Integrand size = 43, antiderivative size = 43 \[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\frac {2}{5} \arctan \left (x \sqrt [4]{-b x+a x^6}\right )-\frac {2}{5} \text {arctanh}\left (x \sqrt [4]{-b x+a x^6}\right ) \]
\[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx \]
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 2.13 (sec) , antiderivative size = 419, normalized size of antiderivative = 9.74, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2027, 2467, 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.
\(\displaystyle \int \frac {2 a x^8-b x^3}{\sqrt [4]{a x^6-b x} \left (a x^{10}-b x^5-1\right )} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x^3 \left (2 a x^5-b\right )}{\sqrt [4]{a x^6-b x} \left (a x^{10}-b x^5-1\right )}dx\) |
\(\Big \downarrow \) 2467 |
\(\displaystyle \frac {\sqrt [4]{x} \sqrt [4]{a x^5-b} \int \frac {x^{11/4} \left (b-2 a x^5\right )}{\sqrt [4]{a x^5-b} \left (-a x^{10}+b x^5+1\right )}dx}{\sqrt [4]{a x^6-b x}}\) |
\(\Big \downarrow \) 2035 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^5-b} \int \frac {x^{7/2} \left (b-2 a x^5\right )}{\sqrt [4]{a x^5-b} \left (-a x^{10}+b x^5+1\right )}d\sqrt [4]{x}}{\sqrt [4]{a x^6-b x}}\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^5-b} \int \left (\frac {2 a x^{17/2}}{\sqrt [4]{a x^5-b} \left (a x^{10}-b x^5-1\right )}+\frac {b x^{7/2}}{\sqrt [4]{a x^5-b} \left (-a x^{10}+b x^5+1\right )}\right )d\sqrt [4]{x}}{\sqrt [4]{a x^6-b x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4 \sqrt [4]{x} \sqrt [4]{a x^5-b} \left (\frac {2 a x^{15/4} \sqrt [4]{1-\frac {a x^5}{b}} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},\frac {2 a x^5}{b-\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \sqrt {4 a+b^2} \sqrt [4]{a x^5-b}}+\frac {2 a b x^{15/4} \sqrt [4]{1-\frac {a x^5}{b}} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},\frac {2 a x^5}{b-\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \left (-b \sqrt {4 a+b^2}+4 a+b^2\right ) \sqrt [4]{a x^5-b}}-\frac {2 a x^{15/4} \sqrt [4]{1-\frac {a x^5}{b}} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},\frac {2 a x^5}{b+\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \sqrt {4 a+b^2} \sqrt [4]{a x^5-b}}+\frac {2 a b x^{15/4} \sqrt [4]{1-\frac {a x^5}{b}} \operatorname {AppellF1}\left (\frac {3}{4},1,\frac {1}{4},\frac {7}{4},\frac {2 a x^5}{b+\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \left (b \left (\sqrt {4 a+b^2}+b\right )+4 a\right ) \sqrt [4]{a x^5-b}}\right )}{\sqrt [4]{a x^6-b x}}\) |
(4*x^(1/4)*(-b + a*x^5)^(1/4)*((2*a*x^(15/4)*(1 - (a*x^5)/b)^(1/4)*AppellF 1[3/4, 1, 1/4, 7/4, (2*a*x^5)/(b - Sqrt[4*a + b^2]), (a*x^5)/b])/(15*Sqrt[ 4*a + b^2]*(-b + a*x^5)^(1/4)) + (2*a*b*x^(15/4)*(1 - (a*x^5)/b)^(1/4)*App ellF1[3/4, 1, 1/4, 7/4, (2*a*x^5)/(b - Sqrt[4*a + b^2]), (a*x^5)/b])/(15*( 4*a + b^2 - b*Sqrt[4*a + b^2])*(-b + a*x^5)^(1/4)) - (2*a*x^(15/4)*(1 - (a *x^5)/b)^(1/4)*AppellF1[3/4, 1, 1/4, 7/4, (2*a*x^5)/(b + Sqrt[4*a + b^2]), (a*x^5)/b])/(15*Sqrt[4*a + b^2]*(-b + a*x^5)^(1/4)) + (2*a*b*x^(15/4)*(1 - (a*x^5)/b)^(1/4)*AppellF1[3/4, 1, 1/4, 7/4, (2*a*x^5)/(b + Sqrt[4*a + b^ 2]), (a*x^5)/b])/(15*(4*a + b*(b + Sqrt[4*a + b^2]))*(-b + a*x^5)^(1/4)))) /(-(b*x) + a*x^6)^(1/4)
3.6.63.3.1 Defintions of rubi rules used
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
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]
\[\int \frac {2 a \,x^{8}-b \,x^{3}}{\left (a \,x^{6}-b x \right )^{\frac {1}{4}} \left (a \,x^{10}-b \,x^{5}-1\right )}d x\]
Timed out. \[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\text {Timed out} \]
\[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\int { \frac {2 \, a x^{8} - b x^{3}}{{\left (a x^{10} - b x^{5} - 1\right )} {\left (a x^{6} - b x\right )}^{\frac {1}{4}}} \,d x } \]
\[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\int { \frac {2 \, a x^{8} - b x^{3}}{{\left (a x^{10} - b x^{5} - 1\right )} {\left (a x^{6} - b x\right )}^{\frac {1}{4}}} \,d x } \]
Timed out. \[ \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx=\int -\frac {2\,a\,x^8-b\,x^3}{{\left (a\,x^6-b\,x\right )}^{1/4}\,\left (-a\,x^{10}+b\,x^5+1\right )} \,d x \]