Integrand size = 17, antiderivative size = 98 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\frac {x}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 x}{8 a^2 \left (a+b \left (c x^n\right )^{2/n}\right )}+\frac {3 x \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {b}} \]
1/4*x/a/(a+b*(c*x^n)^(2/n))^2+3/8*x/a^2/(a+b*(c*x^n)^(2/n))+3/8*x*arctan(( c*x^n)^(1/n)*b^(1/2)/a^(1/2))/a^(5/2)/((c*x^n)^(1/n))/b^(1/2)
Time = 0.31 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\frac {x \left (\frac {\sqrt {a} \left (5 a+3 b \left (c x^n\right )^{2/n}\right )}{\left (a+b \left (c x^n\right )^{2/n}\right )^2}+\frac {3 \left (c x^n\right )^{-1/n} \arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{\sqrt {b}}\right )}{8 a^{5/2}} \]
(x*((Sqrt[a]*(5*a + 3*b*(c*x^n)^(2/n)))/(a + b*(c*x^n)^(2/n))^2 + (3*ArcTa n[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]])/(Sqrt[b]*(c*x^n)^n^(-1))))/(8*a^(5/2) )
Time = 0.20 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {786, 215, 215, 218}
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.
| \(\displaystyle \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx\) |
| \(\Big \downarrow \) 786 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \int \frac {1}{\left (b \left (c x^n\right )^{2/n}+a\right )^3}d\left (c x^n\right )^{\frac {1}{n}}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {3 \int \frac {1}{\left (b \left (c x^n\right )^{2/n}+a\right )^2}d\left (c x^n\right )^{\frac {1}{n}}}{4 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}\right )\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {3 \left (\frac {\int \frac {1}{b \left (c x^n\right )^{2/n}+a}d\left (c x^n\right )^{\frac {1}{n}}}{2 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}\right )}{4 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}\right )\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {3 \left (\frac {\arctan \left (\frac {\sqrt {b} \left (c x^n\right )^{\frac {1}{n}}}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {b}}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{2 a \left (a+b \left (c x^n\right )^{2/n}\right )}\right )}{4 a}+\frac {\left (c x^n\right )^{\frac {1}{n}}}{4 a \left (a+b \left (c x^n\right )^{2/n}\right )^2}\right )\) |
(x*((c*x^n)^n^(-1)/(4*a*(a + b*(c*x^n)^(2/n))^2) + (3*((c*x^n)^n^(-1)/(2*a *(a + b*(c*x^n)^(2/n))) + ArcTan[(Sqrt[b]*(c*x^n)^n^(-1))/Sqrt[a]]/(2*a^(3 /2)*Sqrt[b])))/(4*a)))/(c*x^n)^n^(-1)
3.31.33.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.68 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.86
| method | result | size |
| risch | \(\frac {x \left (3 b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}+5 a \right )}{8 a^{2} \left (a +b \,c^{\frac {2}{n}} \left (x^{n}\right )^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}\right )^{2}}+\frac {3 \arctan \left (\frac {b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\right )}{8 a^{2} \sqrt {\frac {a b \left (x^{n}\right )^{\frac {2}{n}} c^{\frac {2}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}}{x^{2}}}}\) | \(378\) |
1/8*x*(3*b*c^(2/n)*(x^n)^(2/n)*exp(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)+5*a)/a^2/(a+b*c^(2/n)*(x^n)^(2/n)*ex p(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+3/8/a^2/(a*b/x^2*(x^n)^(2/n)*c^(2/n)*exp(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))^(1/2)*arctan(b/x*(x^n )^(2/n)*c^(2/n)*exp(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/x^2*(x^n)^(2/n)*c^(2/n)*exp(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))^(1/2))
Time = 0.28 (sec) , antiderivative size = 328, normalized size of antiderivative = 3.35 \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\left [\frac {6 \, a b^{2} c^{\frac {4}{n}} x^{3} + 10 \, a^{2} b c^{\frac {2}{n}} x - 3 \, {\left (b^{2} c^{\frac {4}{n}} x^{4} + 2 \, a b c^{\frac {2}{n}} x^{2} + a^{2}\right )} \sqrt {-a b c^{\frac {2}{n}}} \log \left (\frac {b c^{\frac {2}{n}} x^{2} - 2 \, \sqrt {-a b c^{\frac {2}{n}}} x - a}{b c^{\frac {2}{n}} x^{2} + a}\right )}{16 \, {\left (a^{3} b^{3} c^{\frac {6}{n}} x^{4} + 2 \, a^{4} b^{2} c^{\frac {4}{n}} x^{2} + a^{5} b c^{\frac {2}{n}}\right )}}, \frac {3 \, a b^{2} c^{\frac {4}{n}} x^{3} + 5 \, a^{2} b c^{\frac {2}{n}} x + 3 \, {\left (b^{2} c^{\frac {4}{n}} x^{4} + 2 \, a b c^{\frac {2}{n}} x^{2} + a^{2}\right )} \sqrt {a b c^{\frac {2}{n}}} \arctan \left (\frac {\sqrt {a b c^{\frac {2}{n}}} x}{a}\right )}{8 \, {\left (a^{3} b^{3} c^{\frac {6}{n}} x^{4} + 2 \, a^{4} b^{2} c^{\frac {4}{n}} x^{2} + a^{5} b c^{\frac {2}{n}}\right )}}\right ] \]
[1/16*(6*a*b^2*c^(4/n)*x^3 + 10*a^2*b*c^(2/n)*x - 3*(b^2*c^(4/n)*x^4 + 2*a *b*c^(2/n)*x^2 + a^2)*sqrt(-a*b*c^(2/n))*log((b*c^(2/n)*x^2 - 2*sqrt(-a*b* c^(2/n))*x - a)/(b*c^(2/n)*x^2 + a)))/(a^3*b^3*c^(6/n)*x^4 + 2*a^4*b^2*c^( 4/n)*x^2 + a^5*b*c^(2/n)), 1/8*(3*a*b^2*c^(4/n)*x^3 + 5*a^2*b*c^(2/n)*x + 3*(b^2*c^(4/n)*x^4 + 2*a*b*c^(2/n)*x^2 + a^2)*sqrt(a*b*c^(2/n))*arctan(sqr t(a*b*c^(2/n))*x/a))/(a^3*b^3*c^(6/n)*x^4 + 2*a^4*b^2*c^(4/n)*x^2 + a^5*b* c^(2/n))]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\int \frac {1}{\left (a + b \left (c x^{n}\right )^{\frac {2}{n}}\right )^{3}}\, dx \]
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{3}} \,d x } \]
1/8*(3*b*c^(2/n)*x*(x^n)^(2/n) + 5*a*x)/(a^2*b^2*c^(4/n)*(x^n)^(4/n) + 2*a ^3*b*c^(2/n)*(x^n)^(2/n) + a^4) + 3*integrate(1/8/(a^2*b*c^(2/n)*(x^n)^(2/ n) + a^3), x)
\[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\int { \frac {1}{{\left (\left (c x^{n}\right )^{\frac {2}{n}} b + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \left (c x^n\right )^{2/n}\right )^3} \, dx=\int \frac {1}{{\left (a+b\,{\left (c\,x^n\right )}^{2/n}\right )}^3} \,d x \]