\(\int (r+s x)^m (a+b (r+s x)^5)^p \, dx\) [284]

Optimal result

Integrand size = 21, antiderivative size = 80 \[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\frac {(r+s x)^{1+m} \left (a+b (r+s x)^5\right )^p \left (1+\frac {b (r+s x)^5}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{5},-p,\frac {6+m}{5},-\frac {b (r+s x)^5}{a}\right )}{(1+m) s} \] Output:

(s*x+r)^(1+m)*(a+b*(s*x+r)^5)^p*hypergeom([-p, 1/5+1/5*m],[6/5+1/5*m],-b*( 
s*x+r)^5/a)/(1+m)/s/((1+b*(s*x+r)^5/a)^p)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\frac {(r+s x)^{1+m} \left (a+b (r+s x)^5\right )^p \left (1+\frac {b (r+s x)^5}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{5},-p,1+\frac {1+m}{5},-\frac {b (r+s x)^5}{a}\right )}{(1+m) s} \] Input:

Integrate[(r + s*x)^m*(a + b*(r + s*x)^5)^p,x]
 

Output:

((r + s*x)^(1 + m)*(a + b*(r + s*x)^5)^p*Hypergeometric2F1[(1 + m)/5, -p, 
1 + (1 + m)/5, -((b*(r + s*x)^5)/a)])/((1 + m)*s*(1 + (b*(r + s*x)^5)/a)^p 
)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {895, 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.

\(\displaystyle \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx\)

\(\Big \downarrow \) 895

\(\displaystyle \frac {\int (r+s x)^m \left (b (r+s x)^5+a\right )^pd(r+s x)}{s}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {\left (a+b (r+s x)^5\right )^p \left (\frac {b (r+s x)^5}{a}+1\right )^{-p} \int (r+s x)^m \left (\frac {b (r+s x)^5}{a}+1\right )^pd(r+s x)}{s}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(r+s x)^{m+1} \left (a+b (r+s x)^5\right )^p \left (\frac {b (r+s x)^5}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{5},-p,\frac {m+6}{5},-\frac {b (r+s x)^5}{a}\right )}{(m+1) s}\)

Input:

Int[(r + s*x)^m*(a + b*(r + s*x)^5)^p,x]
 

Output:

((r + s*x)^(1 + m)*(a + b*(r + s*x)^5)^p*Hypergeometric2F1[(1 + m)/5, -p, 
(6 + m)/5, -((b*(r + s*x)^5)/a)])/((1 + m)*s*(1 + (b*(r + s*x)^5)/a)^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 895

Int[(u_)^(m_.)*((a_) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Simp[u^m/(Coeff 
icient[v, x, 1]*v^m)   Subst[Int[x^m*(a + b*x^n)^p, x], x, v], x] /; FreeQ[ 
{a, b, m, n, p}, x] && LinearPairQ[u, v, x]
 

Maple [F]

\[\int \left (s x +r \right )^{m} \left (a +b \left (s x +r \right )^{5}\right )^{p}d x\]

Input:

int((s*x+r)^m*(a+b*(s*x+r)^5)^p,x)
 

Output:

int((s*x+r)^m*(a+b*(s*x+r)^5)^p,x)
 

Fricas [F]

\[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\int { {\left ({\left (s x + r\right )}^{5} b + a\right )}^{p} {\left (s x + r\right )}^{m} \,d x } \] Input:

integrate((s*x+r)^m*(a+b*(s*x+r)^5)^p,x, algorithm="fricas")
 

Output:

integral((b*s^5*x^5 + 5*b*r*s^4*x^4 + 10*b*r^2*s^3*x^3 + 10*b*r^3*s^2*x^2 
+ 5*b*r^4*s*x + b*r^5 + a)^p*(s*x + r)^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\text {Timed out} \] Input:

integrate((s*x+r)**m*(a+b*(s*x+r)**5)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\int { {\left ({\left (s x + r\right )}^{5} b + a\right )}^{p} {\left (s x + r\right )}^{m} \,d x } \] Input:

integrate((s*x+r)^m*(a+b*(s*x+r)^5)^p,x, algorithm="maxima")
 

Output:

integrate(((s*x + r)^5*b + a)^p*(s*x + r)^m, x)
 

Giac [F]

\[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\int { {\left ({\left (s x + r\right )}^{5} b + a\right )}^{p} {\left (s x + r\right )}^{m} \,d x } \] Input:

integrate((s*x+r)^m*(a+b*(s*x+r)^5)^p,x, algorithm="giac")
 

Output:

integrate(((s*x + r)^5*b + a)^p*(s*x + r)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\int {\left (r+s\,x\right )}^m\,{\left (a+b\,{\left (r+s\,x\right )}^5\right )}^p \,d x \] Input:

int((r + s*x)^m*(a + b*(r + s*x)^5)^p,x)
 

Output:

int((r + s*x)^m*(a + b*(r + s*x)^5)^p, x)
 

Reduce [F]

\[ \int (r+s x)^m \left (a+b (r+s x)^5\right )^p \, dx=\text {too large to display} \] Input:

int((s*x+r)^m*(a+b*(s*x+r)^5)^p,x)
 

Output:

((r + s*x)**m*(a + b*r**5 + 5*b*r**4*s*x + 10*b*r**3*s**2*x**2 + 10*b*r**2 
*s**3*x**3 + 5*b*r*s**4*x**4 + b*s**5*x**5)**p*a*p + (r + s*x)**m*(a + b*r 
**5 + 5*b*r**4*s*x + 10*b*r**3*s**2*x**2 + 10*b*r**2*s**3*x**3 + 5*b*r*s** 
4*x**4 + b*s**5*x**5)**p*b*m*r**5 + (r + s*x)**m*(a + b*r**5 + 5*b*r**4*s* 
x + 10*b*r**3*s**2*x**2 + 10*b*r**2*s**3*x**3 + 5*b*r*s**4*x**4 + b*s**5*x 
**5)**p*b*m*r**4*s*x + 5*(r + s*x)**m*(a + b*r**5 + 5*b*r**4*s*x + 10*b*r* 
*3*s**2*x**2 + 10*b*r**2*s**3*x**3 + 5*b*r*s**4*x**4 + b*s**5*x**5)**p*b*p 
*r**5 + 5*(r + s*x)**m*(a + b*r**5 + 5*b*r**4*s*x + 10*b*r**3*s**2*x**2 + 
10*b*r**2*s**3*x**3 + 5*b*r*s**4*x**4 + b*s**5*x**5)**p*b*p*r**4*s*x - int 
(((r + s*x)**m*(a + b*r**5 + 5*b*r**4*s*x + 10*b*r**3*s**2*x**2 + 10*b*r** 
2*s**3*x**3 + 5*b*r*s**4*x**4 + b*s**5*x**5)**p*x**5)/(a*m**2*r + a*m**2*s 
*x + 10*a*m*p*r + 10*a*m*p*s*x + a*m*r + a*m*s*x + 25*a*p**2*r + 25*a*p**2 
*s*x + 5*a*p*r + 5*a*p*s*x + b*m**2*r**6 + 6*b*m**2*r**5*s*x + 15*b*m**2*r 
**4*s**2*x**2 + 20*b*m**2*r**3*s**3*x**3 + 15*b*m**2*r**2*s**4*x**4 + 6*b* 
m**2*r*s**5*x**5 + b*m**2*s**6*x**6 + 10*b*m*p*r**6 + 60*b*m*p*r**5*s*x + 
150*b*m*p*r**4*s**2*x**2 + 200*b*m*p*r**3*s**3*x**3 + 150*b*m*p*r**2*s**4* 
x**4 + 60*b*m*p*r*s**5*x**5 + 10*b*m*p*s**6*x**6 + b*m*r**6 + 6*b*m*r**5*s 
*x + 15*b*m*r**4*s**2*x**2 + 20*b*m*r**3*s**3*x**3 + 15*b*m*r**2*s**4*x**4 
 + 6*b*m*r*s**5*x**5 + b*m*s**6*x**6 + 25*b*p**2*r**6 + 150*b*p**2*r**5*s* 
x + 375*b*p**2*r**4*s**2*x**2 + 500*b*p**2*r**3*s**3*x**3 + 375*b*p**2*...