Integrand size = 15, antiderivative size = 61 \[ \int (a+b x)^m (c+d x)^n \, dx=\frac {(a+b x)^{1+m} (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,2+m+n,2+m,-\frac {d (a+b x)}{b c-a d}\right )}{(b c-a d) (1+m)} \] Output:
(b*x+a)^(1+m)*(d*x+c)^(1+n)*hypergeom([1, 2+m+n],[2+m],-d*(b*x+a)/(-a*d+b* c))/(-a*d+b*c)/(1+m)
Time = 0.04 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int (a+b x)^m (c+d x)^n \, dx=\frac {(a+b x)^{1+m} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,\frac {d (a+b x)}{-b c+a d}\right )}{b (1+m)} \] Input:
Integrate[(a + b*x)^m*(c + d*x)^n,x]
Output:
((a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, (d*(a + b*x))/(-(b*c) + a*d)])/(b*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n)
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {80, 79}
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.
\(\displaystyle \int (a+b x)^m (c+d x)^n \, dx\) |
\(\Big \downarrow \) 80 |
\(\displaystyle (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \int (a+b x)^m \left (\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}\right )^ndx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle \frac {(a+b x)^{m+1} (c+d x)^n \left (\frac {b (c+d x)}{b c-a d}\right )^{-n} \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d (a+b x)}{b c-a d}\right )}{b (m+1)}\) |
Input:
Int[(a + b*x)^m*(c + d*x)^n,x]
Output:
((a + b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*(a + b*x))/(b*c - a*d))])/(b*(1 + m)*((b*(c + d*x))/(b*c - a*d))^n)
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 , m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) ^FracPart[n]) Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) ), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && !IntegerQ[m] && !Integ erQ[n] && (RationalQ[m] || !SimplerQ[n + 1, m + 1])
\[\int \left (b x +a \right )^{m} \left (x d +c \right )^{n}d x\]
Input:
int((b*x+a)^m*(d*x+c)^n,x)
Output:
int((b*x+a)^m*(d*x+c)^n,x)
\[ \int (a+b x)^m (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^n,x, algorithm="fricas")
Output:
integral((b*x + a)^m*(d*x + c)^n, x)
Exception generated. \[ \int (a+b x)^m (c+d x)^n \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:
integrate((b*x+a)**m*(d*x+c)**n,x)
Output:
Exception raised: HeuristicGCDFailed >> no luck
\[ \int (a+b x)^m (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^n,x, algorithm="maxima")
Output:
integrate((b*x + a)^m*(d*x + c)^n, x)
\[ \int (a+b x)^m (c+d x)^n \, dx=\int { {\left (b x + a\right )}^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x+a)^m*(d*x+c)^n,x, algorithm="giac")
Output:
integrate((b*x + a)^m*(d*x + c)^n, x)
Timed out. \[ \int (a+b x)^m (c+d x)^n \, dx=\int {\left (a+b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \] Input:
int((a + b*x)^m*(c + d*x)^n,x)
Output:
int((a + b*x)^m*(c + d*x)^n, x)
\[ \int (a+b x)^m (c+d x)^n \, dx=\text {too large to display} \] Input:
int((b*x+a)^m*(d*x+c)^n,x)
Output:
((c + d*x)**n*(a + b*x)**m*a*c*m + (c + d*x)**n*(a + b*x)**m*a*c*n + (c + d*x)**n*(a + b*x)**m*a*d*n*x + (c + d*x)**n*(a + b*x)**m*b*c*m*x + int(((c + d*x)**n*(a + b*x)**m*x)/(a**2*c*d*m*n + a**2*c*d*n**2 + a**2*c*d*n + a* *2*d**2*m*n*x + a**2*d**2*n**2*x + a**2*d**2*n*x + a*b*c**2*m**2 + a*b*c** 2*m*n + a*b*c**2*m + a*b*c*d*m**2*x + 2*a*b*c*d*m*n*x + a*b*c*d*m*x + a*b* c*d*n**2*x + a*b*c*d*n*x + a*b*d**2*m*n*x**2 + a*b*d**2*n**2*x**2 + a*b*d* *2*n*x**2 + b**2*c**2*m**2*x + b**2*c**2*m*n*x + b**2*c**2*m*x + b**2*c*d* m**2*x**2 + b**2*c*d*m*n*x**2 + b**2*c*d*m*x**2),x)*a**3*d**3*m**2*n**2 + int(((c + d*x)**n*(a + b*x)**m*x)/(a**2*c*d*m*n + a**2*c*d*n**2 + a**2*c*d *n + a**2*d**2*m*n*x + a**2*d**2*n**2*x + a**2*d**2*n*x + a*b*c**2*m**2 + a*b*c**2*m*n + a*b*c**2*m + a*b*c*d*m**2*x + 2*a*b*c*d*m*n*x + a*b*c*d*m*x + a*b*c*d*n**2*x + a*b*c*d*n*x + a*b*d**2*m*n*x**2 + a*b*d**2*n**2*x**2 + a*b*d**2*n*x**2 + b**2*c**2*m**2*x + b**2*c**2*m*n*x + b**2*c**2*m*x + b* *2*c*d*m**2*x**2 + b**2*c*d*m*n*x**2 + b**2*c*d*m*x**2),x)*a**3*d**3*m*n** 3 + int(((c + d*x)**n*(a + b*x)**m*x)/(a**2*c*d*m*n + a**2*c*d*n**2 + a**2 *c*d*n + a**2*d**2*m*n*x + a**2*d**2*n**2*x + a**2*d**2*n*x + a*b*c**2*m** 2 + a*b*c**2*m*n + a*b*c**2*m + a*b*c*d*m**2*x + 2*a*b*c*d*m*n*x + a*b*c*d *m*x + a*b*c*d*n**2*x + a*b*c*d*n*x + a*b*d**2*m*n*x**2 + a*b*d**2*n**2*x* *2 + a*b*d**2*n*x**2 + b**2*c**2*m**2*x + b**2*c**2*m*n*x + b**2*c**2*m*x + b**2*c*d*m**2*x**2 + b**2*c*d*m*n*x**2 + b**2*c*d*m*x**2),x)*a**3*d**...