\(\int \frac {x^7 (a+b x^8)^p}{c+d x^4} \, dx\) [23]

Optimal result

Integrand size = 22, antiderivative size = 131 \[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=-\frac {d x^{12} \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,1,\frac {5}{2},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{12 c^2}+\frac {c \left (a+b x^8\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {d^2 \left (a+b x^8\right )}{b c^2+a d^2}\right )}{8 \left (b c^2+a d^2\right ) (1+p)} \] Output:

-1/12*d*x^12*(b*x^8+a)^p*AppellF1(3/2,1,-p,5/2,d^2*x^8/c^2,-b*x^8/a)/c^2/( 
(1+b*x^8/a)^p)+1/8*c*(b*x^8+a)^(p+1)*hypergeom([1, p+1],[2+p],d^2*(b*x^8+a 
)/(a*d^2+b*c^2))/(a*d^2+b*c^2)/(p+1)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.42 \[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\frac {\left (a+b x^8\right )^p \left (-\frac {c \left (\frac {d \left (-\sqrt {-\frac {a}{b}}+x^4\right )}{c+d x^4}\right )^{-p} \left (\frac {d \left (\sqrt {-\frac {a}{b}}+x^4\right )}{c+d x^4}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {c-\sqrt {-\frac {a}{b}} d}{c+d x^4},\frac {c+\sqrt {-\frac {a}{b}} d}{c+d x^4}\right )}{p}+2 d x^4 \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^8}{a}\right )\right )}{8 d^2} \] Input:

Integrate[(x^7*(a + b*x^8)^p)/(c + d*x^4),x]
 

Output:

((a + b*x^8)^p*(-((c*AppellF1[-2*p, -p, -p, 1 - 2*p, (c - Sqrt[-(a/b)]*d)/ 
(c + d*x^4), (c + Sqrt[-(a/b)]*d)/(c + d*x^4)])/(p*((d*(-Sqrt[-(a/b)] + x^ 
4))/(c + d*x^4))^p*((d*(Sqrt[-(a/b)] + x^4))/(c + d*x^4))^p)) + (2*d*x^4*H 
ypergeometric2F1[1/2, -p, 3/2, -((b*x^8)/a)])/(1 + (b*x^8)/a)^p))/(8*d^2)
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1803, 621, 353, 78, 395, 394}

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.

\(\displaystyle \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx\)

\(\Big \downarrow \) 1803

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

\(\Big \downarrow \) 621

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

\(\Big \downarrow \) 353

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

\(\Big \downarrow \) 78

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

\(\displaystyle \frac {1}{4} \left (\frac {c \left (a+b x^8\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {d^2 \left (b x^8+a\right )}{b c^2+a d^2}\right )}{2 (p+1) \left (a d^2+b c^2\right )}-\frac {d x^{12} \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},-p,1,\frac {5}{2},-\frac {b x^8}{a},\frac {d^2 x^8}{c^2}\right )}{3 c^2}\right )\)

Input:

Int[(x^7*(a + b*x^8)^p)/(c + d*x^4),x]
 

Output:

(-1/3*(d*x^12*(a + b*x^8)^p*AppellF1[3/2, -p, 1, 5/2, -((b*x^8)/a), (d^2*x 
^8)/c^2])/(c^2*(1 + (b*x^8)/a)^p) + (c*(a + b*x^8)^(1 + p)*Hypergeometric2 
F1[1, 1 + p, 2 + p, (d^2*(a + b*x^8))/(b*c^2 + a*d^2)])/(2*(b*c^2 + a*d^2) 
*(1 + p)))/4
 

Defintions of rubi rules used

rule 78

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b 
*c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m 
 + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] 
 &&  !IntegerQ[m] && IntegerQ[n]
 

rule 353

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

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 621

Int[((x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_))/((c_) + (d_.)*(x_)), x_Symbol] 
:> Simp[c   Int[x^m*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] - Simp[d   Int[ 
x^(m + 1)*((a + b*x^2)^p/(c^2 - d^2*x^2)), x], x] /; FreeQ[{a, b, c, d, m, 
p}, x]
 

rule 1803

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x 
)^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && 
 EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
 

Maple [F]

\[\int \frac {x^{7} \left (b \,x^{8}+a \right )^{p}}{x^{4} d +c}d x\]

Input:

int(x^7*(b*x^8+a)^p/(d*x^4+c),x)
 

Output:

int(x^7*(b*x^8+a)^p/(d*x^4+c),x)
 

Fricas [F]

\[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x^{7}}{d x^{4} + c} \,d x } \] Input:

integrate(x^7*(b*x^8+a)^p/(d*x^4+c),x, algorithm="fricas")
 

Output:

integral((b*x^8 + a)^p*x^7/(d*x^4 + c), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\text {Timed out} \] Input:

integrate(x**7*(b*x**8+a)**p/(d*x**4+c),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x^{7}}{d x^{4} + c} \,d x } \] Input:

integrate(x^7*(b*x^8+a)^p/(d*x^4+c),x, algorithm="maxima")
 

Output:

integrate((b*x^8 + a)^p*x^7/(d*x^4 + c), x)
 

Giac [F]

\[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int { \frac {{\left (b x^{8} + a\right )}^{p} x^{7}}{d x^{4} + c} \,d x } \] Input:

integrate(x^7*(b*x^8+a)^p/(d*x^4+c),x, algorithm="giac")
 

Output:

integrate((b*x^8 + a)^p*x^7/(d*x^4 + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\int \frac {x^7\,{\left (b\,x^8+a\right )}^p}{d\,x^4+c} \,d x \] Input:

int((x^7*(a + b*x^8)^p)/(c + d*x^4),x)
 

Output:

int((x^7*(a + b*x^8)^p)/(c + d*x^4), x)
 

Reduce [F]

\[ \int \frac {x^7 \left (a+b x^8\right )^p}{c+d x^4} \, dx=\frac {\left (b \,x^{8}+a \right )^{p} a d +\left (b \,x^{8}+a \right )^{p} b c \,x^{4}-16 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{11}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) a b \,d^{2} p^{2}-8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{11}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) a b \,d^{2} p -16 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{11}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) b^{2} c^{2} p^{2}-16 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{11}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) b^{2} c^{2} p -4 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{11}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) b^{2} c^{2}-8 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{3}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) a b \,c^{2} p -4 \left (\int \frac {\left (b \,x^{8}+a \right )^{p} x^{3}}{2 b d p \,x^{12}+b d \,x^{12}+2 b c p \,x^{8}+b c \,x^{8}+2 a d p \,x^{4}+a d \,x^{4}+2 a c p +a c}d x \right ) a b \,c^{2}}{4 b c d \left (2 p +1\right )} \] Input:

int(x^7*(b*x^8+a)^p/(d*x^4+c),x)
                                                                                    
                                                                                    
 

Output:

((a + b*x**8)**p*a*d + (a + b*x**8)**p*b*c*x**4 - 16*int(((a + b*x**8)**p* 
x**11)/(2*a*c*p + a*c + 2*a*d*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 
+ 2*b*d*p*x**12 + b*d*x**12),x)*a*b*d**2*p**2 - 8*int(((a + b*x**8)**p*x** 
11)/(2*a*c*p + a*c + 2*a*d*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 + 2 
*b*d*p*x**12 + b*d*x**12),x)*a*b*d**2*p - 16*int(((a + b*x**8)**p*x**11)/( 
2*a*c*p + a*c + 2*a*d*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 + 2*b*d* 
p*x**12 + b*d*x**12),x)*b**2*c**2*p**2 - 16*int(((a + b*x**8)**p*x**11)/(2 
*a*c*p + a*c + 2*a*d*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 + 2*b*d*p 
*x**12 + b*d*x**12),x)*b**2*c**2*p - 4*int(((a + b*x**8)**p*x**11)/(2*a*c* 
p + a*c + 2*a*d*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 + 2*b*d*p*x**1 
2 + b*d*x**12),x)*b**2*c**2 - 8*int(((a + b*x**8)**p*x**3)/(2*a*c*p + a*c 
+ 2*a*d*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 + 2*b*d*p*x**12 + b*d* 
x**12),x)*a*b*c**2*p - 4*int(((a + b*x**8)**p*x**3)/(2*a*c*p + a*c + 2*a*d 
*p*x**4 + a*d*x**4 + 2*b*c*p*x**8 + b*c*x**8 + 2*b*d*p*x**12 + b*d*x**12), 
x)*a*b*c**2)/(4*b*c*d*(2*p + 1))