\(\int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx\) [589]

Optimal result

Integrand size = 22, antiderivative size = 79 \[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\frac {2 (b x)^{3/2} (c+d x)^n \left (1+\frac {d x}{c}\right )^{-n} (e+f x)^p \left (1+\frac {f x}{e}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-n,-p,\frac {5}{2},-\frac {d x}{c},-\frac {f x}{e}\right )}{3 b} \] Output:

2/3*(b*x)^(3/2)*(d*x+c)^n*(f*x+e)^p*AppellF1(3/2,-n,-p,5/2,-d*x/c,-f*x/e)/ 
b/((1+d*x/c)^n)/((1+f*x/e)^p)
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00 \[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\frac {2}{3} x \sqrt {b x} (c+d x)^n \left (\frac {c+d x}{c}\right )^{-n} (e+f x)^p \left (\frac {e+f x}{e}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-n,-p,\frac {5}{2},-\frac {d x}{c},-\frac {f x}{e}\right ) \] Input:

Integrate[Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p,x]
 

Output:

(2*x*Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p*AppellF1[3/2, -n, -p, 5/2, -((d*x)/ 
c), -((f*x)/e)])/(3*((c + d*x)/c)^n*((e + f*x)/e)^p)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {152, 152, 150}

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.

\(\displaystyle \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx\)

\(\Big \downarrow \) 152

\(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} \int \sqrt {b x} \left (\frac {d x}{c}+1\right )^n (e+f x)^pdx\)

\(\Big \downarrow \) 152

\(\displaystyle (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} \int \sqrt {b x} \left (\frac {d x}{c}+1\right )^n \left (\frac {f x}{e}+1\right )^pdx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {2 (b x)^{3/2} (c+d x)^n \left (\frac {d x}{c}+1\right )^{-n} (e+f x)^p \left (\frac {f x}{e}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-n,-p,\frac {5}{2},-\frac {d x}{c},-\frac {f x}{e}\right )}{3 b}\)

Input:

Int[Sqrt[b*x]*(c + d*x)^n*(e + f*x)^p,x]
 

Output:

(2*(b*x)^(3/2)*(c + d*x)^n*(e + f*x)^p*AppellF1[3/2, -n, -p, 5/2, -((d*x)/ 
c), -((f*x)/e)])/(3*b*(1 + (d*x)/c)^n*(1 + (f*x)/e)^p)
 

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 152

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^IntPart[n]*((c + d*x)^FracPart[n]/(1 + d*(x/c))^FracPart[n]) 
Int[(b*x)^m*(1 + d*(x/c))^n*(e + f*x)^p, x], x] /; FreeQ[{b, c, d, e, f, m, 
 n, p}, x] &&  !IntegerQ[m] &&  !IntegerQ[n] &&  !GtQ[c, 0]
 

Maple [F]

\[\int \sqrt {b x}\, \left (x d +c \right )^{n} \left (f x +e \right )^{p}d x\]

Input:

int((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x)
 

Output:

int((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x)
 

Fricas [F]

\[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\int { \sqrt {b x} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \] Input:

integrate((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x, algorithm="fricas")
 

Output:

integral(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p, x)
 

Sympy [F]

\[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\int \sqrt {b x} \left (c + d x\right )^{n} \left (e + f x\right )^{p}\, dx \] Input:

integrate((b*x)**(1/2)*(d*x+c)**n*(f*x+e)**p,x)
 

Output:

Integral(sqrt(b*x)*(c + d*x)**n*(e + f*x)**p, x)
 

Maxima [F]

\[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\int { \sqrt {b x} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \] Input:

integrate((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x, algorithm="maxima")
 

Output:

integrate(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p, x)
 

Giac [F]

\[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\int { \sqrt {b x} {\left (d x + c\right )}^{n} {\left (f x + e\right )}^{p} \,d x } \] Input:

integrate((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x, algorithm="giac")
 

Output:

integrate(sqrt(b*x)*(d*x + c)^n*(f*x + e)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\int {\left (e+f\,x\right )}^p\,\sqrt {b\,x}\,{\left (c+d\,x\right )}^n \,d x \] Input:

int((e + f*x)^p*(b*x)^(1/2)*(c + d*x)^n,x)
 

Output:

int((e + f*x)^p*(b*x)^(1/2)*(c + d*x)^n, x)
 

Reduce [F]

\[ \int \sqrt {b x} (c+d x)^n (e+f x)^p \, dx=\text {too large to display} \] Input:

int((b*x)^(1/2)*(d*x+c)^n*(f*x+e)^p,x)
 

Output:

(2*sqrt(b)*(2*sqrt(x)*(e + f*x)**p*(c + d*x)**n*c*e*n + 2*sqrt(x)*(e + f*x 
)**p*(c + d*x)**n*c*e*p + 2*sqrt(x)*(e + f*x)**p*(c + d*x)**n*c*f*p*x + sq 
rt(x)*(e + f*x)**p*(c + d*x)**n*c*f*x + 2*sqrt(x)*(e + f*x)**p*(c + d*x)** 
n*d*e*n*x + sqrt(x)*(e + f*x)**p*(c + d*x)**n*d*e*x + 8*int((sqrt(x)*(e + 
f*x)**p*(c + d*x)**n*x)/(4*c**2*e*f*n*p + 2*c**2*e*f*n + 4*c**2*e*f*p**2 + 
 8*c**2*e*f*p + 3*c**2*e*f + 4*c**2*f**2*n*p*x + 2*c**2*f**2*n*x + 4*c**2* 
f**2*p**2*x + 8*c**2*f**2*p*x + 3*c**2*f**2*x + 4*c*d*e**2*n**2 + 4*c*d*e* 
*2*n*p + 8*c*d*e**2*n + 2*c*d*e**2*p + 3*c*d*e**2 + 4*c*d*e*f*n**2*x + 8*c 
*d*e*f*n*p*x + 10*c*d*e*f*n*x + 4*c*d*e*f*p**2*x + 10*c*d*e*f*p*x + 6*c*d* 
e*f*x + 4*c*d*f**2*n*p*x**2 + 2*c*d*f**2*n*x**2 + 4*c*d*f**2*p**2*x**2 + 8 
*c*d*f**2*p*x**2 + 3*c*d*f**2*x**2 + 4*d**2*e**2*n**2*x + 4*d**2*e**2*n*p* 
x + 8*d**2*e**2*n*x + 2*d**2*e**2*p*x + 3*d**2*e**2*x + 4*d**2*e*f*n**2*x* 
*2 + 4*d**2*e*f*n*p*x**2 + 8*d**2*e*f*n*x**2 + 2*d**2*e*f*p*x**2 + 3*d**2* 
e*f*x**2),x)*c**3*f**3*n**2*p**2 + 8*int((sqrt(x)*(e + f*x)**p*(c + d*x)** 
n*x)/(4*c**2*e*f*n*p + 2*c**2*e*f*n + 4*c**2*e*f*p**2 + 8*c**2*e*f*p + 3*c 
**2*e*f + 4*c**2*f**2*n*p*x + 2*c**2*f**2*n*x + 4*c**2*f**2*p**2*x + 8*c** 
2*f**2*p*x + 3*c**2*f**2*x + 4*c*d*e**2*n**2 + 4*c*d*e**2*n*p + 8*c*d*e**2 
*n + 2*c*d*e**2*p + 3*c*d*e**2 + 4*c*d*e*f*n**2*x + 8*c*d*e*f*n*p*x + 10*c 
*d*e*f*n*x + 4*c*d*e*f*p**2*x + 10*c*d*e*f*p*x + 6*c*d*e*f*x + 4*c*d*f**2* 
n*p*x**2 + 2*c*d*f**2*n*x**2 + 4*c*d*f**2*p**2*x**2 + 8*c*d*f**2*p*x**2...