\(\int (c x)^m (a x^n+b x^{1+n})^p \, dx\) [455]

Optimal result

Integrand size = 21, antiderivative size = 63 \[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\frac {x^{1-n} (c x)^m \left (a x^n+b x^{1+n}\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2+m+p+n p,2+m+n p,-\frac {b x}{a}\right )}{a (1+m+n p)} \] Output:

x^(1-n)*(c*x)^m*(a*x^n+b*x^(1+n))^(p+1)*hypergeom([1, n*p+m+p+2],[n*p+m+2] 
,-b*x/a)/a/(n*p+m+1)
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.97 \[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\frac {x (c x)^m \left (x^n (a+b x)\right )^p \left (1+\frac {b x}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,1+m+n p,2+m+n p,-\frac {b x}{a}\right )}{1+m+n p} \] Input:

Integrate[(c*x)^m*(a*x^n + b*x^(1 + n))^p,x]
 

Output:

(x*(c*x)^m*(x^n*(a + b*x))^p*Hypergeometric2F1[-p, 1 + m + n*p, 2 + m + n* 
p, -((b*x)/a)])/((1 + m + n*p)*(1 + (b*x)/a)^p)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.03, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1938, 76, 74}

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.

\(\displaystyle \int (c x)^m \left (a x^n+b x^{n+1}\right )^p \, dx\)

\(\Big \downarrow \) 1938

\(\displaystyle (c x)^m (a+b x)^{-p} x^{-m-n p} \left (a x^n+b x^{n+1}\right )^p \int x^{m+n p} (a+b x)^pdx\)

\(\Big \downarrow \) 76

\(\displaystyle (c x)^m \left (\frac {b x}{a}+1\right )^{-p} x^{-m-n p} \left (a x^n+b x^{n+1}\right )^p \int x^{m+n p} \left (\frac {b x}{a}+1\right )^pdx\)

\(\Big \downarrow \) 74

\(\displaystyle \frac {x (c x)^m \left (\frac {b x}{a}+1\right )^{-p} \left (a x^n+b x^{n+1}\right )^p \operatorname {Hypergeometric2F1}\left (-p,m+n p+1,m+n p+2,-\frac {b x}{a}\right )}{m+n p+1}\)

Input:

Int[(c*x)^m*(a*x^n + b*x^(1 + n))^p,x]
 

Output:

(x*(c*x)^m*(a*x^n + b*x^(1 + n))^p*Hypergeometric2F1[-p, 1 + m + n*p, 2 + 
m + n*p, -((b*x)/a)])/((1 + m + n*p)*(1 + (b*x)/a)^p)
 

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 76

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^IntPart 
[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n])   Int[(b*x)^m*(1 + d* 
(x/c))^n, x], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integer 
Q[n] &&  !GtQ[c, 0] &&  !GtQ[-d/(b*c), 0] && ((RationalQ[m] &&  !(EqQ[n, -2 
^(-1)] && EqQ[c^2 - d^2, 0])) ||  !RationalQ[n])
 

rule 1938

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F 
racPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p]))   Int[x^(m + j* 
p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !Inte 
gerQ[p] && NeQ[n, j] && PosQ[n - j]
 

Maple [F]

\[\int \left (c x \right )^{m} \left (a \,x^{n}+b \,x^{1+n}\right )^{p}d x\]

Input:

int((c*x)^m*(a*x^n+b*x^(1+n))^p,x)
 

Output:

int((c*x)^m*(a*x^n+b*x^(1+n))^p,x)
 

Fricas [F]

\[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\int { {\left (b x^{n + 1} + a x^{n}\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(a*x^n+b*x^(1+n))^p,x, algorithm="fricas")
 

Output:

integral((b*x^(n + 1) + a*x^n)^p*(c*x)^m, x)
 

Sympy [F]

\[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\int \left (c x\right )^{m} \left (a x^{n} + b x^{n + 1}\right )^{p}\, dx \] Input:

integrate((c*x)**m*(a*x**n+b*x**(1+n))**p,x)
 

Output:

Integral((c*x)**m*(a*x**n + b*x**(n + 1))**p, x)
 

Maxima [F]

\[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\int { {\left (b x^{n + 1} + a x^{n}\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(a*x^n+b*x^(1+n))^p,x, algorithm="maxima")
 

Output:

integrate((b*x^(n + 1) + a*x^n)^p*(c*x)^m, x)
 

Giac [F]

\[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\int { {\left (b x^{n + 1} + a x^{n}\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(a*x^n+b*x^(1+n))^p,x, algorithm="giac")
 

Output:

integrate((b*x^(n + 1) + a*x^n)^p*(c*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\int {\left (c\,x\right )}^m\,{\left (a\,x^n+b\,x^{n+1}\right )}^p \,d x \] Input:

int((c*x)^m*(a*x^n + b*x^(n + 1))^p,x)
 

Output:

int((c*x)^m*(a*x^n + b*x^(n + 1))^p, x)
 

Reduce [F]

\[ \int (c x)^m \left (a x^n+b x^{1+n}\right )^p \, dx=\text {too large to display} \] Input:

int((c*x)^m*(a*x^n+b*x^(1+n))^p,x)
 

Output:

(c**m*(x**m*(x**n*a + x**n*b*x)**p*a*p + x**m*(x**n*a + x**n*b*x)**p*b*m*x 
 + x**m*(x**n*a + x**n*b*x)**p*b*n*p*x + x**m*(x**n*a + x**n*b*x)**p*b*p*x 
 - int((x**m*(x**n*a + x**n*b*x)**p)/(a*m**2*x + 2*a*m*n*p*x + 2*a*m*p*x + 
 a*m*x + a*n**2*p**2*x + 2*a*n*p**2*x + a*n*p*x + a*p**2*x + a*p*x + b*m** 
2*x**2 + 2*b*m*n*p*x**2 + 2*b*m*p*x**2 + b*m*x**2 + b*n**2*p**2*x**2 + 2*b 
*n*p**2*x**2 + b*n*p*x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**3*p - 3*int 
((x**m*(x**n*a + x**n*b*x)**p)/(a*m**2*x + 2*a*m*n*p*x + 2*a*m*p*x + a*m*x 
 + a*n**2*p**2*x + 2*a*n*p**2*x + a*n*p*x + a*p**2*x + a*p*x + b*m**2*x**2 
 + 2*b*m*n*p*x**2 + 2*b*m*p*x**2 + b*m*x**2 + b*n**2*p**2*x**2 + 2*b*n*p** 
2*x**2 + b*n*p*x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**2*n*p**2 - 2*int( 
(x**m*(x**n*a + x**n*b*x)**p)/(a*m**2*x + 2*a*m*n*p*x + 2*a*m*p*x + a*m*x 
+ a*n**2*p**2*x + 2*a*n*p**2*x + a*n*p*x + a*p**2*x + a*p*x + b*m**2*x**2 
+ 2*b*m*n*p*x**2 + 2*b*m*p*x**2 + b*m*x**2 + b*n**2*p**2*x**2 + 2*b*n*p**2 
*x**2 + b*n*p*x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**2*p**2 - int((x**m 
*(x**n*a + x**n*b*x)**p)/(a*m**2*x + 2*a*m*n*p*x + 2*a*m*p*x + a*m*x + a*n 
**2*p**2*x + 2*a*n*p**2*x + a*n*p*x + a*p**2*x + a*p*x + b*m**2*x**2 + 2*b 
*m*n*p*x**2 + 2*b*m*p*x**2 + b*m*x**2 + b*n**2*p**2*x**2 + 2*b*n*p**2*x**2 
 + b*n*p*x**2 + b*p**2*x**2 + b*p*x**2),x)*a**2*m**2*p - 3*int((x**m*(x**n 
*a + x**n*b*x)**p)/(a*m**2*x + 2*a*m*n*p*x + 2*a*m*p*x + a*m*x + a*n**2*p* 
*2*x + 2*a*n*p**2*x + a*n*p*x + a*p**2*x + a*p*x + b*m**2*x**2 + 2*b*m*...