\(\int (c+d x)^n (a+b x^2)^p \, dx\) [358]

Optimal result

Integrand size = 17, antiderivative size = 152 \[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\frac {(c+d x)^{1+n} \left (a+b x^2\right )^p \left (1-\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \left (1-\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )^{-p} \operatorname {AppellF1}\left (1+n,-p,-p,2+n,\frac {c+d x}{c-\frac {\sqrt {-a} d}{\sqrt {b}}},\frac {c+d x}{c+\frac {\sqrt {-a} d}{\sqrt {b}}}\right )}{d (1+n)} \] Output:

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.03 \[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\frac {\left (\frac {d \left (\sqrt {-\frac {a}{b}}-x\right )}{c+\sqrt {-\frac {a}{b}} d}\right )^{-p} \left (\frac {d \left (\sqrt {-\frac {a}{b}}+x\right )}{-c+\sqrt {-\frac {a}{b}} d}\right )^{-p} (c+d x)^{1+n} \left (a+b x^2\right )^p \operatorname {AppellF1}\left (1+n,-p,-p,2+n,\frac {c+d x}{c-\sqrt {-\frac {a}{b}} d},\frac {c+d x}{c+\sqrt {-\frac {a}{b}} d}\right )}{d (1+n)} \] Input:

Integrate[(c + d*x)^n*(a + b*x^2)^p,x]
 

Output:

((c + d*x)^(1 + n)*(a + b*x^2)^p*AppellF1[1 + n, -p, -p, 2 + n, (c + d*x)/ 
(c - Sqrt[-(a/b)]*d), (c + d*x)/(c + Sqrt[-(a/b)]*d)])/(d*(1 + n)*((d*(Sqr 
t[-(a/b)] - x))/(c + Sqrt[-(a/b)]*d))^p*((d*(Sqrt[-(a/b)] + x))/(-c + Sqrt 
[-(a/b)]*d))^p)
 

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {514, 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.

\(\displaystyle \int \left (a+b x^2\right )^p (c+d x)^n \, dx\)

\(\Big \downarrow \) 514

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

\(\Big \downarrow \) 150

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

Input:

Int[(c + d*x)^n*(a + b*x^2)^p,x]
 

Output:

((c + d*x)^(1 + n)*(a + b*x^2)^p*AppellF1[1 + n, -p, -p, 2 + n, (c + d*x)/ 
(c - (Sqrt[-a]*d)/Sqrt[b]), (c + d*x)/(c + (Sqrt[-a]*d)/Sqrt[b])])/(d*(1 + 
 n)*(1 - (c + d*x)/(c - (Sqrt[-a]*d)/Sqrt[b]))^p*(1 - (c + d*x)/(c + (Sqrt 
[-a]*d)/Sqrt[b]))^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 514

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ 
{q = Rt[-a/b, 2]}, Simp[(a + b*x^2)^p/(d*(1 - (c + d*x)/(c - d*q))^p*(1 - ( 
c + d*x)/(c + d*q))^p)   Subst[Int[x^n*Simp[1 - x/(c + d*q), x]^p*Simp[1 - 
x/(c - d*q), x]^p, x], x, c + d*x], x]] /; FreeQ[{a, b, c, d, n, p}, x] && 
NeQ[b*c^2 + a*d^2, 0]
 

Maple [F]

\[\int \left (d x +c \right )^{n} \left (b \,x^{2}+a \right )^{p}d x\]

Input:

int((d*x+c)^n*(b*x^2+a)^p,x)
 

Output:

int((d*x+c)^n*(b*x^2+a)^p,x)
 

Fricas [F]

\[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^p,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(d*x + c)^n, x)
 

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**n*(b*x**2+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^n, x)
 

Giac [F]

\[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\int { {\left (b x^{2} + a\right )}^{p} {\left (d x + c\right )}^{n} \,d x } \] Input:

integrate((d*x+c)^n*(b*x^2+a)^p,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*(d*x + c)^n, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\int {\left (b\,x^2+a\right )}^p\,{\left (c+d\,x\right )}^n \,d x \] Input:

int((a + b*x^2)^p*(c + d*x)^n,x)
 

Output:

int((a + b*x^2)^p*(c + d*x)^n, x)
 

Reduce [F]

\[ \int (c+d x)^n \left (a+b x^2\right )^p \, dx=\text {too large to display} \] Input:

int((d*x+c)^n*(b*x^2+a)^p,x)
 

Output:

((c + d*x)**n*(a + b*x**2)**p*a*d + (c + d*x)**n*(a + b*x**2)**p*b*c*x - i 
nt(((c + d*x)**n*(a + b*x**2)**p*x**2)/(a*c*n + 2*a*c*p + a*c + a*d*n*x + 
2*a*d*p*x + a*d*x + b*c*n*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*n*x**3 + 2* 
b*d*p*x**3 + b*d*x**3),x)*a*b*d**2*n**2 - 4*int(((c + d*x)**n*(a + b*x**2) 
**p*x**2)/(a*c*n + 2*a*c*p + a*c + a*d*n*x + 2*a*d*p*x + a*d*x + b*c*n*x** 
2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*n*x**3 + 2*b*d*p*x**3 + b*d*x**3),x)*a*b 
*d**2*n*p - int(((c + d*x)**n*(a + b*x**2)**p*x**2)/(a*c*n + 2*a*c*p + a*c 
 + a*d*n*x + 2*a*d*p*x + a*d*x + b*c*n*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b* 
d*n*x**3 + 2*b*d*p*x**3 + b*d*x**3),x)*a*b*d**2*n - 4*int(((c + d*x)**n*(a 
 + b*x**2)**p*x**2)/(a*c*n + 2*a*c*p + a*c + a*d*n*x + 2*a*d*p*x + a*d*x + 
 b*c*n*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*n*x**3 + 2*b*d*p*x**3 + b*d*x* 
*3),x)*a*b*d**2*p**2 - 2*int(((c + d*x)**n*(a + b*x**2)**p*x**2)/(a*c*n + 
2*a*c*p + a*c + a*d*n*x + 2*a*d*p*x + a*d*x + b*c*n*x**2 + 2*b*c*p*x**2 + 
b*c*x**2 + b*d*n*x**3 + 2*b*d*p*x**3 + b*d*x**3),x)*a*b*d**2*p + int(((c + 
 d*x)**n*(a + b*x**2)**p*x**2)/(a*c*n + 2*a*c*p + a*c + a*d*n*x + 2*a*d*p* 
x + a*d*x + b*c*n*x**2 + 2*b*c*p*x**2 + b*c*x**2 + b*d*n*x**3 + 2*b*d*p*x* 
*3 + b*d*x**3),x)*b**2*c**2*n**2 + 2*int(((c + d*x)**n*(a + b*x**2)**p*x** 
2)/(a*c*n + 2*a*c*p + a*c + a*d*n*x + 2*a*d*p*x + a*d*x + b*c*n*x**2 + 2*b 
*c*p*x**2 + b*c*x**2 + b*d*n*x**3 + 2*b*d*p*x**3 + b*d*x**3),x)*b**2*c**2* 
n*p + int(((c + d*x)**n*(a + b*x**2)**p*x**2)/(a*c*n + 2*a*c*p + a*c + ...