Integrand size = 48, antiderivative size = 752 \[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=-\frac {3 \left (2-a x-2 x^2-3 a^2 x^2+a x^3+3 a^2 x^4\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{10 x^2}+\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{1-a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1-a}}-\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {\sqrt [3]{1+a} \left (-2+2 x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{\sqrt {3}}\right )}{2 \sqrt [3]{1+a}}-\frac {\log \left (1+\left (-\sqrt [3]{1-a}+\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1-a}}+\frac {\log \left (-1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}\right )}{2 \sqrt [3]{1+a}}+\frac {\log \left (1+\left (\sqrt [3]{1-a}-\sqrt [3]{1-a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1-a)^{2/3}-2 (1-a)^{2/3} x^2+(1-a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1-a}}-\frac {\log \left (1+\left (-\sqrt [3]{1+a}+\sqrt [3]{1+a} x^2\right ) \sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}+\left ((1+a)^{2/3}-2 (1+a)^{2/3} x^2+(1+a)^{2/3} x^4\right ) \left (\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}\right )^{2/3}\right )}{4 \sqrt [3]{1+a}} \]
-3/10*(3*a^2*x^4-3*a^2*x^2+a*x^3-a*x-2*x^2+2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^ 3-3*x^4-a*x+3*x^2-1))^(1/3)/x^2-1/2*3^(1/2)*arctan(-1/3*3^(1/2)+1/3*(1-a)^ (1/3)*(2*x^2-2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)*3^ (1/2))/(1-a)^(1/3)-1/2*3^(1/2)*arctan(1/3*3^(1/2)+1/3*(1+a)^(1/3)*(2*x^2-2 )*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3)*3^(1/2))/(1+a)^( 1/3)-1/2*ln(1+(-(1-a)^(1/3)+(1-a)^(1/3)*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3 -3*x^4-a*x+3*x^2-1))^(1/3))/(1-a)^(1/3)+1/2*ln(-1+(-(1+a)^(1/3)+(1+a)^(1/3 )*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(1/3))/(1+a)^(1/3 )+1/4*ln(1+((1-a)^(1/3)-(1-a)^(1/3)*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x ^4-a*x+3*x^2-1))^(1/3)+((1-a)^(2/3)-2*(1-a)^(2/3)*x^2+(1-a)^(2/3)*x^4)*(x/ (a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(2/3))/(1-a)^(1/3)-1/4*ln(1 +(-(1+a)^(1/3)+(1+a)^(1/3)*x^2)*(x/(a*x^7-3*a*x^5+x^6+3*a*x^3-3*x^4-a*x+3* x^2-1))^(1/3)+((1+a)^(2/3)-2*(1+a)^(2/3)*x^2+(1+a)^(2/3)*x^4)*(x/(a*x^7-3* a*x^5+x^6+3*a*x^3-3*x^4-a*x+3*x^2-1))^(2/3))/(1+a)^(1/3)
Result contains complex when optimal does not.
Time = 4.36 (sec) , antiderivative size = 568, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=\frac {\sqrt [3]{\frac {x}{(1+a x) \left (-1+x^2\right )^3}} \left (-1+x^2\right ) \left (-12 (1+a x) (-2+3 a x)+\frac {10 \sqrt {-6+6 i \sqrt {3}} x^{5/3} \sqrt [3]{1+a x} \arctan \left (\frac {3 \sqrt [3]{-1+a} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{-1+a} \sqrt [3]{x}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{1+a x}}\right )}{\sqrt [3]{-1+a}}+\frac {10 \sqrt {-6+6 i \sqrt {3}} x^{5/3} \sqrt [3]{1+a x} \arctan \left (\frac {3 \sqrt [3]{1+a} \sqrt [3]{x}}{\sqrt {3} \sqrt [3]{1+a} \sqrt [3]{x}-\left (3 i+\sqrt {3}\right ) \sqrt [3]{1+a x}}\right )}{\sqrt [3]{1+a}}-\frac {10 i \left (-i+\sqrt {3}\right ) x^{5/3} \sqrt [3]{1+a x} \log \left (2 \sqrt [3]{-1+a} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{1+a x}\right )}{\sqrt [3]{-1+a}}-\frac {10 i \left (-i+\sqrt {3}\right ) x^{5/3} \sqrt [3]{1+a x} \log \left (2 \sqrt [3]{1+a} \sqrt [3]{x}+\left (1+i \sqrt {3}\right ) \sqrt [3]{1+a x}\right )}{\sqrt [3]{1+a}}+\frac {5 \left (1+i \sqrt {3}\right ) x^{5/3} \sqrt [3]{1+a x} \log \left (\left (-\sqrt [3]{-1+a} \sqrt [3]{x}+\sqrt [3]{1+a x}\right ) \left (2 i \sqrt [3]{-1+a} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{1+a x}\right )\right )}{\sqrt [3]{-1+a}}+\frac {5 \left (1+i \sqrt {3}\right ) x^{5/3} \sqrt [3]{1+a x} \log \left (\left (-\sqrt [3]{1+a} \sqrt [3]{x}+\sqrt [3]{1+a x}\right ) \left (2 i \sqrt [3]{1+a} \sqrt [3]{x}+\left (i+\sqrt {3}\right ) \sqrt [3]{1+a x}\right )\right )}{\sqrt [3]{1+a}}\right )}{40 x^2} \]
((x/((1 + a*x)*(-1 + x^2)^3))^(1/3)*(-1 + x^2)*(-12*(1 + a*x)*(-2 + 3*a*x) + (10*Sqrt[-6 + (6*I)*Sqrt[3]]*x^(5/3)*(1 + a*x)^(1/3)*ArcTan[(3*(-1 + a) ^(1/3)*x^(1/3))/(Sqrt[3]*(-1 + a)^(1/3)*x^(1/3) - (3*I + Sqrt[3])*(1 + a*x )^(1/3))])/(-1 + a)^(1/3) + (10*Sqrt[-6 + (6*I)*Sqrt[3]]*x^(5/3)*(1 + a*x) ^(1/3)*ArcTan[(3*(1 + a)^(1/3)*x^(1/3))/(Sqrt[3]*(1 + a)^(1/3)*x^(1/3) - ( 3*I + Sqrt[3])*(1 + a*x)^(1/3))])/(1 + a)^(1/3) - ((10*I)*(-I + Sqrt[3])*x ^(5/3)*(1 + a*x)^(1/3)*Log[2*(-1 + a)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(1 + a*x)^(1/3)])/(-1 + a)^(1/3) - ((10*I)*(-I + Sqrt[3])*x^(5/3)*(1 + a*x)^(1 /3)*Log[2*(1 + a)^(1/3)*x^(1/3) + (1 + I*Sqrt[3])*(1 + a*x)^(1/3)])/(1 + a )^(1/3) + (5*(1 + I*Sqrt[3])*x^(5/3)*(1 + a*x)^(1/3)*Log[(-((-1 + a)^(1/3) *x^(1/3)) + (1 + a*x)^(1/3))*((2*I)*(-1 + a)^(1/3)*x^(1/3) + (I + Sqrt[3]) *(1 + a*x)^(1/3))])/(-1 + a)^(1/3) + (5*(1 + I*Sqrt[3])*x^(5/3)*(1 + a*x)^ (1/3)*Log[(-((1 + a)^(1/3)*x^(1/3)) + (1 + a*x)^(1/3))*((2*I)*(1 + a)^(1/3 )*x^(1/3) + (I + Sqrt[3])*(1 + a*x)^(1/3))])/(1 + a)^(1/3)))/(40*x^2)
Time = 0.74 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.42, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.104, Rules used = {7239, 7269, 25, 615, 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.
| \(\displaystyle \int \frac {\sqrt [3]{\frac {x}{a x^7-3 a x^5+3 a x^3-a x+x^6-3 x^4+3 x^2-1}}}{x^3} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {\sqrt [3]{\frac {x}{\left (x^2-1\right )^3 (a x+1)}}}{x^3}dx\) |
| \(\Big \downarrow \) 7269 |
| \(\displaystyle -\frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \int -\frac {1}{x^{8/3} \sqrt [3]{a x+1} \left (1-x^2\right )}dx}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \int \frac {1}{x^{8/3} \sqrt [3]{a x+1} \left (1-x^2\right )}dx}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \int \left (\frac {1}{2 x^{8/3} (x+1) \sqrt [3]{a x+1}}+\frac {1}{2 (1-x) x^{8/3} \sqrt [3]{a x+1}}\right )dx}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\left (1-x^2\right ) \sqrt [3]{a x+1} \sqrt [3]{-\frac {x}{\left (1-x^2\right )^3 (a x+1)}} \left (-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a-1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a-1}}-\frac {\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{a x+1}}{\sqrt {3} \sqrt [3]{a+1} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2 \sqrt [3]{a+1}}-\frac {3 (5-3 a) (a x+1)^{2/3}}{20 x^{2/3}}+\frac {3 (3 a+5) (a x+1)^{2/3}}{20 x^{2/3}}-\frac {3 (a x+1)^{2/3}}{5 x^{5/3}}+\frac {\log (1-x)}{4 \sqrt [3]{a+1}}+\frac {\log (x+1)}{4 \sqrt [3]{a-1}}-\frac {3 \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a-1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a-1}}-\frac {3 \log \left (\frac {\sqrt [3]{a x+1}}{\sqrt [3]{a+1}}-\sqrt [3]{x}\right )}{4 \sqrt [3]{a+1}}\right )}{\sqrt [3]{x}}\) |
((1 + a*x)^(1/3)*(-(x/((1 + a*x)*(1 - x^2)^3)))^(1/3)*(1 - x^2)*((-3*(1 + a*x)^(2/3))/(5*x^(5/3)) - (3*(5 - 3*a)*(1 + a*x)^(2/3))/(20*x^(2/3)) + (3* (5 + 3*a)*(1 + a*x)^(2/3))/(20*x^(2/3)) - (Sqrt[3]*ArcTan[1/Sqrt[3] + (2*( 1 + a*x)^(1/3))/(Sqrt[3]*(-1 + a)^(1/3)*x^(1/3))])/(2*(-1 + a)^(1/3)) - (S qrt[3]*ArcTan[1/Sqrt[3] + (2*(1 + a*x)^(1/3))/(Sqrt[3]*(1 + a)^(1/3)*x^(1/ 3))])/(2*(1 + a)^(1/3)) + Log[1 - x]/(4*(1 + a)^(1/3)) + Log[1 + x]/(4*(-1 + a)^(1/3)) - (3*Log[-x^(1/3) + (1 + a*x)^(1/3)/(-1 + a)^(1/3)])/(4*(-1 + a)^(1/3)) - (3*Log[-x^(1/3) + (1 + a*x)^(1/3)/(1 + a)^(1/3)])/(4*(1 + a)^ (1/3))))/x^(1/3)
3.32.27.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.)*(z_)^(q_.))^(p_), x_Symbol] :> Simp[ a^IntPart[p]*((a*v^m*w^n*z^q)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[ p])*z^(q*FracPart[p]))) Int[u*v^(m*p)*w^(n*p)*z^(p*q), x], x] /; FreeQ[{a , m, n, p, q}, x] && !IntegerQ[p] && !FreeQ[v, x] && !FreeQ[w, x] && !F reeQ[z, x]
\[\int \frac {\left (\frac {x}{a \,x^{7}-3 a \,x^{5}+x^{6}+3 a \,x^{3}-3 x^{4}-a x +3 x^{2}-1}\right )^{\frac {1}{3}}}{x^{3}}d x\]
Time = 0.33 (sec) , antiderivative size = 3627, normalized size of antiderivative = 4.82 \[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=\text {Too large to display} \]
[1/20*(5*sqrt(3)*(a^2 - 1)*x^2*sqrt(-1/(a - 1)^(2/3))*log(-((3*a - 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 - a)*x^5 + (a - 1)*x^ 4 - 2*(a^2 - a)*x^3 - 2*(a - 1)*x^2 + (a^2 - a)*x + a - 1)*(x/(a*x^7 - 3*a *x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x + 1)*(a - 1) ^(1/3))*sqrt(-1/(a - 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a - 1)^(1/3)*( x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/ (x + 1)) + 5*sqrt(3)*(a^2 - 1)*x^2*sqrt(-1/(a + 1)^(2/3))*log(((3*a + 2)*x + sqrt(3)*((a*x^3 - a*x + x^2 - 1)*(a + 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^ 6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) - 2*((a^2 + a)*x^5 + (a + 1) *x^4 - 2*(a^2 + a)*x^3 - 2*(a + 1)*x^2 + (a^2 + a)*x + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(2/3) + (a*x + 1)*(a + 1)^(1/3))*sqrt(-1/(a + 1)^(2/3)) - 3*(a*x^3 - a*x + x^2 - 1)*(a + 1)^(1/3 )*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + 1)/(x - 1)) - 5*(a + 1)^(2/3)*(a - 1)*x^2*log((x^2 - 1)*(a + 1)^(2/3)*(x/( a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a + 1)*x^4 - 2*(a + 1)*x^2 + a + 1)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^ 4 - a*x + 3*x^2 - 1))^(2/3) + (a + 1)^(1/3)) - 5*(a + 1)*(a - 1)^(2/3)*x^2 *log((x^2 - 1)*(a - 1)^(2/3)*(x/(a*x^7 - 3*a*x^5 + x^6 + 3*a*x^3 - 3*x^4 - a*x + 3*x^2 - 1))^(1/3) + ((a - 1)*x^4 - 2*(a - 1)*x^2 + a - 1)*(x/(a*...
\[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=\int \frac {\sqrt [3]{\frac {x}{a x^{7} - 3 a x^{5} + 3 a x^{3} - a x + x^{6} - 3 x^{4} + 3 x^{2} - 1}}}{x^{3}}\, dx \]
Integral((x/(a*x**7 - 3*a*x**5 + 3*a*x**3 - a*x + x**6 - 3*x**4 + 3*x**2 - 1))**(1/3)/x**3, x)
\[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=\int { \frac {\left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=\int { \frac {\left (\frac {x}{a x^{7} - 3 \, a x^{5} + x^{6} + 3 \, a x^{3} - 3 \, x^{4} - a x + 3 \, x^{2} - 1}\right )^{\frac {1}{3}}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt [3]{\frac {x}{-1-a x+3 x^2+3 a x^3-3 x^4-3 a x^5+x^6+a x^7}}}{x^3} \, dx=\int \frac {{\left (-\frac {x}{-a\,x^7-x^6+3\,a\,x^5+3\,x^4-3\,a\,x^3-3\,x^2+a\,x+1}\right )}^{1/3}}{x^3} \,d x \]