\(\int x^m (a x^j+b x^n)^p \, dx\) [422]

Optimal result

Integrand size = 17, antiderivative size = 92 \[ \int x^m \left (a x^j+b x^n\right )^p \, dx=\frac {x^{1+m} \left (1+\frac {a x^{j-n}}{b}\right )^{-p} \left (a x^j+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (-p,\frac {1+m+n p}{j-n},1+\frac {1+m+n p}{j-n},-\frac {a x^{j-n}}{b}\right )}{1+m+n p} \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00 \[ \int x^m \left (a x^j+b x^n\right )^p \, dx=\frac {x^{1+m} \left (1+\frac {a x^{j-n}}{b}\right )^{-p} \left (a x^j+b x^n\right )^p \operatorname {Hypergeometric2F1}\left (-p,\frac {1+m+n p}{j-n},1+\frac {1+m+n p}{j-n},-\frac {a x^{j-n}}{b}\right )}{1+m+n p} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1938, 889, 888}

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.

\(\displaystyle \int x^m \left (a x^j+b x^n\right )^p \, dx\)

\(\Big \downarrow \) 1938

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 889

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I 
ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(c*x) 
^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0 
] &&  !(ILtQ[p, 0] || GtQ[a, 0])
 

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^m \left (a x^j+b x^n\right )^p \, dx=\frac {x^{m} \left (x^{j} a +x^{n} b \right )^{p} x -\left (\int \frac {x^{j +m} \left (x^{j} a +x^{n} b \right )^{p}}{x^{j} a m +x^{j} a n p +x^{j} a +x^{n} b m +x^{n} b n p +x^{n} b}d x \right ) a j m p -\left (\int \frac {x^{j +m} \left (x^{j} a +x^{n} b \right )^{p}}{x^{j} a m +x^{j} a n p +x^{j} a +x^{n} b m +x^{n} b n p +x^{n} b}d x \right ) a j n \,p^{2}-\left (\int \frac {x^{j +m} \left (x^{j} a +x^{n} b \right )^{p}}{x^{j} a m +x^{j} a n p +x^{j} a +x^{n} b m +x^{n} b n p +x^{n} b}d x \right ) a j p +\left (\int \frac {x^{j +m} \left (x^{j} a +x^{n} b \right )^{p}}{x^{j} a m +x^{j} a n p +x^{j} a +x^{n} b m +x^{n} b n p +x^{n} b}d x \right ) a m n p +\left (\int \frac {x^{j +m} \left (x^{j} a +x^{n} b \right )^{p}}{x^{j} a m +x^{j} a n p +x^{j} a +x^{n} b m +x^{n} b n p +x^{n} b}d x \right ) a \,n^{2} p^{2}+\left (\int \frac {x^{j +m} \left (x^{j} a +x^{n} b \right )^{p}}{x^{j} a m +x^{j} a n p +x^{j} a +x^{n} b m +x^{n} b n p +x^{n} b}d x \right ) a n p}{n p +m +1} \] Input:

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

Output:

(x**m*(x**j*a + x**n*b)**p*x - int((x**(j + m)*(x**j*a + x**n*b)**p)/(x**j 
*a*m + x**j*a*n*p + x**j*a + x**n*b*m + x**n*b*n*p + x**n*b),x)*a*j*m*p - 
int((x**(j + m)*(x**j*a + x**n*b)**p)/(x**j*a*m + x**j*a*n*p + x**j*a + x* 
*n*b*m + x**n*b*n*p + x**n*b),x)*a*j*n*p**2 - int((x**(j + m)*(x**j*a + x* 
*n*b)**p)/(x**j*a*m + x**j*a*n*p + x**j*a + x**n*b*m + x**n*b*n*p + x**n*b 
),x)*a*j*p + int((x**(j + m)*(x**j*a + x**n*b)**p)/(x**j*a*m + x**j*a*n*p 
+ x**j*a + x**n*b*m + x**n*b*n*p + x**n*b),x)*a*m*n*p + int((x**(j + m)*(x 
**j*a + x**n*b)**p)/(x**j*a*m + x**j*a*n*p + x**j*a + x**n*b*m + x**n*b*n* 
p + x**n*b),x)*a*n**2*p**2 + int((x**(j + m)*(x**j*a + x**n*b)**p)/(x**j*a 
*m + x**j*a*n*p + x**j*a + x**n*b*m + x**n*b*n*p + x**n*b),x)*a*n*p)/(m + 
n*p + 1)