\(\int (c x)^m (d+e x+f x^2+g x^3) (a+b x^n)^p \, dx\) [31]

Optimal result

Integrand size = 30, antiderivative size = 280 \[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\frac {d (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{c (1+m)}+\frac {e x (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{n},-p,\frac {2+m+n}{n},-\frac {b x^n}{a}\right )}{c (2+m)}+\frac {f x^2 (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3+m}{n},-p,\frac {3+m+n}{n},-\frac {b x^n}{a}\right )}{c (3+m)}+\frac {g x^3 (c x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {4+m}{n},-p,\frac {4+m+n}{n},-\frac {b x^n}{a}\right )}{c (4+m)} \] Output:

d*(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([-p, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/c/ 
(1+m)/((1+b*x^n/a)^p)+e*x*(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([-p, (2+m)/n], 
[(2+m+n)/n],-b*x^n/a)/c/(2+m)/((1+b*x^n/a)^p)+f*x^2*(c*x)^(1+m)*(a+b*x^n)^ 
p*hypergeom([-p, (3+m)/n],[(3+m+n)/n],-b*x^n/a)/c/(3+m)/((1+b*x^n/a)^p)+g* 
x^3*(c*x)^(1+m)*(a+b*x^n)^p*hypergeom([-p, (4+m)/n],[(4+m+n)/n],-b*x^n/a)/ 
c/(4+m)/((1+b*x^n/a)^p)
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.64 \[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=x (c x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (\frac {d \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{1+m}+x \left (\frac {e \operatorname {Hypergeometric2F1}\left (\frac {2+m}{n},-p,\frac {2+m+n}{n},-\frac {b x^n}{a}\right )}{2+m}+x \left (\frac {f \operatorname {Hypergeometric2F1}\left (\frac {3+m}{n},-p,\frac {3+m+n}{n},-\frac {b x^n}{a}\right )}{3+m}+\frac {g x \operatorname {Hypergeometric2F1}\left (\frac {4+m}{n},-p,\frac {4+m+n}{n},-\frac {b x^n}{a}\right )}{4+m}\right )\right )\right ) \] Input:

Integrate[(c*x)^m*(d + e*x + f*x^2 + g*x^3)*(a + b*x^n)^p,x]
 

Output:

(x*(c*x)^m*(a + b*x^n)^p*((d*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/ 
n, -((b*x^n)/a)])/(1 + m) + x*((e*Hypergeometric2F1[(2 + m)/n, -p, (2 + m 
+ n)/n, -((b*x^n)/a)])/(2 + m) + x*((f*Hypergeometric2F1[(3 + m)/n, -p, (3 
 + m + n)/n, -((b*x^n)/a)])/(3 + m) + (g*x*Hypergeometric2F1[(4 + m)/n, -p 
, (4 + m + n)/n, -((b*x^n)/a)])/(4 + m)))))/(1 + (b*x^n)/a)^p
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.98, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2383, 2009}

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.

\(\displaystyle \int (c x)^m \left (a+b x^n\right )^p \left (d+e x+f x^2+g x^3\right ) \, dx\)

\(\Big \downarrow \) 2383

\(\displaystyle \int \left (\frac {g (c x)^{m+3} \left (a+b x^n\right )^p}{c^3}+\frac {f (c x)^{m+2} \left (a+b x^n\right )^p}{c^2}+d (c x)^m \left (a+b x^n\right )^p+\frac {e (c x)^{m+1} \left (a+b x^n\right )^p}{c}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {g (c x)^{m+4} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+4}{n},-p,\frac {m+n+4}{n},-\frac {b x^n}{a}\right )}{c^4 (m+4)}+\frac {f (c x)^{m+3} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+3}{n},-p,\frac {m+n+3}{n},-\frac {b x^n}{a}\right )}{c^3 (m+3)}+\frac {e (c x)^{m+2} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{n},-p,\frac {m+n+2}{n},-\frac {b x^n}{a}\right )}{c^2 (m+2)}+\frac {d (c x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{c (m+1)}\)

Input:

Int[(c*x)^m*(d + e*x + f*x^2 + g*x^3)*(a + b*x^n)^p,x]
 

Output:

(d*(c*x)^(1 + m)*(a + b*x^n)^p*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n 
)/n, -((b*x^n)/a)])/(c*(1 + m)*(1 + (b*x^n)/a)^p) + (e*(c*x)^(2 + m)*(a + 
b*x^n)^p*Hypergeometric2F1[(2 + m)/n, -p, (2 + m + n)/n, -((b*x^n)/a)])/(c 
^2*(2 + m)*(1 + (b*x^n)/a)^p) + (f*(c*x)^(3 + m)*(a + b*x^n)^p*Hypergeomet 
ric2F1[(3 + m)/n, -p, (3 + m + n)/n, -((b*x^n)/a)])/(c^3*(3 + m)*(1 + (b*x 
^n)/a)^p) + (g*(c*x)^(4 + m)*(a + b*x^n)^p*Hypergeometric2F1[(4 + m)/n, -p 
, (4 + m + n)/n, -((b*x^n)/a)])/(c^4*(4 + m)*(1 + (b*x^n)/a)^p)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2383

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> I 
nt[ExpandIntegrand[(c*x)^m*Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, m, n 
, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) &&  !IGtQ[m, 0]
 

Maple [F]

\[\int \left (c x \right )^{m} \left (g \,x^{3}+f \,x^{2}+e x +d \right ) \left (a +b \,x^{n}\right )^{p}d x\]

Input:

int((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x)
 

Output:

int((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x)
 

Fricas [F]

\[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\int { {\left (g x^{3} + f x^{2} + e x + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x, algorithm="fricas")
 

Output:

integral((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 107.75 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.09 \[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\frac {a^{\frac {m}{n} + \frac {1}{n}} a^{- \frac {m}{n} + p - \frac {1}{n}} c^{m} d x^{m + 1} \Gamma \left (\frac {m}{n} + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {1}{n} \\ \frac {m}{n} + 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {1}{n}\right )} + \frac {a^{\frac {m}{n} + \frac {2}{n}} a^{- \frac {m}{n} + p - \frac {2}{n}} c^{m} e x^{m + 2} \Gamma \left (\frac {m}{n} + \frac {2}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {2}{n} \\ \frac {m}{n} + 1 + \frac {2}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {2}{n}\right )} + \frac {a^{\frac {m}{n} + \frac {3}{n}} a^{- \frac {m}{n} + p - \frac {3}{n}} c^{m} f x^{m + 3} \Gamma \left (\frac {m}{n} + \frac {3}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {3}{n} \\ \frac {m}{n} + 1 + \frac {3}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {3}{n}\right )} + \frac {a^{\frac {m}{n} + \frac {4}{n}} a^{- \frac {m}{n} + p - \frac {4}{n}} c^{m} g x^{m + 4} \Gamma \left (\frac {m}{n} + \frac {4}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{n} + \frac {4}{n} \\ \frac {m}{n} + 1 + \frac {4}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (\frac {m}{n} + 1 + \frac {4}{n}\right )} \] Input:

integrate((c*x)**m*(g*x**3+f*x**2+e*x+d)*(a+b*x**n)**p,x)
 

Output:

a**(m/n + 1/n)*a**(-m/n + p - 1/n)*c**m*d*x**(m + 1)*gamma(m/n + 1/n)*hype 
r((-p, m/n + 1/n), (m/n + 1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/ 
n + 1 + 1/n)) + a**(m/n + 2/n)*a**(-m/n + p - 2/n)*c**m*e*x**(m + 2)*gamma 
(m/n + 2/n)*hyper((-p, m/n + 2/n), (m/n + 1 + 2/n,), b*x**n*exp_polar(I*pi 
)/a)/(n*gamma(m/n + 1 + 2/n)) + a**(m/n + 3/n)*a**(-m/n + p - 3/n)*c**m*f* 
x**(m + 3)*gamma(m/n + 3/n)*hyper((-p, m/n + 3/n), (m/n + 1 + 3/n,), b*x** 
n*exp_polar(I*pi)/a)/(n*gamma(m/n + 1 + 3/n)) + a**(m/n + 4/n)*a**(-m/n + 
p - 4/n)*c**m*g*x**(m + 4)*gamma(m/n + 4/n)*hyper((-p, m/n + 4/n), (m/n + 
1 + 4/n,), b*x**n*exp_polar(I*pi)/a)/(n*gamma(m/n + 1 + 4/n))
 

Maxima [F]

\[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\int { {\left (g x^{3} + f x^{2} + e x + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x, algorithm="maxima")
 

Output:

integrate((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m, x)
 

Giac [F]

\[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\int { {\left (g x^{3} + f x^{2} + e x + d\right )} {\left (b x^{n} + a\right )}^{p} \left (c x\right )^{m} \,d x } \] Input:

integrate((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x, algorithm="giac")
 

Output:

integrate((g*x^3 + f*x^2 + e*x + d)*(b*x^n + a)^p*(c*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\int {\left (c\,x\right )}^m\,{\left (a+b\,x^n\right )}^p\,\left (g\,x^3+f\,x^2+e\,x+d\right ) \,d x \] Input:

int((c*x)^m*(a + b*x^n)^p*(d + e*x + f*x^2 + g*x^3),x)
 

Output:

int((c*x)^m*(a + b*x^n)^p*(d + e*x + f*x^2 + g*x^3), x)
 

Reduce [F]

\[ \int (c x)^m \left (d+e x+f x^2+g x^3\right ) \left (a+b x^n\right )^p \, dx=\text {too large to display} \] Input:

int((c*x)^m*(g*x^3+f*x^2+e*x+d)*(a+b*x^n)^p,x)
 

Output:

(c**m*(x**m*(x**n*b + a)**p*d*m**3*x + 3*x**m*(x**n*b + a)**p*d*m**2*n*p*x 
 + 9*x**m*(x**n*b + a)**p*d*m**2*x + 3*x**m*(x**n*b + a)**p*d*m*n**2*p**2* 
x + 18*x**m*(x**n*b + a)**p*d*m*n*p*x + 26*x**m*(x**n*b + a)**p*d*m*x + x* 
*m*(x**n*b + a)**p*d*n**3*p**3*x + 9*x**m*(x**n*b + a)**p*d*n**2*p**2*x + 
26*x**m*(x**n*b + a)**p*d*n*p*x + 24*x**m*(x**n*b + a)**p*d*x + x**m*(x**n 
*b + a)**p*e*m**3*x**2 + 3*x**m*(x**n*b + a)**p*e*m**2*n*p*x**2 + 8*x**m*( 
x**n*b + a)**p*e*m**2*x**2 + 3*x**m*(x**n*b + a)**p*e*m*n**2*p**2*x**2 + 1 
6*x**m*(x**n*b + a)**p*e*m*n*p*x**2 + 19*x**m*(x**n*b + a)**p*e*m*x**2 + x 
**m*(x**n*b + a)**p*e*n**3*p**3*x**2 + 8*x**m*(x**n*b + a)**p*e*n**2*p**2* 
x**2 + 19*x**m*(x**n*b + a)**p*e*n*p*x**2 + 12*x**m*(x**n*b + a)**p*e*x**2 
 + x**m*(x**n*b + a)**p*f*m**3*x**3 + 3*x**m*(x**n*b + a)**p*f*m**2*n*p*x* 
*3 + 7*x**m*(x**n*b + a)**p*f*m**2*x**3 + 3*x**m*(x**n*b + a)**p*f*m*n**2* 
p**2*x**3 + 14*x**m*(x**n*b + a)**p*f*m*n*p*x**3 + 14*x**m*(x**n*b + a)**p 
*f*m*x**3 + x**m*(x**n*b + a)**p*f*n**3*p**3*x**3 + 7*x**m*(x**n*b + a)**p 
*f*n**2*p**2*x**3 + 14*x**m*(x**n*b + a)**p*f*n*p*x**3 + 8*x**m*(x**n*b + 
a)**p*f*x**3 + x**m*(x**n*b + a)**p*g*m**3*x**4 + 3*x**m*(x**n*b + a)**p*g 
*m**2*n*p*x**4 + 6*x**m*(x**n*b + a)**p*g*m**2*x**4 + 3*x**m*(x**n*b + a)* 
*p*g*m*n**2*p**2*x**4 + 12*x**m*(x**n*b + a)**p*g*m*n*p*x**4 + 11*x**m*(x* 
*n*b + a)**p*g*m*x**4 + x**m*(x**n*b + a)**p*g*n**3*p**3*x**4 + 6*x**m*(x* 
*n*b + a)**p*g*n**2*p**2*x**4 + 11*x**m*(x**n*b + a)**p*g*n*p*x**4 + 6*...