Integrand size = 26, antiderivative size = 164 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{x+x^3}}\right )}{2 c}-\frac {a \log \left (-x+\sqrt [3]{x+x^3}\right )}{2 c}+\frac {a \log \left (x^2+x \sqrt [3]{x+x^3}+\left (x+x^3\right )^{2/3}\right )}{4 c}+\frac {(-b c+a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {-\log (x)+\log \left (\sqrt [3]{x+x^3}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{6 c d} \]
Result contains complex when optimal does not.
Time = 16.52 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.17 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=-\frac {\sqrt [3]{1+\frac {1}{x^2}} x \left (3 a d \left (i \left (i+\sqrt {3}\right ) \log \left (\sqrt {2-2 i \sqrt {3}}-2 i \sqrt [3]{1+\frac {1}{x^2}}\right )+\left (-1-i \sqrt {3}\right ) \log \left (\sqrt {2+2 i \sqrt {3}}+2 i \sqrt [3]{1+\frac {1}{x^2}}\right )+2 \log \left (-1+\sqrt [3]{1+\frac {1}{x^2}}\right )\right )+2 (b c-a d) \text {RootSum}\left [c-d+3 d \text {$\#$1}^3-3 d \text {$\#$1}^6+d \text {$\#$1}^9\&,\frac {\log \left (\sqrt [3]{1+\frac {1}{x^2}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ]\right )}{12 c d \sqrt [3]{x+x^3}} \]
-1/12*((1 + x^(-2))^(1/3)*x*(3*a*d*(I*(I + Sqrt[3])*Log[Sqrt[2 - (2*I)*Sqr t[3]] - (2*I)*(1 + x^(-2))^(1/3)] + (-1 - I*Sqrt[3])*Log[Sqrt[2 + (2*I)*Sq rt[3]] + (2*I)*(1 + x^(-2))^(1/3)] + 2*Log[-1 + (1 + x^(-2))^(1/3)]) + 2*( b*c - a*d)*RootSum[c - d + 3*d*#1^3 - 3*d*#1^6 + d*#1^9 & , Log[(1 + x^(-2 ))^(1/3) - #1]/#1 & ]))/(c*d*(x + x^3)^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(3465\) vs. \(2(164)=328\).
Time = 5.98 (sec) , antiderivative size = 3465, normalized size of antiderivative = 21.13, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2467, 2035, 7266, 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.
| \(\displaystyle \int \frac {a x^6+b}{\sqrt [3]{x^3+x} \left (c x^6+d\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {a x^6+b}{\sqrt [3]{x} \sqrt [3]{x^2+1} \left (c x^6+d\right )}dx}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {\sqrt [3]{x} \left (a x^6+b\right )}{\sqrt [3]{x^2+1} \left (c x^6+d\right )}d\sqrt [3]{x}}{\sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7266 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \frac {a x^3+b}{\sqrt [3]{x+1} \left (c x^3+d\right )}dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \int \left (\frac {a}{c \sqrt [3]{x+1}}+\frac {b c-a d}{c \sqrt [3]{x+1} \left (c x^3+d\right )}\right )dx^{2/3}}{2 \sqrt [3]{x^3+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sqrt [3]{x} \sqrt [3]{x^2+1} \left (-\frac {(-1)^{2/3} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}+\frac {\sqrt [3]{-1} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}-\frac {(b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {\sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}+\frac {(-1)^{7/9} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}-\frac {(-1)^{4/9} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}+\frac {\sqrt [9]{-1} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,\frac {\sqrt [3]{-1} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}-\frac {(-1)^{8/9} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}+\frac {(-1)^{5/9} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}-\frac {(-1)^{2/9} (b c-a d) x^{2/3} \operatorname {AppellF1}\left (\frac {2}{3},\frac {1}{3},1,\frac {5}{3},-x,-\frac {(-1)^{2/3} \sqrt [3]{c} x}{\sqrt [3]{d}}\right )}{18 c^{8/9} d^{10/9}}+\frac {a \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{\sqrt {3} c}-\frac {(b c-a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x+1}}}{\sqrt {3}}\right )}{3 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \arctan \left (\frac {\frac {2 \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d} \sqrt [3]{x+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3} c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{2/3} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [3]{-1} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{7/9} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{4/9} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [9]{-1} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{8/9} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{5/9} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{2/9} (b c-a d) \arctan \left (\frac {\frac {2 \sqrt [9]{c} \sqrt [3]{x+1}}{\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}}+1}{\sqrt {3}}\right )}{9 \sqrt {3} c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(b c-a d) \log \left (-\sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{8/9} (b c-a d) \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{5/9} (b c-a d) \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{2/9} (b c-a d) \log \left (-\sqrt [3]{c} x-(-1)^{2/3} \sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{2/3} (b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {\sqrt [3]{-1} (b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(b c-a d) \log \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (\sqrt [3]{-1} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (-(-1)^{2/3} \sqrt [3]{c} x-\sqrt [3]{d}\right )}{18 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9}}+\frac {(-1)^{7/9} (b c-a d) \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{4/9} (b c-a d) \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {\sqrt [9]{-1} (b c-a d) \log \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{54 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (-\frac {\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{x+1}\right )}{6 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(b c-a d) \log \left (\frac {\sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{x+1}\right )}{6 c \sqrt [3]{\sqrt [3]{-1} \sqrt [3]{c}+\sqrt [3]{d}} d^{8/9}}-\frac {(b c-a d) \log \left (\frac {\sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} x^{2/3}}{\sqrt [9]{d}}-\sqrt [3]{x+1}\right )}{6 c \sqrt [3]{\sqrt [3]{d}-(-1)^{2/3} \sqrt [3]{c}} d^{8/9}}-\frac {a \log \left (\sqrt [3]{x+1}-x^{2/3}\right )}{2 c}+\frac {(-1)^{2/3} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [3]{-1} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}+\frac {(b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-\sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{7/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{4/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}-\frac {\sqrt [9]{-1} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}+\sqrt [3]{-1} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{8/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}-\frac {(-1)^{5/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}+\frac {(-1)^{2/9} (b c-a d) \log \left (\sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}}-\sqrt [9]{c} \sqrt [3]{x+1}\right )}{18 c \sqrt [3]{\sqrt [3]{c}-(-1)^{2/3} \sqrt [3]{d}} d^{8/9}}\right )}{2 \sqrt [3]{x^3+x}}\) |
(3*x^(1/3)*(1 + x^2)^(1/3)*(-1/18*((b*c - a*d)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -((c^(1/3)*x)/d^(1/3))])/(c^(8/9)*d^(10/9)) + ((-1)^(1/3)*(b*c - a*d)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -((c^(1/3)*x)/d^(1/3))])/(1 8*c^(8/9)*d^(10/9)) - ((-1)^(2/3)*(b*c - a*d)*x^(2/3)*AppellF1[2/3, 1/3, 1 , 5/3, -x, -((c^(1/3)*x)/d^(1/3))])/(18*c^(8/9)*d^(10/9)) + ((-1)^(1/9)*(b *c - a*d)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, ((-1)^(1/3)*c^(1/3)*x)/d^ (1/3)])/(18*c^(8/9)*d^(10/9)) - ((-1)^(4/9)*(b*c - a*d)*x^(2/3)*AppellF1[2 /3, 1/3, 1, 5/3, -x, ((-1)^(1/3)*c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9) ) + ((-1)^(7/9)*(b*c - a*d)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, ((-1)^( 1/3)*c^(1/3)*x)/d^(1/3)])/(18*c^(8/9)*d^(10/9)) - ((-1)^(2/9)*(b*c - a*d)* x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))]) /(18*c^(8/9)*d^(10/9)) + ((-1)^(5/9)*(b*c - a*d)*x^(2/3)*AppellF1[2/3, 1/3 , 1, 5/3, -x, -(((-1)^(2/3)*c^(1/3)*x)/d^(1/3))])/(18*c^(8/9)*d^(10/9)) - ((-1)^(8/9)*(b*c - a*d)*x^(2/3)*AppellF1[2/3, 1/3, 1, 5/3, -x, -(((-1)^(2/ 3)*c^(1/3)*x)/d^(1/3))])/(18*c^(8/9)*d^(10/9)) + (a*ArcTan[(1 + (2*x^(2/3) )/(1 + x)^(1/3))/Sqrt[3]])/(Sqrt[3]*c) - ((b*c - a*d)*ArcTan[(1 - (2*(c^(1 /3) - d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x)^(1/3)))/Sqrt[3]])/(3*Sqrt[3 ]*c*(c^(1/3) - d^(1/3))^(1/3)*d^(8/9)) + ((b*c - a*d)*ArcTan[(1 + (2*((-1) ^(1/3)*c^(1/3) + d^(1/3))^(1/3)*x^(2/3))/(d^(1/9)*(1 + x)^(1/3)))/Sqrt[3]] )/(3*Sqrt[3]*c*((-1)^(1/3)*c^(1/3) + d^(1/3))^(1/3)*d^(8/9)) + ((b*c - ...
3.23.13.3.1 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_)*(x_)^(m_.), x_Symbol] :> Simp[1/(m + 1) Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /; FreeQ[m, x] && NeQ[m, -1] && Function OfQ[x^(m + 1), u, 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 = 0.60 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.90
| method | result | size |
| pseudoelliptic | \(\frac {\left (a d -b c \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{9}-3 d \,\textit {\_Z}^{6}+3 d \,\textit {\_Z}^{3}+c -d \right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}}{x}\right )}{\textit {\_R}}\right )-3 \left (\sqrt {3}\, \arctan \left (\frac {\left (2 {\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}}-x}{x}\right )-\frac {\ln \left (\frac {{\left (x \left (x^{2}+1\right )\right )}^{\frac {2}{3}}+{\left (x \left (x^{2}+1\right )\right )}^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}\right ) d a}{6 d c}\) | \(148\) |
1/6*((a*d-b*c)*sum(ln((-_R*x+(x*(x^2+1))^(1/3))/x)/_R,_R=RootOf(_Z^9*d-3*_ Z^6*d+3*_Z^3*d+c-d))-3*(3^(1/2)*arctan(1/3*(2*(x*(x^2+1))^(1/3)+x)*3^(1/2) /x)+ln(((x*(x^2+1))^(1/3)-x)/x)-1/2*ln(((x*(x^2+1))^(2/3)+(x*(x^2+1))^(1/3 )*x+x^2)/x^2))*d*a)/d/c
Exception generated. \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Not integrable
Time = 25.63 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.15 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int \frac {a x^{6} + b}{\sqrt [3]{x \left (x^{2} + 1\right )} \left (c x^{6} + d\right )}\, dx \]
Not integrable
Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int { \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 3.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.02 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int { \frac {a x^{6} + b}{{\left (c x^{6} + d\right )} {\left (x^{3} + x\right )}^{\frac {1}{3}}} \,d x } \]
Not integrable
Time = 6.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.16 \[ \int \frac {b+a x^6}{\sqrt [3]{x+x^3} \left (d+c x^6\right )} \, dx=\int \frac {a\,x^6+b}{\left (c\,x^6+d\right )\,{\left (x^3+x\right )}^{1/3}} \,d x \]