Integrand size = 50, antiderivative size = 60 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\frac {1}{3} \text {RootSum}\left [c+3 a^2 b^2 \text {$\#$1}^2+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt {-b^2 x+a^2 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
Time = 0.00 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\frac {1}{3} \text {RootSum}\left [c+3 a^2 b^2 \text {$\#$1}^2+\text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt {-b^2 x+a^2 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]
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
Leaf count is larger than twice the leaf count of optimal. \(720\) vs. \(2(60)=120\).
Time = 7.87 (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 |
\(\displaystyle -\frac {2 \sqrt {x} \sqrt {a^2 x^2-b^2} \left (-\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{\sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{-c-\sqrt {4 a^6 b^6+c^2}}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (-\frac {\sqrt [3]{-2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {\sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}+\frac {\sqrt {b} \sqrt {1-\frac {a^2 x^2}{b^2}} \operatorname {EllipticPi}\left (\frac {(-1)^{2/3} \sqrt [3]{2} a b}{\sqrt [3]{\sqrt {4 a^6 b^6+c^2}-c}},\arcsin \left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right ),-1\right )}{3 \sqrt {a} \sqrt {a^2 x^2-b^2}}\right )}{\sqrt {a^2 x^3-b^2 x}}\) |
(-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]
3.8.87.3.1 Defintions of rubi rules used
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]
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]
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]
Time = 1.76 (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 | \(\frac {b \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {2 \left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {\sqrt {2}}{2}\right )}{a \sqrt {a^{2} x^{3}-b^{2} x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (a^{6} \textit {\_Z}^{6}-b^{6}+c \,\textit {\_Z}^{3}\right )}{\sum }\frac {\left (-2 b^{6}+\underline {\hspace {1.25 ex}}\alpha ^{3} c \right ) \left (a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c \right ) \sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {\left (x -\frac {b}{a}\right ) a}{b}}\, \sqrt {-\frac {a x}{b}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {b}{a}\right ) a}{b}}, \frac {a^{8} \underline {\hspace {1.25 ex}}\alpha ^{5}-a^{7} \underline {\hspace {1.25 ex}}\alpha ^{4} b +a^{6} \underline {\hspace {1.25 ex}}\alpha ^{3} b^{2}-a^{5} \underline {\hspace {1.25 ex}}\alpha ^{2} b^{3}+a^{4} b^{4} \underline {\hspace {1.25 ex}}\alpha -a^{3} b^{5}+\underline {\hspace {1.25 ex}}\alpha ^{2} a^{2} c -\underline {\hspace {1.25 ex}}\alpha a b c +b^{2} c}{b^{2} c}, \frac {\sqrt {2}}{2}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{2} \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{6}+c \right ) \sqrt {x \left (a^{2} x^{2}-b^{2}\right )}}\right )}{3 b^{2} c}\) | \(371\) |
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.56 (sec) , antiderivative size = 11793, normalized size of antiderivative = 196.55 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6} + c x^{3}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
Not integrable
Time = 1.50 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int { \frac {a^{6} x^{6} + b^{6}}{{\left (a^{6} x^{6} - b^{6} + c x^{3}\right )} \sqrt {a^{2} x^{3} - b^{2} x}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.83 \[ \int \frac {b^6+a^6 x^6}{\sqrt {-b^2 x+a^2 x^3} \left (-b^6+c x^3+a^6 x^6\right )} \, dx=\int \frac {a^6\,x^6+b^6}{\sqrt {a^2\,x^3-b^2\,x}\,\left (a^6\,x^6-b^6+c\,x^3\right )} \,d x \]