\(\int \frac {(e x)^m (A+B x^n) (c+d x^n)^3}{a+b x^n} \, dx\) [37]

Optimal result

Integrand size = 31, antiderivative size = 279 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=-\frac {\left (a^3 B d^3+3 a b^2 c d (B c+A d)-a^2 b d^2 (3 B c+A d)-b^3 c^2 (B c+3 A d)\right ) (e x)^{1+m}}{b^4 e (1+m)}+\frac {d \left (a^2 B d^2+3 b^2 c (B c+A d)-a b d (3 B c+A d)\right ) x^n (e x)^{1+m}}{b^3 e (1+m+n)}+\frac {d^2 (3 b B c+A b d-a B d) x^{2 n} (e x)^{1+m}}{b^2 e (1+m+2 n)}+\frac {B d^3 x^{3 n} (e x)^{1+m}}{b e (1+m+3 n)}+\frac {(A b-a B) (b c-a d)^3 (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a b^4 e (1+m)} \] Output:

-(a^3*B*d^3+3*a*b^2*c*d*(A*d+B*c)-a^2*b*d^2*(A*d+3*B*c)-b^3*c^2*(3*A*d+B*c 
))*(e*x)^(1+m)/b^4/e/(1+m)+d*(a^2*B*d^2+3*b^2*c*(A*d+B*c)-a*b*d*(A*d+3*B*c 
))*x^n*(e*x)^(1+m)/b^3/e/(1+m+n)+d^2*(A*b*d-B*a*d+3*B*b*c)*x^(2*n)*(e*x)^( 
1+m)/b^2/e/(1+m+2*n)+B*d^3*x^(3*n)*(e*x)^(1+m)/b/e/(1+m+3*n)+(A*b-B*a)*(-a 
*d+b*c)^3*(e*x)^(1+m)*hypergeom([1, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/a/b^4/e 
/(1+m)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.82 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\frac {x (e x)^m \left (\frac {-a^3 B d^3-3 a b^2 c d (B c+A d)+a^2 b d^2 (3 B c+A d)+b^3 c^2 (B c+3 A d)}{1+m}+\frac {b d \left (a^2 B d^2+3 b^2 c (B c+A d)-a b d (3 B c+A d)\right ) x^n}{1+m+n}+\frac {b^2 d^2 (3 b B c+A b d-a B d) x^{2 n}}{1+m+2 n}+\frac {b^3 B d^3 x^{3 n}}{1+m+3 n}+\frac {(-A b+a B) (-b c+a d)^3 \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a (1+m)}\right )}{b^4} \] Input:

Integrate[((e*x)^m*(A + B*x^n)*(c + d*x^n)^3)/(a + b*x^n),x]
 

Output:

(x*(e*x)^m*((-(a^3*B*d^3) - 3*a*b^2*c*d*(B*c + A*d) + a^2*b*d^2*(3*B*c + A 
*d) + b^3*c^2*(B*c + 3*A*d))/(1 + m) + (b*d*(a^2*B*d^2 + 3*b^2*c*(B*c + A* 
d) - a*b*d*(3*B*c + A*d))*x^n)/(1 + m + n) + (b^2*d^2*(3*b*B*c + A*b*d - a 
*B*d)*x^(2*n))/(1 + m + 2*n) + (b^3*B*d^3*x^(3*n))/(1 + m + 3*n) + ((-(A*b 
) + a*B)*(-(b*c) + a*d)^3*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, - 
((b*x^n)/a)])/(a*(1 + m))))/b^4
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1040, 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.

\(\displaystyle \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx\)

\(\Big \downarrow \) 1040

\(\displaystyle \int \left (\frac {d x^n (e x)^m \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3}+\frac {(e x)^m \left (-a^3 B d^3+a^2 b d^2 (A d+3 B c)-3 a b^2 c d (A d+B c)+b^3 c^2 (3 A d+B c)\right )}{b^4}+\frac {(e x)^m (A b-a B) (b c-a d)^3}{b^4 \left (a+b x^n\right )}+\frac {d^2 x^{2 n} (e x)^m (-a B d+A b d+3 b B c)}{b^2}+\frac {B d^3 x^{3 n} (e x)^m}{b}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d x^{n+1} (e x)^m \left (a^2 B d^2-a b d (A d+3 B c)+3 b^2 c (A d+B c)\right )}{b^3 (m+n+1)}-\frac {(e x)^{m+1} \left (a^3 B d^3-a^2 b d^2 (A d+3 B c)+3 a b^2 c d (A d+B c)+b^3 \left (-c^2\right ) (3 A d+B c)\right )}{b^4 e (m+1)}+\frac {(e x)^{m+1} (A b-a B) (b c-a d)^3 \operatorname {Hypergeometric2F1}\left (1,\frac {m+1}{n},\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{a b^4 e (m+1)}+\frac {d^2 x^{2 n+1} (e x)^m (-a B d+A b d+3 b B c)}{b^2 (m+2 n+1)}+\frac {B d^3 x^{3 n+1} (e x)^m}{b (m+3 n+1)}\)

Input:

Int[((e*x)^m*(A + B*x^n)*(c + d*x^n)^3)/(a + b*x^n),x]
 

Output:

(d*(a^2*B*d^2 + 3*b^2*c*(B*c + A*d) - a*b*d*(3*B*c + A*d))*x^(1 + n)*(e*x) 
^m)/(b^3*(1 + m + n)) + (d^2*(3*b*B*c + A*b*d - a*B*d)*x^(1 + 2*n)*(e*x)^m 
)/(b^2*(1 + m + 2*n)) + (B*d^3*x^(1 + 3*n)*(e*x)^m)/(b*(1 + m + 3*n)) - (( 
a^3*B*d^3 + 3*a*b^2*c*d*(B*c + A*d) - a^2*b*d^2*(3*B*c + A*d) - b^3*c^2*(B 
*c + 3*A*d))*(e*x)^(1 + m))/(b^4*e*(1 + m)) + ((A*b - a*B)*(b*c - a*d)^3*( 
e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/n, (1 + m + n)/n, -((b*x^n)/a)]) 
/(a*b^4*e*(1 + m))
 

Defintions of rubi rules used

rule 1040

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[ 
(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
 

rule 2009

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

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )^{3}}{a +b \,x^{n}}d x\]

Input:

int((e*x)^m*(A+B*x^n)*(c+d*x^n)^3/(a+b*x^n),x)
 

Output:

int((e*x)^m*(A+B*x^n)*(c+d*x^n)^3/(a+b*x^n),x)
 

Fricas [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{3} \left (e x\right )^{m}}{b x^{n} + a} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^3/(a+b*x^n),x, algorithm="fricas")
 

Output:

integral((B*d^3*x^(4*n) + A*c^3 + (3*B*c*d^2 + A*d^3)*x^(3*n) + 3*(B*c^2*d 
 + A*c*d^2)*x^(2*n) + (B*c^3 + 3*A*c^2*d)*x^n)*(e*x)^m/(b*x^n + a), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 29.43 (sec) , antiderivative size = 1933, normalized size of antiderivative = 6.93 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\text {Too large to display} \] Input:

integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**3/(a+b*x**n),x)
 

Output:

A*a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c**3*e**m*m*x**(m + 1)*lerchphi(b*x** 
n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n + 1 + 
1/n)) + A*a**(m/n + 1/n)*a**(-m/n - 1 - 1/n)*c**3*e**m*x**(m + 1)*lerchphi 
(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1/n)*gamma(m/n + 1/n)/(n**2*gamma(m/n 
+ 1 + 1/n)) + A*a**(-m/n - 4 - 1/n)*a**(m/n + 3 + 1/n)*d**3*e**m*m*x**(m + 
 3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 
 3 + 1/n)/(n**2*gamma(m/n + 4 + 1/n)) + 3*A*a**(-m/n - 4 - 1/n)*a**(m/n + 
3 + 1/n)*d**3*e**m*x**(m + 3*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, 
m/n + 3 + 1/n)*gamma(m/n + 3 + 1/n)/(n*gamma(m/n + 4 + 1/n)) + A*a**(-m/n 
- 4 - 1/n)*a**(m/n + 3 + 1/n)*d**3*e**m*x**(m + 3*n + 1)*lerchphi(b*x**n*e 
xp_polar(I*pi)/a, 1, m/n + 3 + 1/n)*gamma(m/n + 3 + 1/n)/(n**2*gamma(m/n + 
 4 + 1/n)) + 3*A*a**(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c*d**2*e**m*m*x**( 
m + 2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/ 
n + 2 + 1/n)/(n**2*gamma(m/n + 3 + 1/n)) + 6*A*a**(-m/n - 3 - 1/n)*a**(m/n 
 + 2 + 1/n)*c*d**2*e**m*x**(m + 2*n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a 
, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n*gamma(m/n + 3 + 1/n)) + 3*A*a* 
*(-m/n - 3 - 1/n)*a**(m/n + 2 + 1/n)*c*d**2*e**m*x**(m + 2*n + 1)*lerchphi 
(b*x**n*exp_polar(I*pi)/a, 1, m/n + 2 + 1/n)*gamma(m/n + 2 + 1/n)/(n**2*ga 
mma(m/n + 3 + 1/n)) + 3*A*a**(-m/n - 2 - 1/n)*a**(m/n + 1 + 1/n)*c**2*d*e* 
*m*m*x**(m + n + 1)*lerchphi(b*x**n*exp_polar(I*pi)/a, 1, m/n + 1 + 1/n...
 

Maxima [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{3} \left (e x\right )^{m}}{b x^{n} + a} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^3/(a+b*x^n),x, algorithm="maxima")
 

Output:

((b^4*c^3*e^m - 3*a*b^3*c^2*d*e^m + 3*a^2*b^2*c*d^2*e^m - a^3*b*d^3*e^m)*A 
 - (a*b^3*c^3*e^m - 3*a^2*b^2*c^2*d*e^m + 3*a^3*b*c*d^2*e^m - a^4*d^3*e^m) 
*B)*integrate(x^m/(b^5*x^n + a*b^4), x) + ((m^3 + 3*m^2*(n + 1) + (2*n^2 + 
 6*n + 3)*m + 2*n^2 + 3*n + 1)*B*b^3*d^3*e^m*x*e^(m*log(x) + 3*n*log(x)) + 
 ((3*(m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n 
 + 1)*b^3*c^2*d*e^m - 3*(m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 
3)*m + 11*n^2 + 6*n + 1)*a*b^2*c*d^2*e^m + (m^3 + 3*m^2*(2*n + 1) + 6*n^3 
+ (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*a^2*b*d^3*e^m)*A + ((m^3 + 3*m 
^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*b^3*c^3*e 
^m - 3*(m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6 
*n + 1)*a*b^2*c^2*d*e^m + 3*(m^3 + 3*m^2*(2*n + 1) + 6*n^3 + (11*n^2 + 12* 
n + 3)*m + 11*n^2 + 6*n + 1)*a^2*b*c*d^2*e^m - (m^3 + 3*m^2*(2*n + 1) + 6* 
n^3 + (11*n^2 + 12*n + 3)*m + 11*n^2 + 6*n + 1)*a^3*d^3*e^m)*B)*x*x^m + (( 
m^3 + m^2*(4*n + 3) + (3*n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*A*b^3*d^3*e^m 
 + (3*(m^3 + m^2*(4*n + 3) + (3*n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*b^3*c* 
d^2*e^m - (m^3 + m^2*(4*n + 3) + (3*n^2 + 8*n + 3)*m + 3*n^2 + 4*n + 1)*a* 
b^2*d^3*e^m)*B)*x*e^(m*log(x) + 2*n*log(x)) + ((3*(m^3 + m^2*(5*n + 3) + ( 
6*n^2 + 10*n + 3)*m + 6*n^2 + 5*n + 1)*b^3*c*d^2*e^m - (m^3 + m^2*(5*n + 3 
) + (6*n^2 + 10*n + 3)*m + 6*n^2 + 5*n + 1)*a*b^2*d^3*e^m)*A + (3*(m^3 + m 
^2*(5*n + 3) + (6*n^2 + 10*n + 3)*m + 6*n^2 + 5*n + 1)*b^3*c^2*d*e^m - ...
 

Giac [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{3} \left (e x\right )^{m}}{b x^{n} + a} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^3/(a+b*x^n),x, algorithm="giac")
 

Output:

integrate((B*x^n + A)*(d*x^n + c)^3*(e*x)^m/(b*x^n + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (c+d\,x^n\right )}^3}{a+b\,x^n} \,d x \] Input:

int(((e*x)^m*(A + B*x^n)*(c + d*x^n)^3)/(a + b*x^n),x)
 

Output:

int(((e*x)^m*(A + B*x^n)*(c + d*x^n)^3)/(a + b*x^n), x)
 

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.72 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3}{a+b x^n} \, dx=\frac {x^{m} e^{m} x \left (3 x^{3 n} d^{3} m^{2}+3 x^{3 n} d^{3} m +2 x^{3 n} d^{3} n^{2}+3 x^{3 n} d^{3} n +3 x^{2 n} c \,d^{2}+3 x^{n} c^{2} d +6 c^{3} m^{2} n +11 c^{3} m \,n^{2}+12 c^{3} m n +c^{3}+12 x^{2 n} c \,d^{2} m^{2} n +9 x^{2 n} c \,d^{2} m \,n^{2}+24 x^{2 n} c \,d^{2} m n +15 x^{n} c^{2} d \,m^{2} n +18 x^{n} c^{2} d m \,n^{2}+30 x^{n} c^{2} d m n +x^{3 n} d^{3}+c^{3} m^{3}+3 x^{3 n} d^{3} m^{2} n +x^{3 n} d^{3} m^{3}+2 x^{3 n} d^{3} m \,n^{2}+6 x^{3 n} d^{3} m n +3 x^{2 n} c \,d^{2} m^{3}+9 x^{2 n} c \,d^{2} m^{2}+9 x^{2 n} c \,d^{2} m +9 x^{2 n} c \,d^{2} n^{2}+12 x^{2 n} c \,d^{2} n +3 x^{n} c^{2} d \,m^{3}+9 x^{n} c^{2} d \,m^{2}+9 x^{n} c^{2} d m +18 x^{n} c^{2} d \,n^{2}+15 x^{n} c^{2} d n +3 c^{3} m^{2}+3 c^{3} m +6 c^{3} n^{3}+11 c^{3} n^{2}+6 c^{3} n \right )}{m^{4}+6 m^{3} n +11 m^{2} n^{2}+6 m \,n^{3}+4 m^{3}+18 m^{2} n +22 m \,n^{2}+6 n^{3}+6 m^{2}+18 m n +11 n^{2}+4 m +6 n +1} \] Input:

int((e*x)^m*(A+B*x^n)*(c+d*x^n)^3/(a+b*x^n),x)
 

Output:

(x**m*e**m*x*(x**(3*n)*d**3*m**3 + 3*x**(3*n)*d**3*m**2*n + 3*x**(3*n)*d** 
3*m**2 + 2*x**(3*n)*d**3*m*n**2 + 6*x**(3*n)*d**3*m*n + 3*x**(3*n)*d**3*m 
+ 2*x**(3*n)*d**3*n**2 + 3*x**(3*n)*d**3*n + x**(3*n)*d**3 + 3*x**(2*n)*c* 
d**2*m**3 + 12*x**(2*n)*c*d**2*m**2*n + 9*x**(2*n)*c*d**2*m**2 + 9*x**(2*n 
)*c*d**2*m*n**2 + 24*x**(2*n)*c*d**2*m*n + 9*x**(2*n)*c*d**2*m + 9*x**(2*n 
)*c*d**2*n**2 + 12*x**(2*n)*c*d**2*n + 3*x**(2*n)*c*d**2 + 3*x**n*c**2*d*m 
**3 + 15*x**n*c**2*d*m**2*n + 9*x**n*c**2*d*m**2 + 18*x**n*c**2*d*m*n**2 + 
 30*x**n*c**2*d*m*n + 9*x**n*c**2*d*m + 18*x**n*c**2*d*n**2 + 15*x**n*c**2 
*d*n + 3*x**n*c**2*d + c**3*m**3 + 6*c**3*m**2*n + 3*c**3*m**2 + 11*c**3*m 
*n**2 + 12*c**3*m*n + 3*c**3*m + 6*c**3*n**3 + 11*c**3*n**2 + 6*c**3*n + c 
**3))/(m**4 + 6*m**3*n + 4*m**3 + 11*m**2*n**2 + 18*m**2*n + 6*m**2 + 6*m* 
n**3 + 22*m*n**2 + 18*m*n + 4*m + 6*n**3 + 11*n**2 + 6*n + 1)