\(\int \frac {(c+d x^4) (a+b x^8)^p}{x^4} \, dx\) [11]

Optimal result

Integrand size = 20, antiderivative size = 96 \[ \int \frac {\left (c+d x^4\right ) \left (a+b x^8\right )^p}{x^4} \, dx=-\frac {c \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {3}{8},-p,\frac {5}{8},-\frac {b x^8}{a}\right )}{3 x^3}+d x \left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right ) \] Output:

-1/3*c*(b*x^8+a)^p*hypergeom([-3/8, -p],[5/8],-b*x^8/a)/x^3/((1+b*x^8/a)^p 
)+d*x*(b*x^8+a)^p*hypergeom([1/8, -p],[9/8],-b*x^8/a)/((1+b*x^8/a)^p)
 

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c+d x^4\right ) \left (a+b x^8\right )^p}{x^4} \, dx=\frac {\left (a+b x^8\right )^p \left (1+\frac {b x^8}{a}\right )^{-p} \left (-c \operatorname {Hypergeometric2F1}\left (-\frac {3}{8},-p,\frac {5}{8},-\frac {b x^8}{a}\right )+3 d x^4 \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )\right )}{3 x^3} \] Input:

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

Output:

((a + b*x^8)^p*(-(c*Hypergeometric2F1[-3/8, -p, 5/8, -((b*x^8)/a)]) + 3*d* 
x^4*Hypergeometric2F1[1/8, -p, 9/8, -((b*x^8)/a)]))/(3*x^3*(1 + (b*x^8)/a) 
^p)
 

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1865, 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.

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

\(\Big \downarrow \) 1865

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

\(\Big \downarrow \) 2009

\(\displaystyle d x \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{8},-p,\frac {9}{8},-\frac {b x^8}{a}\right )-\frac {c \left (a+b x^8\right )^p \left (\frac {b x^8}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {3}{8},-p,\frac {5}{8},-\frac {b x^8}{a}\right )}{3 x^3}\)

Input:

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

Output:

-1/3*(c*(a + b*x^8)^p*Hypergeometric2F1[-3/8, -p, 5/8, -((b*x^8)/a)])/(x^3 
*(1 + (b*x^8)/a)^p) + (d*x*(a + b*x^8)^p*Hypergeometric2F1[1/8, -p, 9/8, - 
((b*x^8)/a)])/(1 + (b*x^8)/a)^p
 

Defintions of rubi rules used

rule 1865

Int[((f_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^ 
(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^(2*n))^p, (f*x)^m*(d 
 + e*x^n)^q, x], x] /; FreeQ[{a, c, d, e, f, m, q}, x] && EqQ[n2, 2*n] && I 
GtQ[n, 0] && IGtQ[q, 0]
 

rule 2009

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 123.76 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.81 \[ \int \frac {\left (c+d x^4\right ) \left (a+b x^8\right )^p}{x^4} \, dx=\frac {a^{p} c \Gamma \left (- \frac {3}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{8}, - p \\ \frac {5}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 x^{3} \Gamma \left (\frac {5}{8}\right )} + \frac {a^{p} d x \Gamma \left (\frac {1}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{8}, - p \\ \frac {9}{8} \end {matrix}\middle | {\frac {b x^{8} e^{i \pi }}{a}} \right )}}{8 \Gamma \left (\frac {9}{8}\right )} \] Input:

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

Output:

a**p*c*gamma(-3/8)*hyper((-3/8, -p), (5/8,), b*x**8*exp_polar(I*pi)/a)/(8* 
x**3*gamma(5/8)) + a**p*d*x*gamma(1/8)*hyper((1/8, -p), (9/8,), b*x**8*exp 
_polar(I*pi)/a)/(8*gamma(9/8))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (c+d x^4\right ) \left (a+b x^8\right )^p}{x^4} \, dx=\frac {8 \left (b \,x^{8}+a \right )^{p} c p +\left (b \,x^{8}+a \right )^{p} c +8 \left (b \,x^{8}+a \right )^{p} d p \,x^{4}-3 \left (b \,x^{8}+a \right )^{p} d \,x^{4}+4096 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{12}-16 b p \,x^{12}-3 b \,x^{12}+64 a \,p^{2} x^{4}-16 a p \,x^{4}-3 a \,x^{4}}d x \right ) a c \,p^{4} x^{3}-512 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{12}-16 b p \,x^{12}-3 b \,x^{12}+64 a \,p^{2} x^{4}-16 a p \,x^{4}-3 a \,x^{4}}d x \right ) a c \,p^{3} x^{3}-320 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{12}-16 b p \,x^{12}-3 b \,x^{12}+64 a \,p^{2} x^{4}-16 a p \,x^{4}-3 a \,x^{4}}d x \right ) a c \,p^{2} x^{3}-24 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{12}-16 b p \,x^{12}-3 b \,x^{12}+64 a \,p^{2} x^{4}-16 a p \,x^{4}-3 a \,x^{4}}d x \right ) a c p \,x^{3}+4096 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}-16 b p \,x^{8}-3 b \,x^{8}+64 a \,p^{2}-16 a p -3 a}d x \right ) a d \,p^{4} x^{3}-2560 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}-16 b p \,x^{8}-3 b \,x^{8}+64 a \,p^{2}-16 a p -3 a}d x \right ) a d \,p^{3} x^{3}+192 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}-16 b p \,x^{8}-3 b \,x^{8}+64 a \,p^{2}-16 a p -3 a}d x \right ) a d \,p^{2} x^{3}+72 \left (\int \frac {\left (b \,x^{8}+a \right )^{p}}{64 b \,p^{2} x^{8}-16 b p \,x^{8}-3 b \,x^{8}+64 a \,p^{2}-16 a p -3 a}d x \right ) a d p \,x^{3}}{x^{3} \left (64 p^{2}-16 p -3\right )} \] Input:

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

Output:

(8*(a + b*x**8)**p*c*p + (a + b*x**8)**p*c + 8*(a + b*x**8)**p*d*p*x**4 - 
3*(a + b*x**8)**p*d*x**4 + 4096*int((a + b*x**8)**p/(64*a*p**2*x**4 - 16*a 
*p*x**4 - 3*a*x**4 + 64*b*p**2*x**12 - 16*b*p*x**12 - 3*b*x**12),x)*a*c*p* 
*4*x**3 - 512*int((a + b*x**8)**p/(64*a*p**2*x**4 - 16*a*p*x**4 - 3*a*x**4 
 + 64*b*p**2*x**12 - 16*b*p*x**12 - 3*b*x**12),x)*a*c*p**3*x**3 - 320*int( 
(a + b*x**8)**p/(64*a*p**2*x**4 - 16*a*p*x**4 - 3*a*x**4 + 64*b*p**2*x**12 
 - 16*b*p*x**12 - 3*b*x**12),x)*a*c*p**2*x**3 - 24*int((a + b*x**8)**p/(64 
*a*p**2*x**4 - 16*a*p*x**4 - 3*a*x**4 + 64*b*p**2*x**12 - 16*b*p*x**12 - 3 
*b*x**12),x)*a*c*p*x**3 + 4096*int((a + b*x**8)**p/(64*a*p**2 - 16*a*p - 3 
*a + 64*b*p**2*x**8 - 16*b*p*x**8 - 3*b*x**8),x)*a*d*p**4*x**3 - 2560*int( 
(a + b*x**8)**p/(64*a*p**2 - 16*a*p - 3*a + 64*b*p**2*x**8 - 16*b*p*x**8 - 
 3*b*x**8),x)*a*d*p**3*x**3 + 192*int((a + b*x**8)**p/(64*a*p**2 - 16*a*p 
- 3*a + 64*b*p**2*x**8 - 16*b*p*x**8 - 3*b*x**8),x)*a*d*p**2*x**3 + 72*int 
((a + b*x**8)**p/(64*a*p**2 - 16*a*p - 3*a + 64*b*p**2*x**8 - 16*b*p*x**8 
- 3*b*x**8),x)*a*d*p*x**3)/(x**3*(64*p**2 - 16*p - 3))