\(\int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} (-b^6+c x^3+a^6 x^6)} \, dx\) [787]

Optimal result

Integrand size = 50, antiderivative size = 60 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\frac {1}{3} \text {RootSum}\left [c+3 a^2 b^2 \text {$\#$1}^2+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt {-b^2 x+a^2 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Output:

Unintegrable
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\frac {1}{3} \text {RootSum}\left [c+3 a^2 b^2 \text {$\#$1}^2+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt {-b^2 x+a^2 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \] Input:

Integrate[(b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^6 + c*x^3 + a^6*x^ 
6)),x]
 

Output:

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

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(720\) vs. \(2(60)=120\).

Time = 7.70 (sec) , antiderivative size = 720, normalized size of antiderivative = 12.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2467, 25, 2035, 7279, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {a^6 x^6+b^6}{\sqrt {a^2 x^3-b^2 x} \left (a^6 x^6-b^6+c x^3\right )} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int -\frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6-c x^3\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^6+a^6 x^6}{\sqrt {x} \sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6-c x^3\right )}dx}{\sqrt {a^2 x^3-b^2 x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \frac {b^6+a^6 x^6}{\sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6-c x^3\right )}d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\)

\(\Big \downarrow \) 7279

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \int \left (\frac {2 b^6-c x^3}{\sqrt {a^2 x^2-b^2} \left (b^6-a^6 x^6-c x^3\right )}-\frac {1}{\sqrt {a^2 x^2-b^2}}\right )d\sqrt {x}}{\sqrt {a^2 x^3-b^2 x}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \left (-\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}\right )}{\sqrt {a^2 x^3-b^2 x}}\)

Input:

Int[(b^6 + a^6*x^6)/(Sqrt[-(b^2*x) + a^2*x^3]*(-b^6 + c*x^3 + a^6*x^6)),x]
 

Output:

(-2*Sqrt[x]*Sqrt[-b^2 + a^2*x^2]*(-((Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*Ellip 
ticF[ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(Sqrt[a]*Sqrt[-b^2 + a^2*x^2] 
)) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[-(((-2)^(1/3)*a*b)/(-c - 
Sqrt[4*a^6*b^6 + c^2])^(1/3)), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3* 
Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticP 
i[(2^(1/3)*a*b)/(-c - Sqrt[4*a^6*b^6 + c^2])^(1/3), ArcSin[(Sqrt[a]*Sqrt[x 
])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^ 
2*x^2)/b^2]*EllipticPi[((-1)^(2/3)*2^(1/3)*a*b)/(-c - Sqrt[4*a^6*b^6 + c^2 
])^(1/3), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a 
^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*EllipticPi[-(((-2)^(1/3)*a*b)/ 
(-c + Sqrt[4*a^6*b^6 + c^2])^(1/3)), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1 
])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 - (a^2*x^2)/b^2]*Ell 
ipticPi[(2^(1/3)*a*b)/(-c + Sqrt[4*a^6*b^6 + c^2])^(1/3), ArcSin[(Sqrt[a]* 
Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b^2 + a^2*x^2]) + (Sqrt[b]*Sqrt[1 
 - (a^2*x^2)/b^2]*EllipticPi[((-1)^(2/3)*2^(1/3)*a*b)/(-c + Sqrt[4*a^6*b^6 
 + c^2])^(1/3), ArcSin[(Sqrt[a]*Sqrt[x])/Sqrt[b]], -1])/(3*Sqrt[a]*Sqrt[-b 
^2 + a^2*x^2])))/Sqrt[-(b^2*x) + a^2*x^3]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7279

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ 
{v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su 
mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
 

Maple [N/A] (verified)

Time = 0.94 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+3 a^{2} b^{2} \textit {\_Z}^{2}+c \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {a^{2} x^{3}-b^{2} x}}{x}\right )}{\textit {\_R}}\right )}{3}\)	\(53\)
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+3 a^{2} b^{2} \textit {\_Z}^{2}+c \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\sqrt {a^{2} x^{3}-b^{2} x}}{x}\right )}{\textit {\_R}}\right )}{3}\)	\(53\)
elliptic	\(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{6} \textit {\_Z}^{6}-b^{6}+\textit {\_Z}^{3} c \right )}{\sum }\frac {\left (-2 b^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3} c \right ) \left (a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} c -a b c \underline {\hspace {1.25 ex}}\alpha +b^{2} c \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+a^{2} \underline {\hspace {1.25 ex}}\alpha ^{2} c -a b c \underline {\hspace {1.25 ex}}\alpha +b^{2} c}{b^{2} c}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{6}+c \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b^{2} c}\)	\(371\)

Input:

int((a^6*x^6+b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6-b^6+c*x^3),x,method=_RETU 
RNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

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

Time = 1.42 (sec) , antiderivative size = 11793, normalized size of antiderivative = 196.55 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate((a^6*x^6+b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6-b^6+c*x^3),x, algor 
ithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\text {Timed out} \] Input:

integrate((a**6*x**6+b**6)/(a**2*x**3-b**2*x)**(1/2)/(a**6*x**6-b**6+c*x** 
3),x)
 

Output:

Timed out
 

Maxima [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6} + c x^{3}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \] Input:

integrate((a^6*x^6+b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6-b^6+c*x^3),x, algor 
ithm="maxima")
 

Output:

integrate((a^6*x^6 + b^6)/((a^6*x^6 - b^6 + c*x^3)*sqrt(a^2*x^3 - b^2*x)), 
 x)
                                                                                    
                                                                                    
 

Giac [N/A]

Not integrable

Time = 2.18 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6} + c x^{3}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \] Input:

integrate((a^6*x^6+b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6-b^6+c*x^3),x, algor 
ithm="giac")
 

Output:

integrate((a^6*x^6 + b^6)/((a^6*x^6 - b^6 + c*x^3)*sqrt(a^2*x^3 - b^2*x)), 
 x)
 

Mupad [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int \frac {a^6\,x^6+b^6}{\sqrt {a^2\,x^3-b^2\,x}\,\left (a^6\,x^6-b^6+c\,x^3\right )} \,d x \] Input:

int((b^6 + a^6*x^6)/((a^2*x^3 - b^2*x)^(1/2)*(c*x^3 - b^6 + a^6*x^6)),x)
 

Output:

int((b^6 + a^6*x^6)/((a^2*x^3 - b^2*x)^(1/2)*(c*x^3 - b^6 + a^6*x^6)), x)
 

Reduce [N/A]

Not integrable

Time = 200.02 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int \frac {a^{6} x^{6}+b^{6}}{\sqrt {a^{2} x^{3}-b^{2} x}\, \left (a^{6} x^{6}-b^{6}+c \,x^{3}\right )}d x \] Input:

int((a^6*x^6+b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6-b^6+c*x^3),x)
 

Output:

int((a^6*x^6+b^6)/(a^2*x^3-b^2*x)^(1/2)/(a^6*x^6-b^6+c*x^3),x)