\(\int (e x)^m (a+b x^4)^p (c+d x^4)^2 \, dx\) [292]

Optimal result

Integrand size = 24, antiderivative size = 207 \[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right )^2 \, dx=-\frac {d (a d (5+m)-2 b c (9+m+4 p)) (e x)^{1+m} \left (a+b x^4\right )^{1+p}}{b^2 e (5+m+4 p) (9+m+4 p)}+\frac {d^2 (e x)^{5+m} \left (a+b x^4\right )^{1+p}}{b e^5 (9+m+4 p)}+\frac {\left (\frac {c^2}{1+m}+\frac {a d (a d (5+m)-2 b c (9+m+4 p))}{b^2 (5+m+4 p) (9+m+4 p)}\right ) (e x)^{1+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-p,\frac {5+m}{4},-\frac {b x^4}{a}\right )}{e} \] Output:

-d*(a*d*(5+m)-2*b*c*(9+m+4*p))*(e*x)^(1+m)*(b*x^4+a)^(p+1)/b^2/e/(5+m+4*p) 
/(9+m+4*p)+d^2*(e*x)^(5+m)*(b*x^4+a)^(p+1)/b/e^5/(9+m+4*p)+(c^2/(1+m)+a*d* 
(a*d*(5+m)-2*b*c*(9+m+4*p))/b^2/(5+m+4*p)/(9+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] (warning: unable to verify)

Time = 97.51 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.28 \[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right )^2 \, dx=\frac {x (e x)^m \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (a (13+m) \left (c^2 \left (45+14 m+m^2\right )+2 c d \left (9+10 m+m^2\right ) x^4+d^2 \left (5+6 m+m^2\right ) x^8\right ) \operatorname {Gamma}(-p) \operatorname {Hypergeometric2F1}\left (\frac {1+m}{4},-p,\frac {13+m}{4},-\frac {b x^4}{a}\right )-8 b (1+m) x^4 \left (c+d x^4\right ) \left (c (7+m)+d (3+m) x^4\right ) \operatorname {Gamma}(1-p) \operatorname {Hypergeometric2F1}\left (\frac {5+m}{4},1-p,\frac {17+m}{4},-\frac {b x^4}{a}\right )-16 b (1+m) x^4 \left (c+d x^4\right )^2 \operatorname {Gamma}(1-p) \, _3F_2\left (2,\frac {5}{4}+\frac {m}{4},1-p;1,\frac {17}{4}+\frac {m}{4};-\frac {b x^4}{a}\right )\right )}{a (1+m) (5+m) (9+m) (13+m) \operatorname {Gamma}(-p)} \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^4)^p*(a*(13 + m)*(c^2*(45 + 14*m + m^2) + 2*c*d*(9 + 1 
0*m + m^2)*x^4 + d^2*(5 + 6*m + m^2)*x^8)*Gamma[-p]*Hypergeometric2F1[(1 + 
 m)/4, -p, (13 + m)/4, -((b*x^4)/a)] - 8*b*(1 + m)*x^4*(c + d*x^4)*(c*(7 + 
 m) + d*(3 + m)*x^4)*Gamma[1 - p]*Hypergeometric2F1[(5 + m)/4, 1 - p, (17 
+ m)/4, -((b*x^4)/a)] - 16*b*(1 + m)*x^4*(c + d*x^4)^2*Gamma[1 - p]*Hyperg 
eometricPFQ[{2, 5/4 + m/4, 1 - p}, {1, 17/4 + m/4}, -((b*x^4)/a)]))/(a*(1 
+ m)*(5 + m)*(9 + m)*(13 + m)*(1 + (b*x^4)/a)^p*Gamma[-p])
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {964, 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.

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

\(\Big \downarrow \) 964

\(\displaystyle \frac {\int (e x)^m \left (b x^4+a\right )^p \left (b c^2 (m+4 p+9)-d (a d (m+5)-2 b c (m+4 p+9)) x^4\right )dx}{b (m+4 p+9)}+\frac {d^2 (e x)^{m+5} \left (a+b x^4\right )^{p+1}}{b e^5 (m+4 p+9)}\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {\left (\frac {a d (m+1) (a d (m+5)-2 b c (m+4 p+9))}{b (m+4 p+5)}+b c^2 (m+4 p+9)\right ) \int (e x)^m \left (b x^4+a\right )^pdx-\frac {d (e x)^{m+1} \left (a+b x^4\right )^{p+1} (a d (m+5)-2 b c (m+4 p+9))}{b e (m+4 p+5)}}{b (m+4 p+9)}+\frac {d^2 (e x)^{m+5} \left (a+b x^4\right )^{p+1}}{b e^5 (m+4 p+9)}\)

\(\Big \downarrow \) 889

\(\displaystyle \frac {\left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (\frac {a d (m+1) (a d (m+5)-2 b c (m+4 p+9))}{b (m+4 p+5)}+b c^2 (m+4 p+9)\right ) \int (e x)^m \left (\frac {b x^4}{a}+1\right )^pdx-\frac {d (e x)^{m+1} \left (a+b x^4\right )^{p+1} (a d (m+5)-2 b c (m+4 p+9))}{b e (m+4 p+5)}}{b (m+4 p+9)}+\frac {d^2 (e x)^{m+5} \left (a+b x^4\right )^{p+1}}{b e^5 (m+4 p+9)}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {\frac {(e x)^{m+1} \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (\frac {a d (m+1) (a d (m+5)-2 b c (m+4 p+9))}{b (m+4 p+5)}+b c^2 (m+4 p+9)\right ) \operatorname {Hypergeometric2F1}\left (\frac {m+1}{4},-p,\frac {m+5}{4},-\frac {b x^4}{a}\right )}{e (m+1)}-\frac {d (e x)^{m+1} \left (a+b x^4\right )^{p+1} (a d (m+5)-2 b c (m+4 p+9))}{b e (m+4 p+5)}}{b (m+4 p+9)}+\frac {d^2 (e x)^{m+5} \left (a+b x^4\right )^{p+1}}{b e^5 (m+4 p+9)}\)

Input:

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

Output:

(d^2*(e*x)^(5 + m)*(a + b*x^4)^(1 + p))/(b*e^5*(9 + m + 4*p)) + (-((d*(a*d 
*(5 + m) - 2*b*c*(9 + m + 4*p))*(e*x)^(1 + m)*(a + b*x^4)^(1 + p))/(b*e*(5 
 + m + 4*p))) + ((b*c^2*(9 + m + 4*p) + (a*d*(1 + m)*(a*d*(5 + m) - 2*b*c* 
(9 + m + 4*p)))/(b*(5 + m + 4*p)))*(e*x)^(1 + m)*(a + b*x^4)^p*Hypergeomet 
ric2F1[(1 + m)/4, -p, (5 + m)/4, -((b*x^4)/a)])/(e*(1 + m)*(1 + (b*x^4)/a) 
^p))/(b*(9 + m + 4*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]
 

rule 964

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right )^2 \, dx=\text {too large to display} \] Input:

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

Output:

(e**m*( - 4*x**m*(a + b*x**4)**p*a**2*d**2*m*p*x - 20*x**m*(a + b*x**4)**p 
*a**2*d**2*p*x + 8*x**m*(a + b*x**4)**p*a*b*c*d*m*p*x + 32*x**m*(a + b*x** 
4)**p*a*b*c*d*p**2*x + 72*x**m*(a + b*x**4)**p*a*b*c*d*p*x + 4*x**m*(a + b 
*x**4)**p*a*b*d**2*m*p*x**5 + 16*x**m*(a + b*x**4)**p*a*b*d**2*p**2*x**5 + 
 4*x**m*(a + b*x**4)**p*a*b*d**2*p*x**5 + x**m*(a + b*x**4)**p*b**2*c**2*m 
**2*x + 8*x**m*(a + b*x**4)**p*b**2*c**2*m*p*x + 14*x**m*(a + b*x**4)**p*b 
**2*c**2*m*x + 16*x**m*(a + b*x**4)**p*b**2*c**2*p**2*x + 56*x**m*(a + b*x 
**4)**p*b**2*c**2*p*x + 45*x**m*(a + b*x**4)**p*b**2*c**2*x + 2*x**m*(a + 
b*x**4)**p*b**2*c*d*m**2*x**5 + 16*x**m*(a + b*x**4)**p*b**2*c*d*m*p*x**5 
+ 20*x**m*(a + b*x**4)**p*b**2*c*d*m*x**5 + 32*x**m*(a + b*x**4)**p*b**2*c 
*d*p**2*x**5 + 80*x**m*(a + b*x**4)**p*b**2*c*d*p*x**5 + 18*x**m*(a + b*x* 
*4)**p*b**2*c*d*x**5 + x**m*(a + b*x**4)**p*b**2*d**2*m**2*x**9 + 8*x**m*( 
a + b*x**4)**p*b**2*d**2*m*p*x**9 + 6*x**m*(a + b*x**4)**p*b**2*d**2*m*x** 
9 + 16*x**m*(a + b*x**4)**p*b**2*d**2*p**2*x**9 + 24*x**m*(a + b*x**4)**p* 
b**2*d**2*p*x**9 + 5*x**m*(a + b*x**4)**p*b**2*d**2*x**9 + 4*int((x**m*(a 
+ b*x**4)**p)/(a*m**3 + 12*a*m**2*p + 15*a*m**2 + 48*a*m*p**2 + 120*a*m*p 
+ 59*a*m + 64*a*p**3 + 240*a*p**2 + 236*a*p + 45*a + b*m**3*x**4 + 12*b*m* 
*2*p*x**4 + 15*b*m**2*x**4 + 48*b*m*p**2*x**4 + 120*b*m*p*x**4 + 59*b*m*x* 
*4 + 64*b*p**3*x**4 + 240*b*p**2*x**4 + 236*b*p*x**4 + 45*b*x**4),x)*a**3* 
d**2*m**5*p + 48*int((x**m*(a + b*x**4)**p)/(a*m**3 + 12*a*m**2*p + 15*...