Integrand size = 67, antiderivative size = 80 \[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\frac {(r+s x)^{1+m} \left (a+b (r+s x)^5\right )^p \left (1+\frac {b (r+s x)^5}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{5},-p,\frac {6+m}{5},-\frac {b (r+s x)^5}{a}\right )}{(1+m) s} \] 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)
Time = 0.01 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02 \[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\frac {(r+s x)^{1+m} \left (a+b (r+s x)^5\right )^p \left (1+\frac {b (r+s x)^5}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{5},-p,1+\frac {1+m}{5},-\frac {b (r+s x)^5}{a}\right )}{(1+m) s} \] Input:
Integrate[(r + s*x)^m*(a + b*r^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]
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 )
Time = 0.53 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {2509, 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.
| \(\displaystyle \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx\) |
| \(\Big \downarrow \) 2509 |
| \(\displaystyle \frac {\int (r+s x)^m \left (b (r+s x)^5+a\right )^pd(r+s x)}{s}\) |
| \(\Big \downarrow \) 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^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]
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)
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])
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])
Int[(Pn_)^(p_.)*((g_) + (h_.)*(x_))^(m_.), x_Symbol] :> With[{Px = Pn /. x -> (x - g)/h}, Simp[1/h Subst[Int[x^m*ExpandToSum[Px, x]^p, x], x, g + h* x], x] /; BinomialQ[Px, x]] /; FreeQ[{g, h, m, p}, x] && PolyQ[Pn, x]
\[\int \left (s x +r \right )^{m} \left (b \,s^{5} x^{5}+5 b r \,s^{4} x^{4}+10 b \,r^{2} s^{3} x^{3}+10 b \,r^{3} s^{2} x^{2}+5 b \,r^{4} s x +b \,r^{5}+a \right )^{p}d x\]
Input:
int((s*x+r)^m*(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,x)
Output:
int((s*x+r)^m*(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,x)
\[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\int { {\left (b s^{5} x^{5} + 5 \, b r s^{4} x^{4} + 10 \, b r^{2} s^{3} x^{3} + 10 \, b r^{3} s^{2} x^{2} + 5 \, b r^{4} s x + b r^{5} + a\right )}^{p} {\left (s x + r\right )}^{m} \,d x } \] Input:
integrate((s*x+r)^m*(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,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)
\[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\int \left (r + s x\right )^{m} \left (a + b r^{5} + 5 b r^{4} s x + 10 b r^{3} s^{2} x^{2} + 10 b r^{2} s^{3} x^{3} + 5 b r s^{4} x^{4} + b s^{5} x^{5}\right )^{p}\, dx \] Input:
integrate((s*x+r)**m*(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,x)
Output:
Integral((r + s*x)**m*(a + b*r**5 + 5*b*r**4*s*x + 10*b*r**3*s**2*x**2 + 1 0*b*r**2*s**3*x**3 + 5*b*r*s**4*x**4 + b*s**5*x**5)**p, x)
\[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\int { {\left (b s^{5} x^{5} + 5 \, b r s^{4} x^{4} + 10 \, b r^{2} s^{3} x^{3} + 10 \, b r^{3} s^{2} x^{2} + 5 \, b r^{4} s x + b r^{5} + a\right )}^{p} {\left (s x + r\right )}^{m} \,d x } \] Input:
integrate((s*x+r)^m*(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,x, algorithm="maxima")
Output:
integrate((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)
\[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\int { {\left (b s^{5} x^{5} + 5 \, b r s^{4} x^{4} + 10 \, b r^{2} s^{3} x^{3} + 10 \, b r^{3} s^{2} x^{2} + 5 \, b r^{4} s x + b r^{5} + a\right )}^{p} {\left (s x + r\right )}^{m} \,d x } \] Input:
integrate((s*x+r)^m*(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,x, algorithm="giac")
Output:
integrate((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)
Timed out. \[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\int {\left (r+s\,x\right )}^m\,{\left (b\,r^5+5\,b\,r^4\,s\,x+10\,b\,r^3\,s^2\,x^2+10\,b\,r^2\,s^3\,x^3+5\,b\,r\,s^4\,x^4+b\,s^5\,x^5+a\right )}^p \,d x \] Input:
int((r + s*x)^m*(a + b*r^5 + b*s^5*x^5 + 10*b*r^3*s^2*x^2 + 10*b*r^2*s^3*x ^3 + 5*b*r^4*s*x + 5*b*r*s^4*x^4)^p,x)
Output:
int((r + s*x)^m*(a + b*r^5 + b*s^5*x^5 + 10*b*r^3*s^2*x^2 + 10*b*r^2*s^3*x ^3 + 5*b*r^4*s*x + 5*b*r*s^4*x^4)^p, x)
\[ \int (r+s x)^m \left (a+b r^5+5 b r^4 s x+10 b r^3 s^2 x^2+10 b r^2 s^3 x^3+5 b r s^4 x^4+b s^5 x^5\right )^p \, dx=\text {too large to display} \] Input:
int((s*x+r)^m*(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,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*...