\(\int (g x)^m (c+d x+e x^2+f x^3) (a+b x^4)^p \, dx\) [109]

Optimal result

Integrand size = 30, antiderivative size = 269 \[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\frac {c (g x)^{1+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-p,\frac {5+m}{4},-\frac {b x^4}{a}\right )}{g (1+m)}+\frac {d (g x)^{2+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {2+m}{4},-p,\frac {6+m}{4},-\frac {b x^4}{a}\right )}{g^2 (2+m)}+\frac {e (g x)^{3+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3+m}{4},-p,\frac {7+m}{4},-\frac {b x^4}{a}\right )}{g^3 (3+m)}+\frac {f (g x)^{4+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {4+m}{4},-p,\frac {8+m}{4},-\frac {b x^4}{a}\right )}{g^4 (4+m)} \] Output:

c*(g*x)^(1+m)*(b*x^4+a)^p*hypergeom([-p, 1/4+1/4*m],[5/4+1/4*m],-b*x^4/a)/ 
g/(1+m)/((1+b*x^4/a)^p)+d*(g*x)^(2+m)*(b*x^4+a)^p*hypergeom([-p, 1/2+1/4*m 
],[3/2+1/4*m],-b*x^4/a)/g^2/(2+m)/((1+b*x^4/a)^p)+e*(g*x)^(3+m)*(b*x^4+a)^ 
p*hypergeom([-p, 3/4+1/4*m],[7/4+1/4*m],-b*x^4/a)/g^3/(3+m)/((1+b*x^4/a)^p 
)+f*(g*x)^(4+m)*(b*x^4+a)^p*hypergeom([-p, 1+1/4*m],[2+1/4*m],-b*x^4/a)/g^ 
4/(4+m)/((1+b*x^4/a)^p)
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.65 \[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=x (g x)^m \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (\frac {c \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-p,\frac {5+m}{4},-\frac {b x^4}{a}\right )}{1+m}+x \left (\frac {d \operatorname {Hypergeometric2F1}\left (\frac {2+m}{4},-p,\frac {6+m}{4},-\frac {b x^4}{a}\right )}{2+m}+x \left (\frac {e \operatorname {Hypergeometric2F1}\left (\frac {3+m}{4},-p,\frac {7+m}{4},-\frac {b x^4}{a}\right )}{3+m}+\frac {f x \operatorname {Hypergeometric2F1}\left (\frac {4+m}{4},-p,\frac {8+m}{4},-\frac {b x^4}{a}\right )}{4+m}\right )\right )\right ) \] Input:

Integrate[(g*x)^m*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]
 

Output:

(x*(g*x)^m*(a + b*x^4)^p*((c*Hypergeometric2F1[(1 + m)/4, -p, (5 + m)/4, - 
((b*x^4)/a)])/(1 + m) + x*((d*Hypergeometric2F1[(2 + m)/4, -p, (6 + m)/4, 
-((b*x^4)/a)])/(2 + m) + x*((e*Hypergeometric2F1[(3 + m)/4, -p, (7 + m)/4, 
 -((b*x^4)/a)])/(3 + m) + (f*x*Hypergeometric2F1[(4 + m)/4, -p, (8 + m)/4, 
 -((b*x^4)/a)])/(4 + m)))))/(1 + (b*x^4)/a)^p
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2372, 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.

\(\displaystyle \int (g x)^m \left (a+b x^4\right )^p \left (c+d x+e x^2+f x^3\right ) \, dx\)

\(\Big \downarrow \) 2372

\(\displaystyle \int \left (\left (c+e x^2\right ) (g x)^m \left (a+b x^4\right )^p+\frac {\left (d+f x^2\right ) (g x)^{m+1} \left (a+b x^4\right )^p}{g}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {c (g x)^{m+1} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{4},-p,\frac {m+5}{4},-\frac {b x^4}{a}\right )}{g (m+1)}+\frac {d (g x)^{m+2} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+2}{4},-p,\frac {m+6}{4},-\frac {b x^4}{a}\right )}{g^2 (m+2)}+\frac {e (g x)^{m+3} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+3}{4},-p,\frac {m+7}{4},-\frac {b x^4}{a}\right )}{g^3 (m+3)}+\frac {f (g x)^{m+4} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+4}{4},-p,\frac {m+8}{4},-\frac {b x^4}{a}\right )}{g^4 (m+4)}\)

Input:

Int[(g*x)^m*(c + d*x + e*x^2 + f*x^3)*(a + b*x^4)^p,x]
 

Output:

(c*(g*x)^(1 + m)*(a + b*x^4)^p*Hypergeometric2F1[(1 + m)/4, -p, (5 + m)/4, 
 -((b*x^4)/a)])/(g*(1 + m)*(1 + (b*x^4)/a)^p) + (d*(g*x)^(2 + m)*(a + b*x^ 
4)^p*Hypergeometric2F1[(2 + m)/4, -p, (6 + m)/4, -((b*x^4)/a)])/(g^2*(2 + 
m)*(1 + (b*x^4)/a)^p) + (e*(g*x)^(3 + m)*(a + b*x^4)^p*Hypergeometric2F1[( 
3 + m)/4, -p, (7 + m)/4, -((b*x^4)/a)])/(g^3*(3 + m)*(1 + (b*x^4)/a)^p) + 
(f*(g*x)^(4 + m)*(a + b*x^4)^p*Hypergeometric2F1[(4 + m)/4, -p, (8 + m)/4, 
 -((b*x^4)/a)])/(g^4*(4 + m)*(1 + (b*x^4)/a)^p)
 

Defintions of rubi rules used

rule 2009

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

rule 2372

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Mo 
dule[{q = Expon[Pq, x], j, k}, Int[Sum[((c*x)^(m + j)/c^j)*Sum[Coeff[Pq, x, 
 j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, 
 n/2 - 1}], x]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0 
] &&  !PolyQ[Pq, x^(n/2)]
 

Maple [F]

\[\int \left (g x \right )^{m} \left (f \,x^{3}+e \,x^{2}+d x +c \right ) \left (b \,x^{4}+a \right )^{p}d x\]

Input:

int((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)
 

Output:

int((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)
 

Fricas [F]

\[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\int { {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m} \,d x } \] Input:

integrate((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x, algorithm="fricas")
 

Output:

integral((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)
 

Sympy [F(-1)]

Timed out. \[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\text {Timed out} \] Input:

integrate((g*x)**m*(f*x**3+e*x**2+d*x+c)*(b*x**4+a)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\int { {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m} \,d x } \] Input:

integrate((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x, algorithm="maxima")
 

Output:

integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)
 

Giac [F]

\[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\int { {\left (f x^{3} + e x^{2} + d x + c\right )} {\left (b x^{4} + a\right )}^{p} \left (g x\right )^{m} \,d x } \] Input:

integrate((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x, algorithm="giac")
 

Output:

integrate((f*x^3 + e*x^2 + d*x + c)*(b*x^4 + a)^p*(g*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\int {\left (g\,x\right )}^m\,{\left (b\,x^4+a\right )}^p\,\left (f\,x^3+e\,x^2+d\,x+c\right ) \,d x \] Input:

int((g*x)^m*(a + b*x^4)^p*(c + d*x + e*x^2 + f*x^3),x)
 

Output:

int((g*x)^m*(a + b*x^4)^p*(c + d*x + e*x^2 + f*x^3), x)
 

Reduce [F]

\[ \int (g x)^m \left (c+d x+e x^2+f x^3\right ) \left (a+b x^4\right )^p \, dx=\text {too large to display} \] Input:

int((g*x)^m*(f*x^3+e*x^2+d*x+c)*(b*x^4+a)^p,x)
 

Output:

(g**m*(4*x**m*(a + b*x**4)**p*a*f*m**3*p + 48*x**m*(a + b*x**4)**p*a*f*m** 
2*p**2 + 24*x**m*(a + b*x**4)**p*a*f*m**2*p + 192*x**m*(a + b*x**4)**p*a*f 
*m*p**3 + 192*x**m*(a + b*x**4)**p*a*f*m*p**2 + 44*x**m*(a + b*x**4)**p*a* 
f*m*p + 256*x**m*(a + b*x**4)**p*a*f*p**4 + 384*x**m*(a + b*x**4)**p*a*f*p 
**3 + 176*x**m*(a + b*x**4)**p*a*f*p**2 + 24*x**m*(a + b*x**4)**p*a*f*p + 
x**m*(a + b*x**4)**p*b*c*m**4*x + 16*x**m*(a + b*x**4)**p*b*c*m**3*p*x + 9 
*x**m*(a + b*x**4)**p*b*c*m**3*x + 96*x**m*(a + b*x**4)**p*b*c*m**2*p**2*x 
 + 108*x**m*(a + b*x**4)**p*b*c*m**2*p*x + 26*x**m*(a + b*x**4)**p*b*c*m** 
2*x + 256*x**m*(a + b*x**4)**p*b*c*m*p**3*x + 432*x**m*(a + b*x**4)**p*b*c 
*m*p**2*x + 208*x**m*(a + b*x**4)**p*b*c*m*p*x + 24*x**m*(a + b*x**4)**p*b 
*c*m*x + 256*x**m*(a + b*x**4)**p*b*c*p**4*x + 576*x**m*(a + b*x**4)**p*b* 
c*p**3*x + 416*x**m*(a + b*x**4)**p*b*c*p**2*x + 96*x**m*(a + b*x**4)**p*b 
*c*p*x + x**m*(a + b*x**4)**p*b*d*m**4*x**2 + 16*x**m*(a + b*x**4)**p*b*d* 
m**3*p*x**2 + 8*x**m*(a + b*x**4)**p*b*d*m**3*x**2 + 96*x**m*(a + b*x**4)* 
*p*b*d*m**2*p**2*x**2 + 96*x**m*(a + b*x**4)**p*b*d*m**2*p*x**2 + 19*x**m* 
(a + b*x**4)**p*b*d*m**2*x**2 + 256*x**m*(a + b*x**4)**p*b*d*m*p**3*x**2 + 
 384*x**m*(a + b*x**4)**p*b*d*m*p**2*x**2 + 152*x**m*(a + b*x**4)**p*b*d*m 
*p*x**2 + 12*x**m*(a + b*x**4)**p*b*d*m*x**2 + 256*x**m*(a + b*x**4)**p*b* 
d*p**4*x**2 + 512*x**m*(a + b*x**4)**p*b*d*p**3*x**2 + 304*x**m*(a + b*x** 
4)**p*b*d*p**2*x**2 + 48*x**m*(a + b*x**4)**p*b*d*p*x**2 + x**m*(a + b*...