\(\int (1-\sqrt {-b} x)^m (1+\sqrt {-b} x)^m (c+d x)^p \, dx\) [1837]

Optimal result

Integrand size = 35, antiderivative size = 132 \[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\frac {(c+d x)^{1+p} \left (1+b x^2\right )^m \left (1-\frac {c+d x}{c+\frac {d}{\sqrt {-b}}}\right )^{-m} \left (1-\frac {c+d x}{c+\frac {b d}{(-b)^{3/2}}}\right )^{-m} \operatorname {AppellF1}\left (1+p,-m,-m,2+p,\frac {c+d x}{c+\frac {b d}{(-b)^{3/2}}},\frac {c+d x}{c+\frac {d}{\sqrt {-b}}}\right )}{d (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.14 \[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\frac {\left (\frac {d \left (\sqrt {-\frac {1}{b}}-x\right )}{c+\sqrt {-\frac {1}{b}} d}\right )^{-m} \left (\frac {d \left (\sqrt {-\frac {1}{b}}+x\right )}{-c+\sqrt {-\frac {1}{b}} d}\right )^{-m} (c+d x)^{1+p} \left (1+b x^2\right )^m \operatorname {AppellF1}\left (1+p,-m,-m,2+p,\frac {c+d x}{c+\left (-\frac {1}{b}\right )^{3/2} b d},\frac {c+d x}{c+\sqrt {-\frac {1}{b}} d}\right )}{d (1+p)} \] Input:

Integrate[(1 - Sqrt[-b]*x)^m*(1 + Sqrt[-b]*x)^m*(c + d*x)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {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.

\(\displaystyle \int \left (1-\sqrt {-b} x\right )^m \left (\sqrt {-b} x+1\right )^m (c+d x)^p \, dx\)

\(\Big \downarrow \) 156

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

\(\Big \downarrow \) 155

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

Input:

Int[(1 - Sqrt[-b]*x)^m*(1 + Sqrt[-b]*x)^m*(c + d*x)^p,x]
 

Output:

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

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]
 

Maple [F]

\[\int \left (1-\sqrt {-b}\, x \right )^{m} \left (1+\sqrt {-b}\, x \right )^{m} \left (x d +c \right )^{p}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\int { {\left (d x + c\right )}^{p} {\left (\sqrt {-b} x + 1\right )}^{m} {\left (-\sqrt {-b} x + 1\right )}^{m} \,d x } \] Input:

integrate((1-(-b)^(1/2)*x)^m*(1+(-b)^(1/2)*x)^m*(d*x+c)^p,x, algorithm="fr 
icas")
 

Output:

integral((d*x + c)^p*(sqrt(-b)*x + 1)^m*(-sqrt(-b)*x + 1)^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\int { {\left (d x + c\right )}^{p} {\left (\sqrt {-b} x + 1\right )}^{m} {\left (-\sqrt {-b} x + 1\right )}^{m} \,d x } \] Input:

integrate((1-(-b)^(1/2)*x)^m*(1+(-b)^(1/2)*x)^m*(d*x+c)^p,x, algorithm="ma 
xima")
 

Output:

integrate((d*x + c)^p*(sqrt(-b)*x + 1)^m*(-sqrt(-b)*x + 1)^m, x)
 

Giac [F]

\[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\int { {\left (d x + c\right )}^{p} {\left (\sqrt {-b} x + 1\right )}^{m} {\left (-\sqrt {-b} x + 1\right )}^{m} \,d x } \] Input:

integrate((1-(-b)^(1/2)*x)^m*(1+(-b)^(1/2)*x)^m*(d*x+c)^p,x, algorithm="gi 
ac")
 

Output:

integrate((d*x + c)^p*(sqrt(-b)*x + 1)^m*(-sqrt(-b)*x + 1)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\int {\left (1-\sqrt {-b}\,x\right )}^m\,{\left (\sqrt {-b}\,x+1\right )}^m\,{\left (c+d\,x\right )}^p \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \left (1-\sqrt {-b} x\right )^m \left (1+\sqrt {-b} x\right )^m (c+d x)^p \, dx=\text {too large to display} \] Input:

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

Output:

((c + d*x)**p*(sqrt(b)*i*x + 1)**m*( - sqrt(b)*i*x + 1)**m*b*c*x + (c + d* 
x)**p*(sqrt(b)*i*x + 1)**m*( - sqrt(b)*i*x + 1)**m*d + 2*int(((c + d*x)**p 
*(sqrt(b)*i*x + 1)**m*( - sqrt(b)*i*x + 1)**m*x**2)/(2*b*c*m*x**2 + b*c*p* 
x**2 + b*c*x**2 + 2*b*d*m*x**3 + b*d*p*x**3 + b*d*x**3 + 2*c*m + c*p + c + 
 2*d*m*x + d*p*x + d*x),x)*b**2*c**2*m*p + int(((c + d*x)**p*(sqrt(b)*i*x 
+ 1)**m*( - sqrt(b)*i*x + 1)**m*x**2)/(2*b*c*m*x**2 + b*c*p*x**2 + b*c*x** 
2 + 2*b*d*m*x**3 + b*d*p*x**3 + b*d*x**3 + 2*c*m + c*p + c + 2*d*m*x + d*p 
*x + d*x),x)*b**2*c**2*p**2 + int(((c + d*x)**p*(sqrt(b)*i*x + 1)**m*( - s 
qrt(b)*i*x + 1)**m*x**2)/(2*b*c*m*x**2 + b*c*p*x**2 + b*c*x**2 + 2*b*d*m*x 
**3 + b*d*p*x**3 + b*d*x**3 + 2*c*m + c*p + c + 2*d*m*x + d*p*x + d*x),x)* 
b**2*c**2*p - 4*int(((c + d*x)**p*(sqrt(b)*i*x + 1)**m*( - sqrt(b)*i*x + 1 
)**m*x**2)/(2*b*c*m*x**2 + b*c*p*x**2 + b*c*x**2 + 2*b*d*m*x**3 + b*d*p*x* 
*3 + b*d*x**3 + 2*c*m + c*p + c + 2*d*m*x + d*p*x + d*x),x)*b*d**2*m**2 - 
4*int(((c + d*x)**p*(sqrt(b)*i*x + 1)**m*( - sqrt(b)*i*x + 1)**m*x**2)/(2* 
b*c*m*x**2 + b*c*p*x**2 + b*c*x**2 + 2*b*d*m*x**3 + b*d*p*x**3 + b*d*x**3 
+ 2*c*m + c*p + c + 2*d*m*x + d*p*x + d*x),x)*b*d**2*m*p - 2*int(((c + d*x 
)**p*(sqrt(b)*i*x + 1)**m*( - sqrt(b)*i*x + 1)**m*x**2)/(2*b*c*m*x**2 + b* 
c*p*x**2 + b*c*x**2 + 2*b*d*m*x**3 + b*d*p*x**3 + b*d*x**3 + 2*c*m + c*p + 
 c + 2*d*m*x + d*p*x + d*x),x)*b*d**2*m - int(((c + d*x)**p*(sqrt(b)*i*x + 
 1)**m*( - sqrt(b)*i*x + 1)**m*x**2)/(2*b*c*m*x**2 + b*c*p*x**2 + b*c*x...