sqrt [3]{1+\frac {2 a x^3}{a^3-\sqrt {a} \sqrt {a^5+4 b}}} \sqrt [3]{1+\frac {2 a x^3}{a^3+\sqrt {a} \sqrt {a^5+4 b}}}\right ) \int \frac {1}{\sqrt [3]{1+\frac {2 a x^3}{a^3-\sqrt {a^6+4 a b}}} \sqrt [3]{1+\frac {2 a x^3}{a^3+\sqrt {a^6+4 a b}}}} \, dx}{\sqrt [3]{-b+a^3 x^3+a x^6}} \\ & = \frac {x \sqrt [3]{1+\frac {2 \sqrt {a} x^3}{a^{5/2}-\sqrt {a^5+4 b}}} \sqrt [3]{1+\frac {2 \sqrt {a} x^3}{a^{5/2}+\sqrt {a^5+4 b}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{3},\frac {1}{3},\frac {4}{3},-\frac {2 a x^3}{a^3-\sqrt {a^6+4 a b}},-\frac {2 a x^3}{a^3+\sqrt {a^6+4 a b}}\right )}{\sqrt [3]{-b+a^3 x^3+a x^6}}-\sqrt {b} \int \frac {1}{\left (\sqrt {b}-\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx-\sqrt {b} \int \frac {1}{\left (\sqrt {b}+\sqrt {-a} x^3\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\frac {1}{6} \text {RootSum}\left [a^6+4 a b-2 a^3 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b+a^3 x^3+a x^6}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[In]

Integrate[(-b + a*x^6)/((b + a*x^6)*(-b + a^3*x^3 + a*x^6)^(1/3)),x]

[Out]

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

Maple [N/A] (verified)

Time = 1.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-2 a^{3} \textit {\_Z}^{3}+a^{6}+4 a b \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (a \,x^{6}+a^{3} x^{3}-b \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )}{6}\)	\(58\)

[In]

int((a*x^6-b)/(a*x^6+b)/(a*x^6+a^3*x^3-b)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a*x^6-b)/(a*x^6+b)/(a*x^6+a^3*x^3-b)^(1/3),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

Sympy [N/A]

Not integrable

Time = 11.54 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.48 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int \frac {a x^{6} - b}{\left (a x^{6} + b\right ) \sqrt [3]{a^{3} x^{3} + a x^{6} - b}}\, dx \]

[In]

integrate((a*x**6-b)/(a*x**6+b)/(a*x**6+a**3*x**3-b)**(1/3),x)

[Out]

Integral((a*x**6 - b)/((a*x**6 + b)*(a**3*x**3 + a*x**6 - b)**(1/3)), x)

Maxima [N/A]

Not integrable

Time = 0.22 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int { \frac {a x^{6} - b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} + b\right )}} \,d x } \]

[In]

integrate((a*x^6-b)/(a*x^6+b)/(a*x^6+a^3*x^3-b)^(1/3),x, algorithm="maxima")

[Out]

integrate((a*x^6 - b)/((a*x^6 + a^3*x^3 - b)^(1/3)*(a*x^6 + b)), x)

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int { \frac {a x^{6} - b}{{\left (a x^{6} + a^{3} x^{3} - b\right )}^{\frac {1}{3}} {\left (a x^{6} + b\right )}} \,d x } \]

[In]

integrate((a*x^6-b)/(a*x^6+b)/(a*x^6+a^3*x^3-b)^(1/3),x, algorithm="giac")

[Out]

integrate((a*x^6 - b)/((a*x^6 + a^3*x^3 - b)^(1/3)*(a*x^6 + b)), x)

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.60 \[ \int \frac {-b+a x^6}{\left (b+a x^6\right ) \sqrt [3]{-b+a^3 x^3+a x^6}} \, dx=\int -\frac {b-a\,x^6}{\left (a\,x^6+b\right )\,{\left (a^3\,x^3+a\,x^6-b\right )}^{1/3}} \,d x \]

[In]

int(-(b - a*x^6)/((b + a*x^6)*(a*x^6 - b + a^3*x^3)^(1/3)),x)

[Out]

int(-(b - a*x^6)/((b + a*x^6)*(a*x^6 - b + a^3*x^3)^(1/3)), x)