\(\int (d x)^m (a^2+2 a b x^4+b^2 x^8)^p \, dx\) [44]

Optimal result

Integrand size = 26, antiderivative size = 77 \[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\frac {(d x)^{1+m} \left (1+\frac {b x^4}{a}\right )^{-2 p} \left (a^2+2 a b x^4+b^2 x^8\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-2 p,\frac {5+m}{4},-\frac {b x^4}{a}\right )}{d (1+m)} \] Output:

(d*x)^(1+m)*(b^2*x^8+2*a*b*x^4+a^2)^p*hypergeom([-2*p, 1/4+1/4*m],[5/4+1/4 
*m],-b*x^4/a)/d/(1+m)/((1+b*x^4/a)^(2*p))
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.86 \[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\frac {x (d x)^m \left (\left (a+b x^4\right )^2\right )^p \left (1+\frac {b x^4}{a}\right )^{-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-2 p,1+\frac {1+m}{4},-\frac {b x^4}{a}\right )}{1+m} \] Input:

Integrate[(d*x)^m*(a^2 + 2*a*b*x^4 + b^2*x^8)^p,x]
 

Output:

(x*(d*x)^m*((a + b*x^4)^2)^p*Hypergeometric2F1[(1 + m)/4, -2*p, 1 + (1 + m 
)/4, -((b*x^4)/a)])/((1 + m)*(1 + (b*x^4)/a)^(2*p))
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1385, 888}

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.

\(\displaystyle \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx\)

\(\Big \downarrow \) 1385

\(\displaystyle \left (\frac {b x^4}{a}+1\right )^{-2 p} \left (a^2+2 a b x^4+b^2 x^8\right )^p \int (d x)^m \left (\frac {b x^4}{a}+1\right )^{2 p}dx\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {(d x)^{m+1} \left (\frac {b x^4}{a}+1\right )^{-2 p} \left (a^2+2 a b x^4+b^2 x^8\right )^p \operatorname {Hypergeometric2F1}\left (\frac {m+1}{4},-2 p,\frac {m+5}{4},-\frac {b x^4}{a}\right )}{d (m+1)}\)

Input:

Int[(d*x)^m*(a^2 + 2*a*b*x^4 + b^2*x^8)^p,x]
 

Output:

((d*x)^(1 + m)*(a^2 + 2*a*b*x^4 + b^2*x^8)^p*Hypergeometric2F1[(1 + m)/4, 
-2*p, (5 + m)/4, -((b*x^4)/a)])/(d*(1 + m)*(1 + (b*x^4)/a)^(2*p))
 

Defintions of rubi rules used

rule 888

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

rule 1385

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S 
imp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/(1 + 2*c*(x^n/b))^(2* 
FracPart[p]))   Int[u*(1 + 2*c*(x^n/b))^(2*p), x], x] /; FreeQ[{a, b, c, n, 
 p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[2*p] && NeQ[u, 
 x^(n - 1)] && NeQ[u, x^(2*n - 1)]
 

Maple [F]

\[\int \left (d x \right )^{m} \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )^{p}d x\]

Input:

int((d*x)^m*(b^2*x^8+2*a*b*x^4+a^2)^p,x)
 

Output:

int((d*x)^m*(b^2*x^8+2*a*b*x^4+a^2)^p,x)
 

Fricas [F]

\[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\int { {\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}^{p} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(b^2*x^8+2*a*b*x^4+a^2)^p,x, algorithm="fricas")
 

Output:

integral((b^2*x^8 + 2*a*b*x^4 + a^2)^p*(d*x)^m, x)
 

Sympy [F]

\[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\int \left (d x\right )^{m} \left (\left (a + b x^{4}\right )^{2}\right )^{p}\, dx \] Input:

integrate((d*x)**m*(b**2*x**8+2*a*b*x**4+a**2)**p,x)
 

Output:

Integral((d*x)**m*((a + b*x**4)**2)**p, x)
 

Maxima [F]

\[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\int { {\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}^{p} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(b^2*x^8+2*a*b*x^4+a^2)^p,x, algorithm="maxima")
 

Output:

integrate((b^2*x^8 + 2*a*b*x^4 + a^2)^p*(d*x)^m, x)
 

Giac [F]

\[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\int { {\left (b^{2} x^{8} + 2 \, a b x^{4} + a^{2}\right )}^{p} \left (d x\right )^{m} \,d x } \] Input:

integrate((d*x)^m*(b^2*x^8+2*a*b*x^4+a^2)^p,x, algorithm="giac")
 

Output:

integrate((b^2*x^8 + 2*a*b*x^4 + a^2)^p*(d*x)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\int {\left (d\,x\right )}^m\,{\left (a^2+2\,a\,b\,x^4+b^2\,x^8\right )}^p \,d x \] Input:

int((d*x)^m*(a^2 + b^2*x^8 + 2*a*b*x^4)^p,x)
 

Output:

int((d*x)^m*(a^2 + b^2*x^8 + 2*a*b*x^4)^p, x)
 

Reduce [F]

\[ \int (d x)^m \left (a^2+2 a b x^4+b^2 x^8\right )^p \, dx=\frac {d^{m} \left (x^{m} \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )^{p} x +8 \left (\int \frac {x^{m} \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )^{p}}{b m \,x^{4}+8 b p \,x^{4}+b \,x^{4}+a m +8 a p +a}d x \right ) a m p +64 \left (\int \frac {x^{m} \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )^{p}}{b m \,x^{4}+8 b p \,x^{4}+b \,x^{4}+a m +8 a p +a}d x \right ) a \,p^{2}+8 \left (\int \frac {x^{m} \left (b^{2} x^{8}+2 a b \,x^{4}+a^{2}\right )^{p}}{b m \,x^{4}+8 b p \,x^{4}+b \,x^{4}+a m +8 a p +a}d x \right ) a p \right )}{m +8 p +1} \] Input:

int((d*x)^m*(b^2*x^8+2*a*b*x^4+a^2)^p,x)
 

Output:

(d**m*(x**m*(a**2 + 2*a*b*x**4 + b**2*x**8)**p*x + 8*int((x**m*(a**2 + 2*a 
*b*x**4 + b**2*x**8)**p)/(a*m + 8*a*p + a + b*m*x**4 + 8*b*p*x**4 + b*x**4 
),x)*a*m*p + 64*int((x**m*(a**2 + 2*a*b*x**4 + b**2*x**8)**p)/(a*m + 8*a*p 
 + a + b*m*x**4 + 8*b*p*x**4 + b*x**4),x)*a*p**2 + 8*int((x**m*(a**2 + 2*a 
*b*x**4 + b**2*x**8)**p)/(a*m + 8*a*p + a + b*m*x**4 + 8*b*p*x**4 + b*x**4 
),x)*a*p))/(m + 8*p + 1)