\(\int \frac {(e x)^m (A+B x^n) (c+d x^n)^2}{(a+b x^n)^3} \, dx\) [32]

Optimal result

Integrand size = 31, antiderivative size = 322 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\frac {d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n)) (e x)^{1+m}}{2 a^2 b^3 e (1+m) n^2}+\frac {(A b-a B) (e x)^{1+m} \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}+\frac {(b c-a d) (e x)^{1+m} \left (c (a B (1+m)-A b (1+m-2 n))-d (A b (1+m)-a B (1+m+2 n)) x^n\right )}{2 a^2 b^2 e n^2 \left (a+b x^n\right )}+\frac {(b c (a B (1+m)-A b (1+m-2 n)) (a d (1+m)-b c (1+m-n))-a d (b c (1+m)-a d (1+m+n)) (A b (1+m)-a B (1+m+2 n))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{2 a^3 b^3 e (1+m) n^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.52 \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\frac {x (e x)^m \left (B d^2+\frac {d (2 b B c+A b d-3 a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a}+\frac {(b c-a d) (b B c+2 A b d-3 a B d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^2}+\frac {(A b-a B) (b c-a d)^2 \operatorname {Hypergeometric2F1}\left (3,\frac {1+m}{n},\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{a^3}\right )}{b^3 (1+m)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1064, 25, 1064, 25, 959, 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.

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

\(\Big \downarrow \) 1064

\(\displaystyle \frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}-\frac {\int -\frac {(e x)^m \left (d x^n+c\right ) \left (c (a B (m+1)-A b (m-2 n+1))-d (A b (m+1)-a B (m+2 n+1)) x^n\right )}{\left (b x^n+a\right )^2}dx}{2 a b n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(e x)^m \left (d x^n+c\right ) \left (c (a B (m+1)-A b (m-2 n+1))-d (A b (m+1)-a B (m+2 n+1)) x^n\right )}{\left (b x^n+a\right )^2}dx}{2 a b n}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 1064

\(\displaystyle \frac {\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{a b e n \left (a+b x^n\right )}-\frac {\int -\frac {(e x)^m \left (d (b c (m+1)-a d (m+n+1)) (A b (m+1)-a B (m+2 n+1)) x^n+c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))\right )}{b x^n+a}dx}{a b n}}{2 a b n}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {(e x)^m \left (d (b c (m+1)-a d (m+n+1)) (A b (m+1)-a B (m+2 n+1)) x^n+c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))\right )}{b x^n+a}dx}{a b n}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{a b e n \left (a+b x^n\right )}}{2 a b n}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 959

\(\displaystyle \frac {\frac {\left (c (a B (m+1)-A b (m-2 n+1)) (a d (m+1)-b c (m-n+1))-\frac {a d (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{b}\right ) \int \frac {(e x)^m}{b x^n+a}dx+\frac {d (e x)^{m+1} (A b (m+1)-a B (m+2 n+1)) (b c (m+1)-a d (m+n+1))}{b e (m+1)}}{a b n}+\frac {(e x)^{m+1} (b c-a d) \left (c (a B (m+1)-A b (m-2 n+1))-d x^n (A b (m+1)-a B (m+2 n+1))\right )}{a b e n \left (a+b x^n\right )}}{2 a b n}+\frac {(e x)^{m+1} (A b-a B) \left (c+d x^n\right )^2}{2 a b e n \left (a+b x^n\right )^2}\)

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 1064

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^( 
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*b*g*n*(p + 1))), x] + Simp[1/( 
a*b*n*(p + 1))   Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c 
*(b*e*n*(p + 1) + (b*e - a*f)*(m + 1)) + d*(b*e*n*(p + 1) + (b*e - a*f)*(m 
+ n*q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x] && LtQ 
[p, -1] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[b*c - a*d, b*e - a*f])
 

Maple [F]

\[\int \frac {\left (e x \right )^{m} \left (A +B \,x^{n}\right ) \left (c +d \,x^{n}\right )^{2}}{\left (a +b \,x^{n}\right )^{3}}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^2/(a+b*x^n)^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^2/(a+b*x^n)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\int { \frac {{\left (B x^{n} + A\right )} {\left (d x^{n} + c\right )}^{2} \left (e x\right )^{m}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \] Input:

integrate((e*x)^m*(A+B*x^n)*(c+d*x^n)^2/(a+b*x^n)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\int \frac {{\left (e\,x\right )}^m\,\left (A+B\,x^n\right )\,{\left (c+d\,x^n\right )}^2}{{\left (a+b\,x^n\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^2}{\left (a+b x^n\right )^3} \, dx=\text {too large to display} \] Input:

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

Output:

(e**m*(2*x**(m + n)*a*c*d*m*x + 2*x**(m + n)*a*c*d*x - x**(m + n)*b*c**2*m 
*x + x**(m + n)*b*c**2*n*x - x**(m + n)*b*c**2*x + x**m*a*c**2*m*x + x**m* 
a*c**2*n*x + x**m*a*c**2*x + x**n*int(x**(m + 2*n)/(x**(2*n)*b**2*m + x**( 
2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*m + 2*x**n*a*b*n + 2*x**n*a*b + a 
**2*m + a**2*n + a**2),x)*a**2*b*d**2*m**3 + 2*x**n*int(x**(m + 2*n)/(x**( 
2*n)*b**2*m + x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*m + 2*x**n*a*b* 
n + 2*x**n*a*b + a**2*m + a**2*n + a**2),x)*a**2*b*d**2*m**2*n + 3*x**n*in 
t(x**(m + 2*n)/(x**(2*n)*b**2*m + x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n 
*a*b*m + 2*x**n*a*b*n + 2*x**n*a*b + a**2*m + a**2*n + a**2),x)*a**2*b*d** 
2*m**2 + x**n*int(x**(m + 2*n)/(x**(2*n)*b**2*m + x**(2*n)*b**2*n + x**(2* 
n)*b**2 + 2*x**n*a*b*m + 2*x**n*a*b*n + 2*x**n*a*b + a**2*m + a**2*n + a** 
2),x)*a**2*b*d**2*m*n**2 + 4*x**n*int(x**(m + 2*n)/(x**(2*n)*b**2*m + x**( 
2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*m + 2*x**n*a*b*n + 2*x**n*a*b + a 
**2*m + a**2*n + a**2),x)*a**2*b*d**2*m*n + 3*x**n*int(x**(m + 2*n)/(x**(2 
*n)*b**2*m + x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*m + 2*x**n*a*b*n 
 + 2*x**n*a*b + a**2*m + a**2*n + a**2),x)*a**2*b*d**2*m + x**n*int(x**(m 
+ 2*n)/(x**(2*n)*b**2*m + x**(2*n)*b**2*n + x**(2*n)*b**2 + 2*x**n*a*b*m + 
 2*x**n*a*b*n + 2*x**n*a*b + a**2*m + a**2*n + a**2),x)*a**2*b*d**2*n**2 + 
 2*x**n*int(x**(m + 2*n)/(x**(2*n)*b**2*m + x**(2*n)*b**2*n + x**(2*n)*b** 
2 + 2*x**n*a*b*m + 2*x**n*a*b*n + 2*x**n*a*b + a**2*m + a**2*n + a**2),...