\(\int (a+b \sqrt {x})^p (c+d \sqrt {x})^q x^m \, dx\) [252]

Optimal result

Integrand size = 26, antiderivative size = 104 \[ \int \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q x^m \, dx=\frac {\left (a+b \sqrt {x}\right )^p \left (1+\frac {b \sqrt {x}}{a}\right )^{-p} \left (c+d \sqrt {x}\right )^q \left (1+\frac {d \sqrt {x}}{c}\right )^{-q} x^{1+m} \operatorname {AppellF1}\left (2 (1+m),-p,-q,3+2 m,-\frac {b \sqrt {x}}{a},-\frac {d \sqrt {x}}{c}\right )}{1+m} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00 \[ \int \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q x^m \, dx=\frac {\left (a+b \sqrt {x}\right )^p \left (1+\frac {b \sqrt {x}}{a}\right )^{-p} \left (c+d \sqrt {x}\right )^q \left (1+\frac {d \sqrt {x}}{c}\right )^{-q} x^{1+m} \operatorname {AppellF1}\left (2+2 m,-p,-q,3+2 m,-\frac {b \sqrt {x}}{a},-\frac {d \sqrt {x}}{c}\right )}{1+m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1000, 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.

\(\displaystyle \int x^m \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q \, dx\)

\(\Big \downarrow \) 1000

\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q x^{\frac {1}{2} (2 m+1)}d\sqrt {x}\)

\(\Big \downarrow \) 152

\(\displaystyle 2 \left (a+b \sqrt {x}\right )^p \left (\frac {b \sqrt {x}}{a}+1\right )^{-p} \int \left (\frac {\sqrt {x} b}{a}+1\right )^p \left (c+d \sqrt {x}\right )^q x^{\frac {1}{2} (2 m+1)}d\sqrt {x}\)

\(\Big \downarrow \) 152

\(\displaystyle 2 \left (a+b \sqrt {x}\right )^p \left (\frac {b \sqrt {x}}{a}+1\right )^{-p} \left (c+d \sqrt {x}\right )^q \left (\frac {d \sqrt {x}}{c}+1\right )^{-q} \int \left (\frac {\sqrt {x} b}{a}+1\right )^p \left (\frac {\sqrt {x} d}{c}+1\right )^q x^{\frac {1}{2} (2 m+1)}d\sqrt {x}\)

\(\Big \downarrow \) 150

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

Input:

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

Output:

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

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]
 

rule 1000

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{g = Denominator[n]}, Simp[g   Subst[Int[x^(g*(m + 1) - 
1)*(a + b*x^(g*n))^p*(c + d*x^(g*n))^q, x], x, x^(1/g)], x]] /; FreeQ[{a, b 
, c, d, m, p, q}, x] && NeQ[b*c - a*d, 0] && FractionQ[n]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

Exception generated. \[ \int \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q x^m \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >>  Error detected within library code:   algl 
ogextint: unimplemented
 

Sympy [F(-2)]

Exception generated. \[ \int \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q x^m \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((b*sqrt(x) + a)^p*(d*sqrt(x) + c)^q*x^m, x)
 

Giac [F]

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

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

Output:

integrate((b*sqrt(x) + a)^p*(d*sqrt(x) + c)^q*x^m, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \left (a+b \sqrt {x}\right )^p \left (c+d \sqrt {x}\right )^q x^m \, dx=\text {too large to display} \] Input:

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

Output:

(4*x**((2*m + 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a**2*d**2*m*p + 
2*x**((2*m + 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a**2*d**2*p*q + 4 
*x**((2*m + 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*c*d*m*p + 4*x* 
*((2*m + 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*c*d*m*q + 2*x**(( 
2*m + 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*c*d*p**2 + 2*x**((2* 
m + 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*c*d*q**2 + 4*x**((2*m 
+ 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*b**2*c**2*m*q + 2*x**((2*m + 
 1)/2)*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*b**2*c**2*p*q - 4*x**m*(sqrt( 
x)*d + c)**q*(sqrt(x)*b + a)**p*a**2*c*d*m*p - 2*x**m*(sqrt(x)*d + c)**q*( 
sqrt(x)*b + a)**p*a**2*c*d*p - 4*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)** 
p*a*b*c**2*m*q - 2*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*c**2*q + 
 8*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*d**2*m**2*x + 4*x**m*(sq 
rt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*d**2*m*p*x + 8*x**m*(sqrt(x)*d + c) 
**q*(sqrt(x)*b + a)**p*a*b*d**2*m*q*x + 4*x**m*(sqrt(x)*d + c)**q*(sqrt(x) 
*b + a)**p*a*b*d**2*m*x + 2*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b 
*d**2*p*q*x + 2*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*d**2*q**2*x 
 + 2*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**p*a*b*d**2*q*x + 8*x**m*(sqr 
t(x)*d + c)**q*(sqrt(x)*b + a)**p*b**2*c*d*m**2*x + 8*x**m*(sqrt(x)*d + c) 
**q*(sqrt(x)*b + a)**p*b**2*c*d*m*p*x + 4*x**m*(sqrt(x)*d + c)**q*(sqrt(x) 
*b + a)**p*b**2*c*d*m*q*x + 4*x**m*(sqrt(x)*d + c)**q*(sqrt(x)*b + a)**...