Integrand size = 32, antiderivative size = 81 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {\text {RootSum}\left [a^9-a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a^3 x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b} \]
Time = 0.00 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\frac {x^{2/3} \sqrt [3]{-b+a^3 x} \text {RootSum}\left [a^9-a^3 b^2-3 a^6 \text {$\#$1}^3+3 a^3 \text {$\#$1}^6-\text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a^3 x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 b \sqrt [3]{x^2 \left (-b+a^3 x\right )}} \]
(x^(2/3)*(-b + a^3*x)^(1/3)*RootSum[a^9 - a^3*b^2 - 3*a^6*#1^3 + 3*a^3*#1^ 6 - #1^9 & , (-Log[x^(1/3)] + Log[(-b + a^3*x)^(1/3) - x^(1/3)*#1])/#1 & ] )/(3*b*(x^2*(-b + a^3*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(3278\) vs. \(2(81)=162\).
Time = 5.82 (sec) , antiderivative size = 3278, normalized size of antiderivative = 40.47, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2467, 25, 2035, 7276, 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.
| \(\displaystyle \int \frac {1}{\left (a^3 x^3-b\right ) \sqrt [3]{a^3 x^3-b x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a^3 x-b} \int -\frac {1}{x^{2/3} \sqrt [3]{a^3 x-b} \left (b-a^3 x^3\right )}dx}{\sqrt [3]{a^3 x^3-b x^2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {x^{2/3} \sqrt [3]{a^3 x-b} \int \frac {1}{x^{2/3} \sqrt [3]{a^3 x-b} \left (b-a^3 x^3\right )}dx}{\sqrt [3]{a^3 x^3-b x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \int \frac {1}{\sqrt [3]{a^3 x-b} \left (b-a^3 x^3\right )}d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3-b x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \int \left (\frac {1}{9 b^{8/9} \left (\sqrt [9]{b}-\sqrt [3]{a} \sqrt [3]{x}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left (\sqrt [9]{-1} \sqrt [3]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left (\sqrt [9]{b}-(-1)^{2/9} \sqrt [3]{a} \sqrt [3]{x}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left (\sqrt [3]{-1} \sqrt [3]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left (\sqrt [9]{b}-(-1)^{4/9} \sqrt [3]{a} \sqrt [3]{x}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left ((-1)^{5/9} \sqrt [3]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left (\sqrt [9]{b}-(-1)^{2/3} \sqrt [3]{a} \sqrt [3]{x}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left ((-1)^{7/9} \sqrt [3]{a} \sqrt [3]{x}+\sqrt [9]{b}\right ) \sqrt [3]{a^3 x-b}}+\frac {1}{9 b^{8/9} \left (\sqrt [9]{b}-(-1)^{8/9} \sqrt [3]{a} \sqrt [3]{x}\right ) \sqrt [3]{a^3 x-b}}\right )d\sqrt [3]{x}}{\sqrt [3]{a^3 x^3-b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 x^{2/3} \sqrt [3]{a^3 x-b} \left (\frac {(-1)^{2/3} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}-\frac {\sqrt [3]{-1} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}+\frac {\sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}-\frac {(-1)^{7/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{-1} a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}+\frac {(-1)^{4/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{-1} a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}-\frac {\sqrt [9]{-1} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},-\frac {\sqrt [3]{-1} a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}+\frac {(-1)^{8/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {(-1)^{2/3} a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}-\frac {(-1)^{5/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {(-1)^{2/3} a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}+\frac {(-1)^{2/9} \sqrt [3]{a} x^{2/3} \sqrt [3]{1-\frac {a^3 x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},1,\frac {1}{3},\frac {5}{3},\frac {(-1)^{2/3} a x}{\sqrt [3]{b}},\frac {a^3 x}{b}\right )}{18 b^{10/9} \sqrt [3]{a^3 x-b}}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b}+\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}}{\sqrt [3]{a^3 x-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b}-\frac {(-1)^{2/3} \arctan \left (\frac {\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{a^2-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}+\frac {\sqrt [3]{-1} \arctan \left (\frac {\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{a^2-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {\arctan \left (\frac {\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{a^2-b^{2/3}}}+\sqrt [9]{b}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {(-1)^{7/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}+\frac {(-1)^{4/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}-\frac {\sqrt [9]{-1} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}+\frac {(-1)^{8/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{b^{2/3}-(-1)^{2/3} a^2}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}-\frac {(-1)^{5/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{b^{2/3}-(-1)^{2/3} a^2}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}+\frac {(-1)^{2/9} \arctan \left (\frac {\sqrt [9]{b}-\frac {2 \sqrt [3]{a^3 x-b}}{\sqrt [3]{b^{2/3}-(-1)^{2/3} a^2}}}{\sqrt {3} \sqrt [9]{b}}\right )}{9 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}+\frac {(-1)^{2/3} \log \left (\sqrt [3]{b}-a x\right )}{54 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{b}-a x\right )}{54 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}+\frac {2 \log \left (\sqrt [3]{b}-a x\right )}{27 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {(-1)^{8/9} \log \left (\sqrt [3]{-1} a x+\sqrt [3]{b}\right )}{54 \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}+\frac {(-1)^{5/9} \log \left (\sqrt [3]{-1} a x+\sqrt [3]{b}\right )}{54 \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}-\frac {(-1)^{2/9} \log \left (\sqrt [3]{-1} a x+\sqrt [3]{b}\right )}{54 \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}+\frac {\log \left (\sqrt [3]{-1} a x+\sqrt [3]{b}\right )}{18 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b}+\frac {(-1)^{7/9} \log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{54 \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}-\frac {(-1)^{4/9} \log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{54 \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}+\frac {\sqrt [9]{-1} \log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{54 \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}+\frac {\log \left (\sqrt [3]{b}-(-1)^{2/3} a x\right )}{18 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b}-\frac {(-1)^{2/3} \log \left (\sqrt [3]{a^2-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}+\frac {\sqrt [3]{-1} \log \left (\sqrt [3]{a^2-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {\log \left (\sqrt [3]{a^2-b^{2/3}} \sqrt [9]{b}-\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-b^{2/3}} b}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2+\sqrt [3]{-1} b^{2/3}} b}-\frac {\log \left (\sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} \sqrt [3]{x}-\sqrt [3]{a^3 x-b}\right )}{6 \sqrt [3]{a} \sqrt [3]{a^2-(-1)^{2/3} b^{2/3}} b}-\frac {(-1)^{7/9} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}}+\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}+\frac {(-1)^{4/9} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}}+\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}-\frac {\sqrt [9]{-1} \log \left (\sqrt [9]{b} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}}+\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{\sqrt [3]{-1} a^2+b^{2/3}} b}+\frac {(-1)^{8/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2}+\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}-\frac {(-1)^{5/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2}+\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}+\frac {(-1)^{2/9} \log \left (\sqrt [9]{b} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2}+\sqrt [3]{a^3 x-b}\right )}{18 \sqrt [3]{a} \sqrt [3]{b^{2/3}-(-1)^{2/3} a^2} b}\right )}{\sqrt [3]{a^3 x^3-b x^2}}\) |
(-3*x^(2/3)*(-b + a^3*x)^(1/3)*((a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*App ellF1[2/3, 1, 1/3, 5/3, (a*x)/b^(1/3), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3* x)^(1/3)) - ((-1)^(1/3)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3 , 1, 1/3, 5/3, (a*x)/b^(1/3), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3*x)^(1/3)) + ((-1)^(2/3)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, (a*x)/b^(1/3), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3*x)^(1/3)) - ((-1)^ (1/9)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, -(( (-1)^(1/3)*a*x)/b^(1/3)), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3*x)^(1/3)) + ( (-1)^(4/9)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3 , -(((-1)^(1/3)*a*x)/b^(1/3)), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3*x)^(1/3) ) - ((-1)^(7/9)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3 , 5/3, -(((-1)^(1/3)*a*x)/b^(1/3)), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3*x)^ (1/3)) + ((-1)^(2/9)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3, 1 , 1/3, 5/3, ((-1)^(2/3)*a*x)/b^(1/3), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3*x )^(1/3)) - ((-1)^(5/9)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/3, 1, 1/3, 5/3, ((-1)^(2/3)*a*x)/b^(1/3), (a^3*x)/b])/(18*b^(10/9)*(-b + a^3 *x)^(1/3)) + ((-1)^(8/9)*a^(1/3)*x^(2/3)*(1 - (a^3*x)/b)^(1/3)*AppellF1[2/ 3, 1, 1/3, 5/3, ((-1)^(2/3)*a*x)/b^(1/3), (a^3*x)/b])/(18*b^(10/9)*(-b + a ^3*x)^(1/3)) + ArcTan[(1 + (2*a^(1/3)*(a^2 - b^(2/3))^(1/3)*x^(1/3))/(-b + a^3*x)^(1/3))/Sqrt[3]]/(3*Sqrt[3]*a^(1/3)*(a^2 - b^(2/3))^(1/3)*b) + A...
3.11.75.3.1 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_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.00 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.88
| method | result | size |
| pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{9}-3 a^{3} \textit {\_Z}^{6}+3 a^{6} \textit {\_Z}^{3}-a^{9}+a^{3} b^{2}\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a^{3} x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}}{3 b}\) | \(71\) |
1/3*sum(ln((-_R*x+(x^2*(a^3*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^9-3*_Z^6*a^3+3 *_Z^3*a^6-a^9+a^3*b^2))/b
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 1.03 (sec) , antiderivative size = 24170, normalized size of antiderivative = 298.40 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\text {Too large to display} \]
Not integrable
Time = 1.67 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{x^{2} \left (a^{3} x - b\right )} \left (a^{3} x^{3} - b\right )}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - b\right )}} \,d x } \]
Not integrable
Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=\int { \frac {1}{{\left (a^{3} x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (a^{3} x^{3} - b\right )}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.41 \[ \int \frac {1}{\left (-b+a^3 x^3\right ) \sqrt [3]{-b x^2+a^3 x^3}} \, dx=-\int \frac {1}{\left (b-a^3\,x^3\right )\,{\left (a^3\,x^3-b\,x^2\right )}^{1/3}} \,d x \]