\(\int x^{-1-4 (1+p)} (a+b x^4)^p (c+d x^4) \, dx\) [192]

Optimal result

Integrand size = 26, antiderivative size = 89 \[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=-\frac {c x^{-4 (1+p)} \left (a+b x^4\right )^{1+p}}{4 a (1+p)}-\frac {d x^{-4 p} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^4}{a}\right )}{4 p} \] Output:

-1/4*c*(b*x^4+a)^(p+1)/a/(p+1)/(x^(4*p+4))-1/4*d*(b*x^4+a)^p*hypergeom([-p 
, -p],[1-p],-b*x^4/a)/p/(x^(4*p))/((1+b*x^4/a)^p)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.91 \[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\frac {1}{4} x^{-4 p} \left (a+b x^4\right )^p \left (-\frac {c \left (a+b x^4\right )}{a (1+p) x^4}-\frac {d \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {b x^4}{a}\right )}{p}\right ) \] Input:

Integrate[x^(-1 - 4*(1 + p))*(a + b*x^4)^p*(c + d*x^4),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {954, 882, 74}

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.

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

\(\Big \downarrow \) 954

\(\displaystyle \frac {d \int x^{-4 p-5} \left (b x^4+a\right )^{p+1}dx}{b}-\frac {x^{-4 (p+1)} (b c-a d) \left (a+b x^4\right )^{p+1}}{4 a b (p+1)}\)

\(\Big \downarrow \) 882

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

\(\Big \downarrow \) 74

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

Input:

Int[x^(-1 - 4*(1 + p))*(a + b*x^4)^p*(c + d*x^4),x]
 

Output:

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

Defintions of rubi rules used

rule 74

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x 
)^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] 
/; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] 
 &&  !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
 

rule 882

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^Simplify[ 
(m + 1)/n + p]*x^m*(a + b*x^n)^p*((x^n/(a + b*x^n))^p/(n*x^Simplify[m + n*p 
]))   Subst[Int[x^((m + 1)/n - 1)/(1 - b*x)^(Simplify[(m + 1)/n + p] + 1), 
x], x, x^n/(a + b*x^n)], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simpli 
fy[(m + 1)/n + p]]
 

rule 954

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[(b*c - a*d)*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*b* 
e*(m + 1))), x] + Simp[d/b   Int[(e*x)^m*(a + b*x^n)^(p + 1), x], x] /; Fre 
eQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n*(p + 1) + 
 1, 0] && NeQ[m, -1]
 

Maple [F]

\[\int x^{-5-4 p} \left (b \,x^{4}+a \right )^{p} \left (d \,x^{4}+c \right )d x\]

Input:

int(x^(-5-4*p)*(b*x^4+a)^p*(d*x^4+c),x)
 

Output:

int(x^(-5-4*p)*(b*x^4+a)^p*(d*x^4+c),x)
 

Fricas [F]

\[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{4} + a\right )}^{p} x^{-4 \, p - 5} \,d x } \] Input:

integrate(x^(-5-4*p)*(b*x^4+a)^p*(d*x^4+c),x, algorithm="fricas")
 

Output:

integral((d*x^4 + c)*(b*x^4 + a)^p*x^(-4*p - 5), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\text {Timed out} \] Input:

integrate(x**(-5-4*p)*(b*x**4+a)**p*(d*x**4+c),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{4} + a\right )}^{p} x^{-4 \, p - 5} \,d x } \] Input:

integrate(x^(-5-4*p)*(b*x^4+a)^p*(d*x^4+c),x, algorithm="maxima")
 

Output:

d*integrate(e^(p*log(b*x^4 + a) - 4*p*log(x))/x, x) - 1/4*(b*x^4 + a)*c*e^ 
(p*log(b*x^4 + a) - 4*p*log(x))/(a*(p + 1)*x^4)
 

Giac [F]

\[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\int { {\left (d x^{4} + c\right )} {\left (b x^{4} + a\right )}^{p} x^{-4 \, p - 5} \,d x } \] Input:

integrate(x^(-5-4*p)*(b*x^4+a)^p*(d*x^4+c),x, algorithm="giac")
 

Output:

integrate((d*x^4 + c)*(b*x^4 + a)^p*x^(-4*p - 5), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-4 (1+p)} \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\int \frac {{\left (b\,x^4+a\right )}^p\,\left (d\,x^4+c\right )}{x^{4\,p+5}} \,d x \] Input:

int(((a + b*x^4)^p*(c + d*x^4))/x^(4*p + 5),x)
 

Output:

int(((a + b*x^4)^p*(c + d*x^4))/x^(4*p + 5), x)
 

Reduce [F]

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

int(x^(-5-4*p)*(b*x^4+a)^p*(d*x^4+c),x)
 

Output:

( - (a + b*x**4)**p*a*c - (a + b*x**4)**p*b*c*x**4 + 4*x**(4*p)*int((a + b 
*x**4)**p/(x**(4*p)*x),x)*a*d*p*x**4 + 4*x**(4*p)*int((a + b*x**4)**p/(x** 
(4*p)*x),x)*a*d*x**4)/(4*x**(4*p)*a*x**4*(p + 1))