\(\int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx\) [237]

Optimal result

Integrand size = 35, antiderivative size = 174 \[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=-\frac {a (2+p+q) x^p (a+b x)^p (a+2 b x)^{1+q} \left (1-\frac {(a+2 b x)^2}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {1+q}{2},\frac {3+q}{2},\frac {(a+2 b x)^2}{a^2}\right )}{2 b (1+q)}+\frac {(3+2 p+q) x^p (a+b x)^p (a+2 b x)^{2+q} \left (1-\frac {(a+2 b x)^2}{a^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,\frac {2+q}{2},\frac {4+q}{2},\frac {(a+2 b x)^2}{a^2}\right )}{2 b (2+q)} \] Output:

-1/2*a*(2+p+q)*x^p*(b*x+a)^p*(2*b*x+a)^(1+q)*hypergeom([-p, 1/2+1/2*q],[3/ 
2+1/2*q],(2*b*x+a)^2/a^2)/b/(1+q)/((1-(2*b*x+a)^2/a^2)^p)+1/2*(3+2*p+q)*x^ 
p*(b*x+a)^p*(2*b*x+a)^(2+q)*hypergeom([-p, 1+1/2*q],[2+1/2*q],(2*b*x+a)^2/ 
a^2)/b/(2+q)/((1-(2*b*x+a)^2/a^2)^p)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.70 \[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\frac {x^{1+p} (a+b x)^p (a+2 b x)^q \left (1+\frac {b x}{a}\right )^{-p} \left (1+\frac {2 b x}{a}\right )^{-q} \left (a (2+p) \operatorname {AppellF1}\left (1+p,-p,-q,2+p,-\frac {b x}{a},-\frac {2 b x}{a}\right )+2 b (3+2 p+q) x \operatorname {AppellF1}\left (2+p,-p,-q,3+p,-\frac {b x}{a},-\frac {2 b x}{a}\right )\right )}{2+p} \] Input:

Integrate[x^p*(a + b*x)^p*(a + 2*b*x)^q*(a*(1 + p) + 2*b*(3 + 2*p + q)*x), 
x]
 

Output:

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

Rubi [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.22, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {177, 146, 152, 152, 150, 279, 278}

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.

\(\displaystyle \int x^p (a+b x)^p (a+2 b x)^q (a (p+1)+2 b x (2 p+q+3)) \, dx\)

\(\Big \downarrow \) 177

\(\displaystyle 2 b (2 p+q+3) \int x^{p+1} (a+b x)^p (a+2 b x)^qdx+a (p+1) \int x^p (a+b x)^p (a+2 b x)^qdx\)

\(\Big \downarrow \) 146

\(\displaystyle \frac {a (p+1) x^p (a+b x)^p \left (a x+b x^2\right )^{-p} \int (a+2 b x)^q \left (\frac {(a+2 b x)^2}{4 b}-\frac {a^2}{4 b}\right )^pd(a+2 b x)}{2 b}+2 b (2 p+q+3) \int x^{p+1} (a+b x)^p (a+2 b x)^qdx\)

\(\Big \downarrow \) 152

\(\displaystyle \frac {a (p+1) x^p (a+b x)^p \left (a x+b x^2\right )^{-p} \int (a+2 b x)^q \left (\frac {(a+2 b x)^2}{4 b}-\frac {a^2}{4 b}\right )^pd(a+2 b x)}{2 b}+2 b (2 p+q+3) (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} \int x^{p+1} (a+2 b x)^q \left (\frac {b x}{a}+1\right )^pdx\)

\(\Big \downarrow \) 152

\(\displaystyle \frac {a (p+1) x^p (a+b x)^p \left (a x+b x^2\right )^{-p} \int (a+2 b x)^q \left (\frac {(a+2 b x)^2}{4 b}-\frac {a^2}{4 b}\right )^pd(a+2 b x)}{2 b}+2 b (2 p+q+3) (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} (a+2 b x)^q \left (\frac {2 b x}{a}+1\right )^{-q} \int x^{p+1} \left (\frac {b x}{a}+1\right )^p \left (\frac {2 b x}{a}+1\right )^qdx\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {a (p+1) x^p (a+b x)^p \left (a x+b x^2\right )^{-p} \int (a+2 b x)^q \left (\frac {(a+2 b x)^2}{4 b}-\frac {a^2}{4 b}\right )^pd(a+2 b x)}{2 b}+\frac {2 b (2 p+q+3) x^{p+2} (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} (a+2 b x)^q \left (\frac {2 b x}{a}+1\right )^{-q} \operatorname {AppellF1}\left (p+2,-p,-q,p+3,-\frac {b x}{a},-\frac {2 b x}{a}\right )}{p+2}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {a (p+1) x^p (a+b x)^p \left (a x+b x^2\right )^{-p} \left (1-\frac {(a+2 b x)^2}{a^2}\right )^{-p} \left (\frac {(a+2 b x)^2}{4 b}-\frac {a^2}{4 b}\right )^p \int (a+2 b x)^q \left (1-\frac {(a+2 b x)^2}{a^2}\right )^pd(a+2 b x)}{2 b}+\frac {2 b (2 p+q+3) x^{p+2} (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} (a+2 b x)^q \left (\frac {2 b x}{a}+1\right )^{-q} \operatorname {AppellF1}\left (p+2,-p,-q,p+3,-\frac {b x}{a},-\frac {2 b x}{a}\right )}{p+2}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {a (p+1) x^p (a+b x)^p \left (a x+b x^2\right )^{-p} \left (1-\frac {(a+2 b x)^2}{a^2}\right )^{-p} \left (\frac {(a+2 b x)^2}{4 b}-\frac {a^2}{4 b}\right )^p (a+2 b x)^{q+1} \operatorname {Hypergeometric2F1}\left (-p,\frac {q+1}{2},\frac {q+3}{2},\frac {(a+2 b x)^2}{a^2}\right )}{2 b (q+1)}+\frac {2 b (2 p+q+3) x^{p+2} (a+b x)^p \left (\frac {b x}{a}+1\right )^{-p} (a+2 b x)^q \left (\frac {2 b x}{a}+1\right )^{-q} \operatorname {AppellF1}\left (p+2,-p,-q,p+3,-\frac {b x}{a},-\frac {2 b x}{a}\right )}{p+2}\)

Input:

Int[x^p*(a + b*x)^p*(a + 2*b*x)^q*(a*(1 + p) + 2*b*(3 + 2*p + q)*x),x]
 

Output:

(2*b*(3 + 2*p + q)*x^(2 + p)*(a + b*x)^p*(a + 2*b*x)^q*AppellF1[2 + p, -p, 
 -q, 3 + p, -((b*x)/a), (-2*b*x)/a])/((2 + p)*(1 + (b*x)/a)^p*(1 + (2*b*x) 
/a)^q) + (a*(1 + p)*x^p*(a + b*x)^p*(a + 2*b*x)^(1 + q)*(-1/4*a^2/b + (a + 
 2*b*x)^2/(4*b))^p*Hypergeometric2F1[-p, (1 + q)/2, (3 + q)/2, (a + 2*b*x) 
^2/a^2])/(2*b*(1 + q)*(a*x + b*x^2)^p*(1 - (a + 2*b*x)^2/a^2)^p)
 

Defintions of rubi rules used

rule 146

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(c + d*x)^n*((e + f*x)^p/(b*(c*e + (d*e + c*f)*x + d*f* 
x^2)^n))   Subst[Int[x^m*(c*e - (d*e + c*f)^2/(4*d*f) + d*f*(x^2/b^2))^n, x 
], x, a + b*x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[p, n] && 
 EqQ[b*d*e + b*c*f - 2*a*d*f, 0]
 

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 177

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h/b   Int[(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p, x], x] + Simp[(b*g - a*h)/b   Int[(a + b*x)^m*(c + d*x)^ 
n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p}, x] && (Su 
mSimplerQ[m, 1] || ( !SumSimplerQ[n, 1] &&  !SumSimplerQ[p, 1]))
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

Maple [F]

\[\int x^{p} \left (b x +a \right )^{p} \left (2 b x +a \right )^{q} \left (a \left (p +1\right )+2 b \left (3+2 p +q \right ) x \right )d x\]

Input:

int(x^p*(b*x+a)^p*(2*b*x+a)^q*(a*(p+1)+2*b*(3+2*p+q)*x),x)
 

Output:

int(x^p*(b*x+a)^p*(2*b*x+a)^q*(a*(p+1)+2*b*(3+2*p+q)*x),x)
 

Fricas [F]

\[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\int { {\left (2 \, b {\left (2 \, p + q + 3\right )} x + a {\left (p + 1\right )}\right )} {\left (2 \, b x + a\right )}^{q} {\left (b x + a\right )}^{p} x^{p} \,d x } \] Input:

integrate(x^p*(b*x+a)^p*(2*b*x+a)^q*(a*(p+1)+2*b*(3+2*p+q)*x),x, algorithm 
="fricas")
 

Output:

integral((a*p + 2*(2*b*p + b*q + 3*b)*x + a)*(2*b*x + a)^q*(b*x + a)^p*x^p 
, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 49.66 (sec) , antiderivative size = 1647, normalized size of antiderivative = 9.47 \[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\text {Too large to display} \] Input:

integrate(x**p*(b*x+a)**p*(2*b*x+a)**q*(a*(p+1)+2*b*(3+2*p+q)*x),x)
 

Output:

a**(2*p + 1)*a**(q + 1)*b**q*b**(-p - q - 1)*p*meijerg(((-p/2 - q/2, -p/2 
- q/2 + 1/2, 1), (1/2 - q/2, -p - q/2, -p - q/2 + 1/2)), ((-p - q/2 - 1/2, 
 -p - q/2, -p/2 - q/2, -p - q/2 + 1/2, -p/2 - q/2 + 1/2), (0,)), a**2/(4*b 
**2*(a/(2*b) + x)**2))/(8*2**(2*p)*pi*gamma(-p)) - a**(2*p + 1)*a**(q + 1) 
*b**q*b**(-p - q - 1)*p*meijerg(((-p/2 - q/2 - 1/2, -p/2 - q/2, 1), (-q/2, 
 -p - q/2 - 1/2, -p - q/2)), ((-p - q/2 - 1/2, -p/2 - q/2 - 1/2, -p - q/2, 
 -p/2 - q/2, -p - q/2 - 1), (0,)), a**2/(4*b**2*(a/(2*b) + x)**2))/(4*2**( 
2*p)*pi*gamma(-p)) - a**(2*p + 1)*a**(q + 1)*b**q*b**(-p - q - 1)*p*meijer 
g(((-q/2 - 1, -q/2 - 1/2, -q/2, -p/2 - q/2 - 1/2, -p/2 - q/2 - 1, 1), ()), 
 ((-p/2 - q/2 - 1/2, -p/2 - q/2 - 1), (-q/2 - 1, -q/2 - 1/2, -p - q/2 - 1, 
 0)), a**2*exp_polar(2*I*pi)/(4*b**2*(a/(2*b) + x)**2))*exp(I*pi*p)*exp(I* 
pi*q)/(4*2**(2*p)*pi*gamma(-p)) - a**(2*p + 1)*a**(q + 1)*b**q*b**(-p - q 
- 1)*p*meijerg(((-q/2 - 1/2, -q/2, 1/2 - q/2, -p/2 - q/2 - 1/2, -p/2 - q/2 
, 1), ()), ((-p/2 - q/2 - 1/2, -p/2 - q/2), (-q/2 - 1/2, -q/2, -p - q/2 - 
1/2, 0)), a**2*exp_polar(2*I*pi)/(4*b**2*(a/(2*b) + x)**2))*exp(I*pi*p)*ex 
p(I*pi*q)/(8*2**(2*p)*pi*gamma(-p)) + a**(2*p + 1)*a**(q + 1)*b**q*b**(-p 
- q - 1)*q*meijerg(((-p/2 - q/2, -p/2 - q/2 + 1/2, 1), (1/2 - q/2, -p - q/ 
2, -p - q/2 + 1/2)), ((-p - q/2 - 1/2, -p - q/2, -p/2 - q/2, -p - q/2 + 1/ 
2, -p/2 - q/2 + 1/2), (0,)), a**2/(4*b**2*(a/(2*b) + x)**2))/(8*2**(2*p)*p 
i*gamma(-p)) - a**(2*p + 1)*a**(q + 1)*b**q*b**(-p - q - 1)*q*meijerg((...
 

Maxima [F]

\[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\int { {\left (2 \, b {\left (2 \, p + q + 3\right )} x + a {\left (p + 1\right )}\right )} {\left (2 \, b x + a\right )}^{q} {\left (b x + a\right )}^{p} x^{p} \,d x } \] Input:

integrate(x^p*(b*x+a)^p*(2*b*x+a)^q*(a*(p+1)+2*b*(3+2*p+q)*x),x, algorithm 
="maxima")
 

Output:

integrate((2*b*(2*p + q + 3)*x + a*(p + 1))*(2*b*x + a)^q*(b*x + a)^p*x^p, 
 x)
 

Giac [F]

\[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\int { {\left (2 \, b {\left (2 \, p + q + 3\right )} x + a {\left (p + 1\right )}\right )} {\left (2 \, b x + a\right )}^{q} {\left (b x + a\right )}^{p} x^{p} \,d x } \] Input:

integrate(x^p*(b*x+a)^p*(2*b*x+a)^q*(a*(p+1)+2*b*(3+2*p+q)*x),x, algorithm 
="giac")
 

Output:

integrate((2*b*(2*p + q + 3)*x + a*(p + 1))*(2*b*x + a)^q*(b*x + a)^p*x^p, 
 x)
 

Mupad [F(-1)]

Timed out. \[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\int x^p\,\left (a\,\left (p+1\right )+2\,b\,x\,\left (2\,p+q+3\right )\right )\,{\left (a+b\,x\right )}^p\,{\left (a+2\,b\,x\right )}^q \,d x \] Input:

int(x^p*(a*(p + 1) + 2*b*x*(2*p + q + 3))*(a + b*x)^p*(a + 2*b*x)^q,x)
 

Output:

int(x^p*(a*(p + 1) + 2*b*x*(2*p + q + 3))*(a + b*x)^p*(a + 2*b*x)^q, x)
 

Reduce [F]

\[ \int x^p (a+b x)^p (a+2 b x)^q (a (1+p)+2 b (3+2 p+q) x) \, dx=\text {too large to display} \] Input:

int(x^p*(b*x+a)^p*(2*b*x+a)^q*(a*(p+1)+2*b*(3+2*p+q)*x),x)
 

Output:

( - 2*x**p*(a + 2*b*x)**q*(a + b*x)**p*a**2*p**3 - x**p*(a + 2*b*x)**q*(a 
+ b*x)**p*a**2*p**2*q - 6*x**p*(a + 2*b*x)**q*(a + b*x)**p*a**2*p**2 - 2*x 
**p*(a + 2*b*x)**q*(a + b*x)**p*a**2*p*q - 4*x**p*(a + 2*b*x)**q*(a + b*x) 
**p*a**2*p - x**p*(a + 2*b*x)**q*(a + b*x)**p*a**2*q + 24*x**p*(a + 2*b*x) 
**q*(a + b*x)**p*a*b*p**3*x + 32*x**p*(a + 2*b*x)**q*(a + b*x)**p*a*b*p**2 
*q*x + 40*x**p*(a + 2*b*x)**q*(a + b*x)**p*a*b*p**2*x + 14*x**p*(a + 2*b*x 
)**q*(a + b*x)**p*a*b*p*q**2*x + 36*x**p*(a + 2*b*x)**q*(a + b*x)**p*a*b*p 
*q*x + 8*x**p*(a + 2*b*x)**q*(a + b*x)**p*a*b*p*x + 2*x**p*(a + 2*b*x)**q* 
(a + b*x)**p*a*b*q**3*x + 8*x**p*(a + 2*b*x)**q*(a + b*x)**p*a*b*q**2*x + 
4*x**p*(a + 2*b*x)**q*(a + b*x)**p*a*b*q*x + 32*x**p*(a + 2*b*x)**q*(a + b 
*x)**p*b**2*p**3*x**2 + 48*x**p*(a + 2*b*x)**q*(a + b*x)**p*b**2*p**2*q*x* 
*2 + 64*x**p*(a + 2*b*x)**q*(a + b*x)**p*b**2*p**2*x**2 + 24*x**p*(a + 2*b 
*x)**q*(a + b*x)**p*b**2*p*q**2*x**2 + 64*x**p*(a + 2*b*x)**q*(a + b*x)**p 
*b**2*p*q*x**2 + 24*x**p*(a + 2*b*x)**q*(a + b*x)**p*b**2*p*x**2 + 4*x**p* 
(a + 2*b*x)**q*(a + b*x)**p*b**2*q**3*x**2 + 16*x**p*(a + 2*b*x)**q*(a + b 
*x)**p*b**2*q**2*x**2 + 12*x**p*(a + 2*b*x)**q*(a + b*x)**p*b**2*q*x**2 - 
64*int((x**p*(a + 2*b*x)**q*(a + b*x)**p*x)/(8*a**2*p**3 + 12*a**2*p**2*q 
+ 12*a**2*p**2 + 6*a**2*p*q**2 + 12*a**2*p*q + 4*a**2*p + a**2*q**3 + 3*a* 
*2*q**2 + 2*a**2*q + 24*a*b*p**3*x + 36*a*b*p**2*q*x + 36*a*b*p**2*x + 18* 
a*b*p*q**2*x + 36*a*b*p*q*x + 12*a*b*p*x + 3*a*b*q**3*x + 9*a*b*q**2*x ...