Integrand size = 25, antiderivative size = 70 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=\frac {a x \left (c x^n\right )^{-1/n}}{3 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}-\frac {x \left (c x^n\right )^{-1/n}}{2 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2} \]
1/3*a*x/b^2/((c*x^n)^(1/n))/(a+b*(c*x^n)^(1/n))^3-1/2*x/b^2/((c*x^n)^(1/n) )/(a+b*(c*x^n)^(1/n))^2
Time = 0.15 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.69 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=-\frac {x \left (c x^n\right )^{-1/n} \left (a+3 b \left (c x^n\right )^{\frac {1}{n}}\right )}{6 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3} \]
Time = 0.19 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {34, 892, 53, 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.
\(\displaystyle \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx\) |
\(\Big \downarrow \) 34 |
\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int \frac {x}{\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^4}dx}{x}\) |
\(\Big \downarrow \) 892 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^4}d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \int \left (\frac {1}{b \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^3}-\frac {a}{b \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^4}\right )d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {a}{3 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^3}-\frac {1}{2 b^2 \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^2}\right )\) |
(x*(a/(3*b^2*(a + b*(c*x^n)^n^(-1))^3) - 1/(2*b^2*(a + b*(c*x^n)^n^(-1))^2 )))/(c*x^n)^n^(-1)
3.31.77.3.1 Defintions of rubi rules used
Int[(u_.)*((a_.)*(x_)^(m_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*x^m)^F racPart[p]/x^(m*FracPart[p])) Int[u*x^(m*p), x], x] /; FreeQ[{a, m, p}, x ] && !IntegerQ[p]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo l] :> Simp[(d*x)^(m + 1)/(d*((c*x^q)^(1/q))^(m + 1)) Subst[Int[x^m*(a + b *x^(n*q))^p, x], x, (c*x^q)^(1/q)], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x ] && IntegerQ[n*q] && NeQ[x, (c*x^q)^(1/q)]
Time = 6.76 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.91
method | result | size |
parallelrisch | \(\frac {b^{4} \left (c \,x^{n}\right )^{\frac {2}{n}} x^{2}+3 x^{2} \left (c \,x^{n}\right )^{\frac {1}{n}} a \,b^{3}}{6 a^{2} b^{3} x {\left (a +b \left (c \,x^{n}\right )^{\frac {1}{n}}\right )}^{3}}\) | \(64\) |
risch | \(\frac {x \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} \left (\left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{n}}+3 a \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}\right )}{6 a^{2} {\left (b \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i c \right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )}^{3}}\) | \(205\) |
Time = 0.28 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=-\frac {3 \, b c^{\left (\frac {1}{n}\right )} x + a}{6 \, {\left (b^{5} c^{\frac {4}{n}} x^{3} + 3 \, a b^{4} c^{\frac {3}{n}} x^{2} + 3 \, a^{2} b^{3} c^{\frac {2}{n}} x + a^{3} b^{2} c^{\left (\frac {1}{n}\right )}\right )}} \]
-1/6*(3*b*c^(1/n)*x + a)/(b^5*c^(4/n)*x^3 + 3*a*b^4*c^(3/n)*x^2 + 3*a^2*b^ 3*c^(2/n)*x + a^3*b^2*c^(1/n))
Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (58) = 116\).
Time = 5.54 (sec) , antiderivative size = 182, normalized size of antiderivative = 2.60 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=\begin {cases} \tilde {\infty } x \left (c x^{n}\right )^{- \frac {3}{n}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {x \left (c x^{n}\right )^{- \frac {3}{n}}}{2 b^{4}} & \text {for}\: a = 0 \\\tilde {\infty } x \left (c x^{n}\right )^{\frac {1}{n}} & \text {for}\: b = - a \left (c x^{n}\right )^{- \frac {1}{n}} \\\frac {3 a x \left (c x^{n}\right )^{\frac {1}{n}}}{6 a^{5} + 18 a^{4} b \left (c x^{n}\right )^{\frac {1}{n}} + 18 a^{3} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 6 a^{2} b^{3} \left (c x^{n}\right )^{\frac {3}{n}}} + \frac {b x \left (c x^{n}\right )^{\frac {2}{n}}}{6 a^{5} + 18 a^{4} b \left (c x^{n}\right )^{\frac {1}{n}} + 18 a^{3} b^{2} \left (c x^{n}\right )^{\frac {2}{n}} + 6 a^{2} b^{3} \left (c x^{n}\right )^{\frac {3}{n}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*x/(c*x**n)**(3/n), Eq(a, 0) & Eq(b, 0)), (-x/(2*b**4*(c*x** n)**(3/n)), Eq(a, 0)), (zoo*x*(c*x**n)**(1/n), Eq(b, -a/(c*x**n)**(1/n))), (3*a*x*(c*x**n)**(1/n)/(6*a**5 + 18*a**4*b*(c*x**n)**(1/n) + 18*a**3*b**2 *(c*x**n)**(2/n) + 6*a**2*b**3*(c*x**n)**(3/n)) + b*x*(c*x**n)**(2/n)/(6*a **5 + 18*a**4*b*(c*x**n)**(1/n) + 18*a**3*b**2*(c*x**n)**(2/n) + 6*a**2*b* *3*(c*x**n)**(3/n)), True))
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.56 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=\frac {b c^{\frac {2}{n}} x {\left (x^{n}\right )}^{\frac {2}{n}} + 3 \, a c^{\left (\frac {1}{n}\right )} x {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )}}{6 \, {\left (a^{2} b^{3} c^{\frac {3}{n}} {\left (x^{n}\right )}^{\frac {3}{n}} + 3 \, a^{3} b^{2} c^{\frac {2}{n}} {\left (x^{n}\right )}^{\frac {2}{n}} + 3 \, a^{4} b c^{\left (\frac {1}{n}\right )} {\left (x^{n}\right )}^{\left (\frac {1}{n}\right )} + a^{5}\right )}} \]
1/6*(b*c^(2/n)*x*(x^n)^(2/n) + 3*a*c^(1/n)*x*(x^n)^(1/n))/(a^2*b^3*c^(3/n) *(x^n)^(3/n) + 3*a^3*b^2*c^(2/n)*(x^n)^(2/n) + 3*a^4*b*c^(1/n)*(x^n)^(1/n) + a^5)
\[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=\int { \frac {\left (c x^{n}\right )^{\left (\frac {1}{n}\right )}}{{\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{4}} \,d x } \]
Time = 5.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.81 \[ \int \frac {\left (c x^n\right )^{\frac {1}{n}}}{\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^4} \, dx=\frac {x}{6\,a^2\,b\,\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}-\frac {x}{3\,b\,\left (b^3\,{\left (c\,x^n\right )}^{3/n}+a^3+3\,a\,b^2\,{\left (c\,x^n\right )}^{2/n}+3\,a^2\,b\,{\left (c\,x^n\right )}^{1/n}\right )}+\frac {x}{6\,a\,b\,\left (b^2\,{\left (c\,x^n\right )}^{2/n}+a^2+2\,a\,b\,{\left (c\,x^n\right )}^{1/n}\right )} \]