\(\int \frac {(a+b x)^m \sqrt {c+d x}}{\sqrt {c-d x}} \, dx\) [131]

Optimal result

Integrand size = 27, antiderivative size = 112 \[ \int \frac {(a+b x)^m \sqrt {c+d x}}{\sqrt {c-d x}} \, dx=-\frac {2 \sqrt {2} (a+b x)^m \left (\frac {d (a+b x)}{b c+a d}\right )^{-m} \sqrt {c-d x} \sqrt {c+d x} \operatorname {AppellF1}\left (\frac {1}{2},-\frac {1}{2},-m,\frac {3}{2},\frac {c-d x}{2 c},\frac {b (c-d x)}{b c+a d}\right )}{d \sqrt {\frac {c+d x}{c}}} \] Output:

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

Mathematica [A] (verified)

Time = 1.39 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^m \sqrt {c+d x}}{\sqrt {c-d x}} \, dx=\frac {(a+b x)^{1+m} \sqrt {\frac {b (c-d x)}{b c+a d}} \sqrt {c+d x} \operatorname {AppellF1}\left (1+m,\frac {1}{2},-\frac {1}{2},2+m,\frac {d (a+b x)}{b c+a d},\frac {d (a+b x)}{-b c+a d}\right )}{b (1+m) \sqrt {c-d x} \sqrt {\frac {b (c+d x)}{b c-a d}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.

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

\(\Big \downarrow \) 157

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

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

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

Input:

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

Output:

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

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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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