\(\int \frac {\sqrt {c+d x} (e+f x)^m}{\sqrt {a+b x}} \, dx\) [1698]

Optimal result

Integrand size = 26, antiderivative size = 121 \[ \int \frac {\sqrt {c+d x} (e+f x)^m}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^m \left (\frac {b (e+f x)}{b e-a f}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}} \] Output:

2*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^m*AppellF1(1/2,-1/2,-m,3/2,-d*(b*x+a 
)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/b/(b*(d*x+c)/(-a*d+b*c))^(1/2)/((b*(f* 
x+e)/(-a*f+b*e))^m)
 

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {c+d x} (e+f x)^m}{\sqrt {a+b x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^m \left (\frac {b (e+f x)}{b e-a f}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}} \] Input:

Integrate[(Sqrt[c + d*x]*(e + f*x)^m)/Sqrt[a + b*x],x]
 

Output:

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^m*AppellF1[1/2, -1/2, -m, 3/2, (d 
*(a + b*x))/(-(b*c) + a*d), (f*(a + b*x))/(-(b*e) + a*f)])/(b*Sqrt[(b*(c + 
 d*x))/(b*c - a*d)]*((b*(e + f*x))/(b*e - a*f))^m)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {157, 156, 155}

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.

\(\displaystyle \int \frac {\sqrt {c+d x} (e+f x)^m}{\sqrt {a+b x}} \, dx\)

\(\Big \downarrow \) 157

\(\displaystyle \frac {\sqrt {c+d x} \int \frac {\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} (e+f x)^m}{\sqrt {a+b x}}dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\)

\(\Big \downarrow \) 156

\(\displaystyle \frac {\sqrt {c+d x} (e+f x)^m \left (\frac {b (e+f x)}{b e-a f}\right )^{-m} \int \frac {\sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \left (\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}\right )^m}{\sqrt {a+b x}}dx}{\sqrt {\frac {b (c+d x)}{b c-a d}}}\)

\(\Big \downarrow \) 155

\(\displaystyle \frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^m \left (\frac {b (e+f x)}{b e-a f}\right )^{-m} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b \sqrt {\frac {b (c+d x)}{b c-a d}}}\)

Input:

Int[(Sqrt[c + d*x]*(e + f*x)^m)/Sqrt[a + b*x],x]
 

Output:

(2*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^m*AppellF1[1/2, -1/2, -m, 3/2, -( 
(d*(a + b*x))/(b*c - a*d)), -((f*(a + b*x))/(b*e - a*f))])/(b*Sqrt[(b*(c + 
 d*x))/(b*c - a*d)]*((b*(e + f*x))/(b*e - a*f))^m)
 

Defintions of rubi rules used

rule 155

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

rule 156

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

rule 157

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(d*x + c)*(f*x + e)^m/sqrt(b*x + a), x)
 

Sympy [F]

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

integrate((d*x+c)**(1/2)*(f*x+e)**m/(b*x+a)**(1/2),x)
 

Output:

Integral(sqrt(c + d*x)*(e + f*x)**m/sqrt(a + b*x), x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(d*x + c)*(f*x + e)^m/sqrt(b*x + a), x)
 

Giac [F]

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

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

Output:

integrate(sqrt(d*x + c)*(f*x + e)^m/sqrt(b*x + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(((e + f*x)**m*sqrt(c + d*x))/sqrt(a + b*x),x)