Integrand size = 18, antiderivative size = 89 \[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\frac {f (b x)^{1+m} (c+d x)^{1+n}}{b d (2+m+n)}+\frac {\left (\frac {e}{c+c m}-\frac {f}{d (2+m+n)}\right ) (b x)^{1+m} (c+d x)^{1+n} \operatorname {Hypergeometric2F1}\left (1,2+m+n,2+m,-\frac {d x}{c}\right )}{b} \] Output:
f*(b*x)^(1+m)*(d*x+c)^(1+n)/b/d/(2+m+n)+(e/(c*m+c)-f/d/(2+m+n))*(b*x)^(1+m )*(d*x+c)^(1+n)*hypergeom([1, 2+m+n],[2+m],-d*x/c)/b
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\frac {x (b x)^m (c+d x)^n \left (f (c+d x)+\frac {(-c f (1+m)+d e (2+m+n)) \left (1+\frac {d x}{c}\right )^{-n} \operatorname {Hypergeometric2F1}\left (1+m,-n,2+m,-\frac {d x}{c}\right )}{1+m}\right )}{d (2+m+n)} \] Input:
Integrate[(b*x)^m*(c + d*x)^n*(e + f*x),x]
Output:
(x*(b*x)^m*(c + d*x)^n*(f*(c + d*x) + ((-(c*f*(1 + m)) + d*e*(2 + m + n))* Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)])/((1 + m)*(1 + (d*x)/c)^n) ))/(d*(2 + m + n))
Time = 0.22 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {90, 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.
| \(\displaystyle \int (b x)^m (e+f x) (c+d x)^n \, dx\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle \left (e-\frac {c f (m+1)}{d (m+n+2)}\right ) \int (b x)^m (c+d x)^ndx+\frac {f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}\) |
| \(\Big \downarrow \) 76 |
| \(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (e-\frac {c f (m+1)}{d (m+n+2)}\right ) \int (b x)^m \left (\frac {d x}{c}+1\right )^ndx+\frac {f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}\) |
| \(\Big \downarrow \) 74 |
| \(\displaystyle \frac {(b x)^{m+1} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \left (e-\frac {c f (m+1)}{d (m+n+2)}\right ) \operatorname {Hypergeometric2F1}\left (m+1,-n,m+2,-\frac {d x}{c}\right )}{b (m+1)}+\frac {f (b x)^{m+1} (c+d x)^{n+1}}{b d (m+n+2)}\) |
Input:
Int[(b*x)^m*(c + d*x)^n*(e + f*x),x]
Output:
(f*(b*x)^(1 + m)*(c + d*x)^(1 + n))/(b*d*(2 + m + n)) + ((e - (c*f*(1 + m) )/(d*(2 + m + n)))*(b*x)^(1 + m)*(c + d*x)^n*Hypergeometric2F1[1 + m, -n, 2 + m, -((d*x)/c)])/(b*(1 + m)*(1 + (d*x)/c)^n)
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])))
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])
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
\[\int \left (b x \right )^{m} \left (x d +c \right )^{n} \left (f x +e \right )d x\]
Input:
int((b*x)^m*(d*x+c)^n*(f*x+e),x)
Output:
int((b*x)^m*(d*x+c)^n*(f*x+e),x)
\[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\int { {\left (f x + e\right )} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x)^m*(d*x+c)^n*(f*x+e),x, algorithm="fricas")
Output:
integral((f*x + e)*(b*x)^m*(d*x + c)^n, x)
Result contains complex when optimal does not.
Time = 4.74 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.90 \[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\frac {b^{m} c^{n} e x^{m + 1} \Gamma \left (m + 1\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 1 \\ m + 2 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 2\right )} + \frac {b^{m} c^{n} f x^{m + 2} \Gamma \left (m + 2\right ) {{}_{2}F_{1}\left (\begin {matrix} - n, m + 2 \\ m + 3 \end {matrix}\middle | {\frac {d x e^{i \pi }}{c}} \right )}}{\Gamma \left (m + 3\right )} \] Input:
integrate((b*x)**m*(d*x+c)**n*(f*x+e),x)
Output:
b**m*c**n*e*x**(m + 1)*gamma(m + 1)*hyper((-n, m + 1), (m + 2,), d*x*exp_p olar(I*pi)/c)/gamma(m + 2) + b**m*c**n*f*x**(m + 2)*gamma(m + 2)*hyper((-n , m + 2), (m + 3,), d*x*exp_polar(I*pi)/c)/gamma(m + 3)
\[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\int { {\left (f x + e\right )} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x)^m*(d*x+c)^n*(f*x+e),x, algorithm="maxima")
Output:
integrate((f*x + e)*(b*x)^m*(d*x + c)^n, x)
\[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\int { {\left (f x + e\right )} \left (b x\right )^{m} {\left (d x + c\right )}^{n} \,d x } \] Input:
integrate((b*x)^m*(d*x+c)^n*(f*x+e),x, algorithm="giac")
Output:
integrate((f*x + e)*(b*x)^m*(d*x + c)^n, x)
Timed out. \[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\int \left (e+f\,x\right )\,{\left (b\,x\right )}^m\,{\left (c+d\,x\right )}^n \,d x \] Input:
int((e + f*x)*(b*x)^m*(c + d*x)^n,x)
Output:
int((e + f*x)*(b*x)^m*(c + d*x)^n, x)
\[ \int (b x)^m (c+d x)^n (e+f x) \, dx=\text {too large to display} \] Input:
int((b*x)^m*(d*x+c)^n*(f*x+e),x)
Output:
(b**m*( - x**m*(c + d*x)**n*c**2*f*m*n - x**m*(c + d*x)**n*c**2*f*n + x**m *(c + d*x)**n*c*d*e*m*n + x**m*(c + d*x)**n*c*d*e*n**2 + 2*x**m*(c + d*x)* *n*c*d*e*n + x**m*(c + d*x)**n*c*d*f*m*n*x + x**m*(c + d*x)**n*c*d*f*n**2* x + x**m*(c + d*x)**n*d**2*e*m**2*x + 2*x**m*(c + d*x)**n*d**2*e*m*n*x + 2 *x**m*(c + d*x)**n*d**2*e*m*x + x**m*(c + d*x)**n*d**2*e*n**2*x + 2*x**m*( c + d*x)**n*d**2*e*n*x + x**m*(c + d*x)**n*d**2*f*m**2*x**2 + 2*x**m*(c + d*x)**n*d**2*f*m*n*x**2 + x**m*(c + d*x)**n*d**2*f*m*x**2 + x**m*(c + d*x) **n*d**2*f*n**2*x**2 + x**m*(c + d*x)**n*d**2*f*n*x**2 + int((x**m*(c + d* x)**n)/(c*m**3*x + 3*c*m**2*n*x + 3*c*m**2*x + 3*c*m*n**2*x + 6*c*m*n*x + 2*c*m*x + c*n**3*x + 3*c*n**2*x + 2*c*n*x + d*m**3*x**2 + 3*d*m**2*n*x**2 + 3*d*m**2*x**2 + 3*d*m*n**2*x**2 + 6*d*m*n*x**2 + 2*d*m*x**2 + d*n**3*x** 2 + 3*d*n**2*x**2 + 2*d*n*x**2),x)*c**3*f*m**5*n + 3*int((x**m*(c + d*x)** n)/(c*m**3*x + 3*c*m**2*n*x + 3*c*m**2*x + 3*c*m*n**2*x + 6*c*m*n*x + 2*c* m*x + c*n**3*x + 3*c*n**2*x + 2*c*n*x + d*m**3*x**2 + 3*d*m**2*n*x**2 + 3* d*m**2*x**2 + 3*d*m*n**2*x**2 + 6*d*m*n*x**2 + 2*d*m*x**2 + d*n**3*x**2 + 3*d*n**2*x**2 + 2*d*n*x**2),x)*c**3*f*m**4*n**2 + 4*int((x**m*(c + d*x)**n )/(c*m**3*x + 3*c*m**2*n*x + 3*c*m**2*x + 3*c*m*n**2*x + 6*c*m*n*x + 2*c*m *x + c*n**3*x + 3*c*n**2*x + 2*c*n*x + d*m**3*x**2 + 3*d*m**2*n*x**2 + 3*d *m**2*x**2 + 3*d*m*n**2*x**2 + 6*d*m*n*x**2 + 2*d*m*x**2 + d*n**3*x**2 + 3 *d*n**2*x**2 + 2*d*n*x**2),x)*c**3*f*m**4*n + 3*int((x**m*(c + d*x)**n)...