3.1.44 \(\int \frac {(e x)^m (a+b x^n)^p (A+B x^n)}{(c+d x^n)^2} \, dx\) [44]

3.1.44.1 Optimal result

Integrand size = 31, antiderivative size = 304 \[ \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\frac {(B c-A d) (e x)^{1+m} \left (a+b x^n\right )^{1+p}}{c (b c-a d) e n \left (c+d x^n\right )}-\frac {(a d (B c (1+m)-A d (1+m-n))+b c (A d (1+m-n (1-p))-B c (1+m+n p))) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{n},-p,1,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c^2 d (b c-a d) e (1+m) n}-\frac {b (B c-A d) (1+m+n p) (e x)^{1+m} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{n},-p,\frac {1+m+n}{n},-\frac {b x^n}{a}\right )}{c d (b c-a d) e (1+m) n} \]

output

(-A*d+B*c)*(e*x)^(1+m)*(a+b*x^n)^(p+1)/c/(-a*d+b*c)/e/n/(c+d*x^n)-(a*d*(B* 
c*(1+m)-A*d*(1+m-n))+b*c*(A*d*(1+m-n*(1-p))-B*c*(n*p+m+1)))*(e*x)^(1+m)*(a 
+b*x^n)^p*AppellF1((1+m)/n,-p,1,(1+m+n)/n,-b*x^n/a,-d*x^n/c)/c^2/d/(-a*d+b 
*c)/e/(1+m)/n/((1+b*x^n/a)^p)-b*(-A*d+B*c)*(n*p+m+1)*(e*x)^(1+m)*(a+b*x^n) 
^p*hypergeom([-p, (1+m)/n],[(1+m+n)/n],-b*x^n/a)/c/d/(-a*d+b*c)/e/(1+m)/n/ 
((1+b*x^n/a)^p)
 

3.1.44.2 Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.45 \[ \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\frac {x (e x)^m \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (A (1+m+n) \operatorname {AppellF1}\left (\frac {1+m}{n},-p,2,\frac {1+m+n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+B (1+m) x^n \operatorname {AppellF1}\left (\frac {1+m+n}{n},-p,2,\frac {1+m+2 n}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )\right )}{c^2 (1+m) (1+m+n)} \]

input

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

output

(x*(e*x)^m*(a + b*x^n)^p*(A*(1 + m + n)*AppellF1[(1 + m)/n, -p, 2, (1 + m 
+ n)/n, -((b*x^n)/a), -((d*x^n)/c)] + B*(1 + m)*x^n*AppellF1[(1 + m + n)/n 
, -p, 2, (1 + m + 2*n)/n, -((b*x^n)/a), -((d*x^n)/c)]))/(c^2*(1 + m)*(1 + 
m + n)*(1 + (b*x^n)/a)^p)
 

3.1.44.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {1065, 25, 1067, 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.

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

\(\Big \downarrow \) 1065

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1067

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {(e x)^{m+1} (B c-A d) \left (a+b x^n\right )^{p+1}}{c e n (b c-a d) \left (c+d x^n\right )}-\frac {\frac {(e x)^{m+1} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} (d (-a A d (m-n+1)+a B c (m+1)-A b c n)-b c (m+n p+1) (B c-A d)) \operatorname {AppellF1}\left (\frac {m+1}{n},-p,1,\frac {m+n+1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{c d e (m+1)}+\frac {b (e x)^{m+1} (m+n p+1) (B c-A d) \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{n},-p,\frac {m+n+1}{n},-\frac {b x^n}{a}\right )}{d e (m+1)}}{c n (b c-a d)}\)

input

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

output

((B*c - A*d)*(e*x)^(1 + m)*(a + b*x^n)^(1 + p))/(c*(b*c - a*d)*e*n*(c + d* 
x^n)) - (((d*(a*B*c*(1 + m) - a*A*d*(1 + m - n) - A*b*c*n) - b*c*(B*c - A* 
d)*(1 + m + n*p))*(e*x)^(1 + m)*(a + b*x^n)^p*AppellF1[(1 + m)/n, -p, 1, ( 
1 + m + n)/n, -((b*x^n)/a), -((d*x^n)/c)])/(c*d*e*(1 + m)*(1 + (b*x^n)/a)^ 
p) + (b*(B*c - A*d)*(1 + m + n*p)*(e*x)^(1 + m)*(a + b*x^n)^p*Hypergeometr 
ic2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)])/(d*e*(1 + m)*(1 + (b*x^ 
n)/a)^p))/(c*(b*c - a*d)*n)
 

3.1.44.3.1 Defintions of rubi rules used

rule 25

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

rule 1065

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 + 1)/(a*g*n*(b*c - a*d)*(p + 1))) 
, x] + Simp[1/(a*n*(b*c - a*d)*(p + 1))   Int[(g*x)^m*(a + b*x^n)^(p + 1)*( 
c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e 
- a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, 
g, m, n, q}, x] && LtQ[p, -1]
 

rule 1067

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

rule 2009

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

3.1.44.4 Maple [F]

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

input

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

output

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

3.1.44.5 Fricas [F]

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

input

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

output

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

3.1.44.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {(e x)^m \left (a+b x^n\right )^p \left (A+B x^n\right )}{\left (c+d x^n\right )^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

input

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

output

Exception raised: HeuristicGCDFailed >> no luck
 

3.1.44.7 Maxima [F]

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

input

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

output

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

3.1.44.8 Giac [F]

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

input

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

output

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

3.1.44.9 Mupad [F(-1)]

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

input

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

output

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