Integrand size = 25, antiderivative size = 79 \[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=-\frac {a x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p}}{b^2 (1+p)}+\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{2+p}}{b^2 (2+p)} \] Output:
-a*x*(a+b*(c*x^n)^(1/n))^(p+1)/b^2/(p+1)/((c*x^n)^(1/n))+x*(a+b*(c*x^n)^(1 /n))^(2+p)/b^2/(2+p)/((c*x^n)^(1/n))
Time = 0.05 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.77 \[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {x \left (c x^n\right )^{-1/n} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{1+p} \left (-a+b (1+p) \left (c x^n\right )^{\frac {1}{n}}\right )}{b^2 (1+p) (2+p)} \] Input:
Integrate[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^p,x]
Output:
(x*(a + b*(c*x^n)^n^(-1))^(1 + p)*(-a + b*(1 + p)*(c*x^n)^n^(-1)))/(b^2*(1 + p)*(2 + p)*(c*x^n)^n^(-1))
Time = 0.33 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.86, 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.
\(\displaystyle \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx\) |
\(\Big \downarrow \) 34 |
\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \int x \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^pdx}{x}\) |
\(\Big \downarrow \) 892 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \int \left (c x^n\right )^{\frac {1}{n}} \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^pd\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \int \left (\frac {\left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^{p+1}}{b}-\frac {a \left (b \left (c x^n\right )^{\frac {1}{n}}+a\right )^p}{b}\right )d\left (c x^n\right )^{\frac {1}{n}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x \left (c x^n\right )^{-1/n} \left (\frac {\left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+2}}{b^2 (p+2)}-\frac {a \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^{p+1}}{b^2 (p+1)}\right )\) |
Input:
Int[(c*x^n)^n^(-1)*(a + b*(c*x^n)^n^(-1))^p,x]
Output:
(x*(-((a*(a + b*(c*x^n)^n^(-1))^(1 + p))/(b^2*(1 + p))) + (a + b*(c*x^n)^n ^(-1))^(2 + p)/(b^2*(2 + p))))/(c*x^n)^n^(-1)
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)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.38 (sec) , antiderivative size = 377, normalized size of antiderivative = 4.77
method | result | size |
risch | \(\frac {a p x \left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )^{p}}{b \left (2+p \right ) \left (p +1\right )}+\frac {\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )^{p} x \left (x^{n}\right )^{\frac {1}{n}} c^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{2+p}-\frac {\left (b \,c^{\frac {1}{n}} \left (x^{n}\right )^{\frac {1}{n}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}+a \right )^{p} a^{2} x \left (x^{n}\right )^{-\frac {1}{n}} c^{-\frac {1}{n}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i c \,x^{n}\right ) \left (-\operatorname {csgn}\left (i x^{n}\right )+\operatorname {csgn}\left (i c \,x^{n}\right )\right ) \left (\operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \,x^{n}\right )\right )}{2 n}}}{\left (2+p \right ) \left (p +1\right ) b^{2}}\) | \(377\) |
Input:
int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x,method=_RETURNVERBOSE)
Output:
a*p/b/(2+p)/(p+1)*x*(b*c^(1/n)*(x^n)^(1/n)*exp(1/2*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)^p+(b*c^(1/n)*(x^n )^(1/n)*exp(1/2*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)^p/(2+p)*x*(x^n)^(1/n)*c^(1/n)*exp(1/2*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)-(b*c^(1/n)* (x^n)^(1/n)*exp(1/2*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)^p*a^2/(2+p)/(p+1)/b^2*x/((x^n)^(1/n))/(c^(1/n))* exp(-1/2*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)
Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.97 \[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (a b c^{\left (\frac {1}{n}\right )} p x + {\left (b^{2} p + b^{2}\right )} c^{\frac {2}{n}} x^{2} - a^{2}\right )} {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p}}{{\left (b^{2} p^{2} + 3 \, b^{2} p + 2 \, b^{2}\right )} c^{\left (\frac {1}{n}\right )}} \] Input:
integrate((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x, algorithm="fricas")
Output:
(a*b*c^(1/n)*p*x + (b^2*p + b^2)*c^(2/n)*x^2 - a^2)*(b*c^(1/n)*x + a)^p/(( b^2*p^2 + 3*b^2*p + 2*b^2)*c^(1/n))
\[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int \left (c x^{n}\right )^{\frac {1}{n}} \left (a + b \left (c x^{n}\right )^{\frac {1}{n}}\right )^{p}\, dx \] Input:
integrate((c*x**n)**(1/n)*(a+b*(c*x**n)**(1/n))**p,x)
Output:
Integral((c*x**n)**(1/n)*(a + b*(c*x**n)**(1/n))**p, x)
\[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int { {\left (\left (c x^{n}\right )^{\left (\frac {1}{n}\right )} b + a\right )}^{p} \left (c x^{n}\right )^{\left (\frac {1}{n}\right )} \,d x } \] Input:
integrate((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x, algorithm="maxima")
Output:
integrate(((c*x^n)^(1/n)*b + a)^p*(c*x^n)^(1/n), x)
Time = 0.19 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.65 \[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {{\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac {2}{n}} p x^{2} + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a b c^{\left (\frac {1}{n}\right )} p x + {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} b^{2} c^{\frac {2}{n}} x^{2} - {\left (b c^{\left (\frac {1}{n}\right )} x + a\right )}^{p} a^{2}}{b^{2} c^{\left (\frac {1}{n}\right )} p^{2} + 3 \, b^{2} c^{\left (\frac {1}{n}\right )} p + 2 \, b^{2} c^{\left (\frac {1}{n}\right )}} \] Input:
integrate((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x, algorithm="giac")
Output:
((b*c^(1/n)*x + a)^p*b^2*c^(2/n)*p*x^2 + (b*c^(1/n)*x + a)^p*a*b*c^(1/n)*p *x + (b*c^(1/n)*x + a)^p*b^2*c^(2/n)*x^2 - (b*c^(1/n)*x + a)^p*a^2)/(b^2*c ^(1/n)*p^2 + 3*b^2*c^(1/n)*p + 2*b^2*c^(1/n))
Timed out. \[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\int {\left (c\,x^n\right )}^{1/n}\,{\left (a+b\,{\left (c\,x^n\right )}^{1/n}\right )}^p \,d x \] Input:
int((c*x^n)^(1/n)*(a + b*(c*x^n)^(1/n))^p,x)
Output:
int((c*x^n)^(1/n)*(a + b*(c*x^n)^(1/n))^p, x)
Time = 0.16 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int \left (c x^n\right )^{\frac {1}{n}} \left (a+b \left (c x^n\right )^{\frac {1}{n}}\right )^p \, dx=\frac {\left (c^{\frac {1}{n}} b x +a \right )^{p} \left (c^{\frac {2}{n}} b^{2} p \,x^{2}+c^{\frac {2}{n}} b^{2} x^{2}+c^{\frac {1}{n}} a b p x -a^{2}\right )}{c^{\frac {1}{n}} b^{2} \left (p^{2}+3 p +2\right )} \] Input:
int((c*x^n)^(1/n)*(a+b*(c*x^n)^(1/n))^p,x)
Output:
((c**(1/n)*b*x + a)**p*(c**(2/n)*b**2*p*x**2 + c**(2/n)*b**2*x**2 + c**(1/ n)*a*b*p*x - a**2))/(c**(1/n)*b**2*(p**2 + 3*p + 2))