\(\int (e x)^m (a+b x^4)^p (c+d x^4) \, dx\) [191]

Optimal result

Integrand size = 22, antiderivative size = 119 \[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\frac {d (e x)^{1+m} \left (a+b x^4\right )^{1+p}}{b e (5+m+4 p)}+\frac {\left (\frac {c}{1+m}-\frac {a d}{b (5+m+4 p)}\right ) (e x)^{1+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-p,\frac {5+m}{4},-\frac {b x^4}{a}\right )}{e} \] Output:

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

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right ) \, dx=\frac {x (e x)^m \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c (5+m) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-p,\frac {5+m}{4},-\frac {b x^4}{a}\right )+d (1+m) x^4 \operatorname {Hypergeometric2F1}\left (\frac {5+m}{4},-p,\frac {9+m}{4},-\frac {b x^4}{a}\right )\right )}{(1+m) (5+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {959, 889, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

(d*(e*x)^(1 + m)*(a + b*x^4)^(1 + p))/(b*e*(5 + m + 4*p)) + ((c - (a*d*(1 
+ m))/(b*(5 + m + 4*p)))*(e*x)^(1 + m)*(a + b*x^4)^p*Hypergeometric2F1[(1 
+ m)/4, -p, (5 + m)/4, -((b*x^4)/a)])/(e*(1 + m)*(1 + (b*x^4)/a)^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 889

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

rule 959

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

integrate((e*x)**m*(b*x**4+a)**p*(d*x**4+c),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*(4*x**m*(a + b*x**4)**p*a*d*p*x + x**m*(a + b*x**4)**p*b*c*m*x + 4*x 
**m*(a + b*x**4)**p*b*c*p*x + 5*x**m*(a + b*x**4)**p*b*c*x + x**m*(a + b*x 
**4)**p*b*d*m*x**5 + 4*x**m*(a + b*x**4)**p*b*d*p*x**5 + x**m*(a + b*x**4) 
**p*b*d*x**5 - 4*int((x**m*(a + b*x**4)**p)/(a*m**2 + 8*a*m*p + 6*a*m + 16 
*a*p**2 + 24*a*p + 5*a + b*m**2*x**4 + 8*b*m*p*x**4 + 6*b*m*x**4 + 16*b*p* 
*2*x**4 + 24*b*p*x**4 + 5*b*x**4),x)*a**2*d*m**3*p - 32*int((x**m*(a + b*x 
**4)**p)/(a*m**2 + 8*a*m*p + 6*a*m + 16*a*p**2 + 24*a*p + 5*a + b*m**2*x** 
4 + 8*b*m*p*x**4 + 6*b*m*x**4 + 16*b*p**2*x**4 + 24*b*p*x**4 + 5*b*x**4),x 
)*a**2*d*m**2*p**2 - 28*int((x**m*(a + b*x**4)**p)/(a*m**2 + 8*a*m*p + 6*a 
*m + 16*a*p**2 + 24*a*p + 5*a + b*m**2*x**4 + 8*b*m*p*x**4 + 6*b*m*x**4 + 
16*b*p**2*x**4 + 24*b*p*x**4 + 5*b*x**4),x)*a**2*d*m**2*p - 64*int((x**m*( 
a + b*x**4)**p)/(a*m**2 + 8*a*m*p + 6*a*m + 16*a*p**2 + 24*a*p + 5*a + b*m 
**2*x**4 + 8*b*m*p*x**4 + 6*b*m*x**4 + 16*b*p**2*x**4 + 24*b*p*x**4 + 5*b* 
x**4),x)*a**2*d*m*p**3 - 128*int((x**m*(a + b*x**4)**p)/(a*m**2 + 8*a*m*p 
+ 6*a*m + 16*a*p**2 + 24*a*p + 5*a + b*m**2*x**4 + 8*b*m*p*x**4 + 6*b*m*x* 
*4 + 16*b*p**2*x**4 + 24*b*p*x**4 + 5*b*x**4),x)*a**2*d*m*p**2 - 44*int((x 
**m*(a + b*x**4)**p)/(a*m**2 + 8*a*m*p + 6*a*m + 16*a*p**2 + 24*a*p + 5*a 
+ b*m**2*x**4 + 8*b*m*p*x**4 + 6*b*m*x**4 + 16*b*p**2*x**4 + 24*b*p*x**4 + 
 5*b*x**4),x)*a**2*d*m*p - 64*int((x**m*(a + b*x**4)**p)/(a*m**2 + 8*a*m*p 
 + 6*a*m + 16*a*p**2 + 24*a*p + 5*a + b*m**2*x**4 + 8*b*m*p*x**4 + 6*b*...