Integrand size = 17, antiderivative size = 183 \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{b}}+\frac {x \left (c x^n\right )^{-1/n} \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}-\frac {x \left (c x^n\right )^{-1/n} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{6 a^{2/3} \sqrt [3]{b}} \]
1/3*x*ln(a^(1/3)+b^(1/3)*(c*x^n)^(1/n))/a^(2/3)/b^(1/3)/((c*x^n)^(1/n))-1/ 6*x*ln(a^(2/3)-a^(1/3)*b^(1/3)*(c*x^n)^(1/n)+b^(2/3)*(c*x^n)^(2/n))/a^(2/3 )/b^(1/3)/((c*x^n)^(1/n))-1/3*x*arctan(1/3*(a^(1/3)-2*b^(1/3)*(c*x^n)^(1/n ))/a^(1/3)*3^(1/2))/a^(2/3)/b^(1/3)/((c*x^n)^(1/n))*3^(1/2)
Time = 0.26 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73 \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \left (2 \sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )+\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )\right )}{6 a^{2/3} \sqrt [3]{b}} \]
-1/6*(x*(2*Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*(c*x^n)^n^(-1))/a^(1/3))/Sqrt[3] ] - 2*Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)] + Log[a^(2/3) - a^(1/3)*b^(1/3 )*(c*x^n)^n^(-1) + b^(2/3)*(c*x^n)^(2/n)]))/(a^(2/3)*b^(1/3)*(c*x^n)^n^(-1 ))
Time = 0.30 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {786, 750, 16, 1142, 25, 27, 1082, 217, 1103}
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.
| \(\displaystyle \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx\) |
| \(\Big \downarrow \) 786 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \int \frac {1}{b \left (c x^n\right )^{3/n}+a}d\left (c x^n\right )^{\frac {1}{n}}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+\sqrt [3]{a}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}+\frac {3 \int \frac {1}{-\left (c x^n\right )^{2/n}-3}d\left (1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{b^{2/3} \left (c x^n\right )^{2/n}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+a^{2/3}}d\left (c x^n\right )^{\frac {1}{n}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}+b^{2/3} \left (c x^n\right )^{2/n}\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} \left (c x^n\right )^{\frac {1}{n}}\right )}{3 a^{2/3} \sqrt [3]{b}}\right )\) |
(x*(Log[a^(1/3) + b^(1/3)*(c*x^n)^n^(-1)]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3 ]*ArcTan[(1 - (2*b^(1/3)*(c*x^n)^n^(-1))/a^(1/3))/Sqrt[3]])/b^(1/3)) - Log [a^(2/3) - a^(1/3)*b^(1/3)*(c*x^n)^n^(-1) + b^(2/3)*(c*x^n)^(2/n)]/(2*b^(1 /3)))/(3*a^(2/3))))/(c*x^n)^n^(-1)
3.31.43.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> Simp[x/(c*x^q )^(1/q) Subst[Int[(a + b*x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{ a, b, c, n, p, q}, x] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.22 (sec) , antiderivative size = 821, normalized size of antiderivative = 4.49
1/3/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n )+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)/(a/b/(c^(3/n))/((x^n)^(3/n)) *x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-c sgn(I*c*x^n))/n))^(2/3)*ln(x+(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*P i*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)) ^(1/3))-1/6/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-cs gn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)/(a/b/(c^(3/n))/((x^n )^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csg n(I*c)-csgn(I*c*x^n))/n))^(2/3)*ln(x^2-(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*ex p(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c *x^n))/n))^(1/3)*x+(a/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c *x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n))^(2/3))+1/ 3/b/(c^(3/n))/((x^n)^(3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+ csgn(I*c*x^n))*(csgn(I*c)-csgn(I*c*x^n))/n)/(a/b/(c^(3/n))/((x^n)^(3/n))*x ^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn(I*c)-csg n(I*c*x^n))/n))^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b/(c^(3/n))/((x^n)^ (3/n))*x^3*exp(-3/2*I*Pi*csgn(I*c*x^n)*(-csgn(I*x^n)+csgn(I*c*x^n))*(csgn( I*c)-csgn(I*c*x^n))/n))^(1/3)*x-1))
Time = 0.30 (sec) , antiderivative size = 549, normalized size of antiderivative = 3.00 \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} a b c^{\frac {3}{n}} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \log \left (\frac {2 \, a b c^{\frac {3}{n}} x^{3} - 3 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b c^{\frac {3}{n}} x^{2} + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{b c^{\frac {3}{n}} x^{3} + a}\right ) - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b c^{\frac {3}{n}}}, \frac {6 \, \sqrt {\frac {1}{3}} a b c^{\frac {3}{n}} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}}}{b c^{\frac {3}{n}}}}}{a^{2}}\right ) - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x^{2} - \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}} \log \left (a b c^{\frac {3}{n}} x + \left (a^{2} b c^{\frac {3}{n}}\right )^{\frac {2}{3}}\right )}{6 \, a^{2} b c^{\frac {3}{n}}}\right ] \]
[1/6*(3*sqrt(1/3)*a*b*c^(3/n)*sqrt(-(a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))*log ((2*a*b*c^(3/n)*x^3 - 3*(a^2*b*c^(3/n))^(1/3)*a*x - a^2 + 3*sqrt(1/3)*(2*a *b*c^(3/n)*x^2 + (a^2*b*c^(3/n))^(2/3)*x - (a^2*b*c^(3/n))^(1/3)*a)*sqrt(- (a^2*b*c^(3/n))^(1/3)/(b*c^(3/n))))/(b*c^(3/n)*x^3 + a)) - (a^2*b*c^(3/n)) ^(2/3)*log(a*b*c^(3/n)*x^2 - (a^2*b*c^(3/n))^(2/3)*x + (a^2*b*c^(3/n))^(1/ 3)*a) + 2*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x + (a^2*b*c^(3/n))^(2/3)) )/(a^2*b*c^(3/n)), 1/6*(6*sqrt(1/3)*a*b*c^(3/n)*sqrt((a^2*b*c^(3/n))^(1/3) /(b*c^(3/n)))*arctan(sqrt(1/3)*(2*(a^2*b*c^(3/n))^(2/3)*x - (a^2*b*c^(3/n) )^(1/3)*a)*sqrt((a^2*b*c^(3/n))^(1/3)/(b*c^(3/n)))/a^2) - (a^2*b*c^(3/n))^ (2/3)*log(a*b*c^(3/n)*x^2 - (a^2*b*c^(3/n))^(2/3)*x + (a^2*b*c^(3/n))^(1/3 )*a) + 2*(a^2*b*c^(3/n))^(2/3)*log(a*b*c^(3/n)*x + (a^2*b*c^(3/n))^(2/3))) /(a^2*b*c^(3/n))]
\[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=\int \frac {1}{a + b \left (c x^{n}\right )^{\frac {3}{n}}}\, dx \]
\[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=\int { \frac {1}{\left (c x^{n}\right )^{\frac {3}{n}} b + a} \,d x } \]
\[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=\int { \frac {1}{\left (c x^{n}\right )^{\frac {3}{n}} b + a} \,d x } \]
Timed out. \[ \int \frac {1}{a+b \left (c x^n\right )^{3/n}} \, dx=\int \frac {1}{a+b\,{\left (c\,x^n\right )}^{3/n}} \,d x \]