Integrand size = 29, antiderivative size = 129 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\frac {3 \left (5 b^2+6 a b x+9 a^2 x^2\right ) \left (-b x^2+a x^3\right )^{2/3}}{40 b^3 d x^4}+\frac {c \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{-b x^2+a x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{3 d^2} \] Output:
Unintegrable
Time = 0.48 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\frac {-9 d \left (5 b^3+a b^2 x+3 a^2 b x^2-9 a^3 x^3\right )+40 b^3 c x^{8/3} \sqrt [3]{-b+a x} \text {RootSum}\left [b^3 c+a^3 d-3 a^2 d \text {$\#$1}^3+3 a d \text {$\#$1}^6-d \text {$\#$1}^9\&,\frac {-\log \left (\sqrt [3]{x}\right )+\log \left (\sqrt [3]{-b+a x}-\sqrt [3]{x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{120 b^3 d^2 x^2 \sqrt [3]{x^2 (-b+a x)}} \] Input:
Integrate[1/(x^3*(-(b*x^2) + a*x^3)^(1/3)*(d + c*x^3)),x]
Output:
(-9*d*(5*b^3 + a*b^2*x + 3*a^2*b*x^2 - 9*a^3*x^3) + 40*b^3*c*x^(8/3)*(-b + a*x)^(1/3)*RootSum[b^3*c + a^3*d - 3*a^2*d*#1^3 + 3*a*d*#1^6 - d*#1^9 & , (-Log[x^(1/3)] + Log[(-b + a*x)^(1/3) - x^(1/3)*#1])/#1 & ])/(120*b^3*d^2 *x^2*(x^2*(-b + a*x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(3566\) vs. \(2(129)=258\).
Time = 7.08 (sec) , antiderivative size = 3566, normalized size of antiderivative = 27.64, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2467, 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.
| \(\displaystyle \int \frac {1}{x^3 \sqrt [3]{a x^3-b x^2} \left (c x^3+d\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {x^{2/3} \sqrt [3]{a x-b} \int \frac {1}{x^{11/3} \sqrt [3]{a x-b} \left (c x^3+d\right )}dx}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a x-b} \int \frac {1}{x^3 \sqrt [3]{a x-b} \left (c x^3+d\right )}d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a x-b} \int \left (\frac {1}{d x^3 \sqrt [3]{a x-b}}-\frac {c}{d \sqrt [3]{a x-b} \left (c x^3+d\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{a x^3-b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 x^{2/3} \sqrt [3]{a x-b} \left (\frac {9 (a x-b)^{2/3} a^2}{40 b^3 d x^{2/3}}+\frac {3 (a x-b)^{2/3} a}{20 b^2 d x^{5/3}}+\frac {(-1)^{2/3} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}-\frac {\sqrt [3]{-1} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}+\frac {c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}-\frac {(-1)^{7/9} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}+\frac {(-1)^{4/9} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}-\frac {\sqrt [9]{-1} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}+\frac {(-1)^{8/9} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}-\frac {(-1)^{5/9} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}+\frac {(-1)^{2/9} c^{10/9} x^{2/3} \sqrt [3]{1-\frac {a x}{b}} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},\frac {a x}{b},-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 d^{19/9} \sqrt [3]{a x-b}}+\frac {c \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt [3]{x}}{\sqrt [9]{d} \sqrt [3]{a x-b}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{17/9}}+\frac {c \arctan \left (\frac {1-\frac {2 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} \sqrt [3]{x}}{\sqrt [9]{d} \sqrt [3]{a x-b}}}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{17/9}}-\frac {c \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} \sqrt [3]{x}}{\sqrt [9]{d} \sqrt [3]{a x-b}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {(-1)^{2/3} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {\sqrt [3]{-1} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{7/9} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}+\frac {(-1)^{4/9} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}-\frac {\sqrt [9]{-1} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}+\frac {(-1)^{8/9} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{5/9} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {(-1)^{2/9} c \arctan \left (\frac {1-\frac {2 \sqrt [9]{c} \sqrt [3]{a x-b}}{\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}}}{\sqrt {3}}\right )}{9 \sqrt {3} \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {c \log \left (-\sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{8/9} c \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {(-1)^{5/9} c \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{2/9} c \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{2/3} c \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {\sqrt [3]{-1} c \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {c \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {c \log \left (\sqrt [3]{-1} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{17/9}}+\frac {c \log \left (-(-1)^{2/3} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{17/9}}+\frac {(-1)^{7/9} c \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}-\frac {(-1)^{4/9} c \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}+\frac {\sqrt [9]{-1} c \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}-\frac {c \log \left (-\frac {\sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} \sqrt [3]{x}}{\sqrt [9]{d}}-\sqrt [3]{a x-b}\right )}{6 \sqrt [3]{\sqrt [3]{-1} b \sqrt [3]{c}-a \sqrt [3]{d}} d^{17/9}}-\frac {c \log \left (-\frac {\sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} \sqrt [3]{x}}{\sqrt [9]{d}}-\sqrt [3]{a x-b}\right )}{6 \sqrt [3]{-\sqrt [3]{d} a-(-1)^{2/3} b \sqrt [3]{c}} d^{17/9}}+\frac {c \log \left (\frac {\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} \sqrt [3]{x}}{\sqrt [9]{d}}-\sqrt [3]{a x-b}\right )}{6 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {(-1)^{2/3} c \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {\sqrt [3]{-1} c \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {c \log \left (\sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{\sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{7/9} c \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}+\frac {(-1)^{4/9} c \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}-\frac {\sqrt [9]{-1} c \log \left (\sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{b \sqrt [3]{c}-\sqrt [3]{-1} a \sqrt [3]{d}} d^{17/9}}+\frac {(-1)^{8/9} c \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}-\frac {(-1)^{5/9} c \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {(-1)^{2/9} c \log \left (\sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}}+\sqrt [9]{c} \sqrt [3]{a x-b}\right )}{18 \sqrt [3]{(-1)^{2/3} \sqrt [3]{d} a+b \sqrt [3]{c}} d^{17/9}}+\frac {(a x-b)^{2/3}}{8 b d x^{8/3}}\right )}{\sqrt [3]{a x^3-b x^2}}\) |
Input:
Int[1/(x^3*(-(b*x^2) + a*x^3)^(1/3)*(d + c*x^3)),x]
Output:
(3*x^(2/3)*(-b + a*x)^(1/3)*((-b + a*x)^(2/3)/(8*b*d*x^(8/3)) + (3*a*(-b + a*x)^(2/3))/(20*b^2*d*x^(5/3)) + (9*a^2*(-b + a*x)^(2/3))/(40*b^3*d*x^(2/ 3)) + (c^(10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a* x)/b, -((c^(1/3)*x)/d^(1/3))])/(18*d^(19/9)*(-b + a*x)^(1/3)) - ((-1)^(1/3 )*c^(10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, -((c^(1/3)*x)/d^(1/3))])/(18*d^(19/9)*(-b + a*x)^(1/3)) + ((-1)^(2/3)*c^( 10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, -((c ^(1/3)*x)/d^(1/3))])/(18*d^(19/9)*(-b + a*x)^(1/3)) - ((-1)^(1/9)*c^(10/9) *x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, ((-1)^(1/ 3)*c^(1/3)*x)/d^(1/3)])/(18*d^(19/9)*(-b + a*x)^(1/3)) + ((-1)^(4/9)*c^(10 /9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, ((-1)^ (1/3)*c^(1/3)*x)/d^(1/3)])/(18*d^(19/9)*(-b + a*x)^(1/3)) - ((-1)^(7/9)*c^ (10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, ((- 1)^(1/3)*c^(1/3)*x)/d^(1/3)])/(18*d^(19/9)*(-b + a*x)^(1/3)) + ((-1)^(2/9) *c^(10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a*x)/b, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*d^(19/9)*(-b + a*x)^(1/3)) - ((-1) ^(5/9)*c^(10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, (a* x)/b, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*d^(19/9)*(-b + a*x)^(1/3)) + ((-1)^(8/9)*c^(10/9)*x^(2/3)*(1 - (a*x)/b)^(1/3)*AppellF1[2/3, 1/3, 1, 5/ 3, (a*x)/b, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*d^(19/9)*(-b + a*x)...
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 = 1.00 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {40 c \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 a d \,\textit {\_Z}^{6}+3 a^{2} d \,\textit {\_Z}^{3}-a^{3} d -b^{3} c \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a x -b \right )\right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right ) b^{3} x^{4}+81 d \left (x^{2} \left (a x -b \right )\right )^{\frac {2}{3}} \left (a^{2} x^{2}+\frac {2}{3} a b x +\frac {5}{9} b^{2}\right )}{120 x^{4} b^{3} d^{2}}\) | \(121\) |
Input:
int(1/x^3/(a*x^3-b*x^2)^(1/3)/(c*x^3+d),x,method=_RETURNVERBOSE)
Output:
1/120*(40*c*sum(ln((-_R*x+(x^2*(a*x-b))^(1/3))/x)/_R,_R=RootOf(_Z^9*d-3*_Z ^6*a*d+3*_Z^3*a^2*d-a^3*d-b^3*c))*b^3*x^4+81*d*(x^2*(a*x-b))^(2/3)*(a^2*x^ 2+2/3*a*b*x+5/9*b^2))/x^4/b^3/d^2
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 18.40 (sec) , antiderivative size = 30549, normalized size of antiderivative = 236.81 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x^3/(a*x^3-b*x^2)^(1/3)/(c*x^3+d),x, algorithm="fricas")
Output:
Too large to include
Not integrable
Time = 3.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.19 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\int \frac {1}{x^{3} \sqrt [3]{x^{2} \left (a x - b\right )} \left (c x^{3} + d\right )}\, dx \] Input:
integrate(1/x**3/(a*x**3-b*x**2)**(1/3)/(c*x**3+d),x)
Output:
Integral(1/(x**3*(x**2*(a*x - b))**(1/3)*(c*x**3 + d)), x)
Not integrable
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\int { \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (c x^{3} + d\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a*x^3-b*x^2)^(1/3)/(c*x^3+d),x, algorithm="maxima")
Output:
integrate(1/((a*x^3 - b*x^2)^(1/3)*(c*x^3 + d)*x^3), x)
Not integrable
Time = 13.98 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\int { \frac {1}{{\left (a x^{3} - b x^{2}\right )}^{\frac {1}{3}} {\left (c x^{3} + d\right )} x^{3}} \,d x } \] Input:
integrate(1/x^3/(a*x^3-b*x^2)^(1/3)/(c*x^3+d),x, algorithm="giac")
Output:
sage0*x
Not integrable
Time = 9.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.22 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\int \frac {1}{x^3\,\left (c\,x^3+d\right )\,{\left (a\,x^3-b\,x^2\right )}^{1/3}} \,d x \] Input:
int(1/(x^3*(d + c*x^3)*(a*x^3 - b*x^2)^(1/3)),x)
Output:
int(1/(x^3*(d + c*x^3)*(a*x^3 - b*x^2)^(1/3)), x)
Not integrable
Time = 7.60 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \sqrt [3]{-b x^2+a x^3} \left (d+c x^3\right )} \, dx=\int \frac {1}{x^{\frac {20}{3}} \left (a x -b \right )^{\frac {1}{3}} c +x^{\frac {11}{3}} \left (a x -b \right )^{\frac {1}{3}} d}d x \] Input:
int(1/x^3/(a*x^3-b*x^2)^(1/3)/(c*x^3+d),x)
Output:
int(1/(x**(2/3)*(a*x - b)**(1/3)*c*x**6 + x**(2/3)*(a*x - b)**(1/3)*d*x**3 ),x)