\(\int \frac {(a+b x^n)^p (c+d x^n)}{(e x)^{5/2}} \, dx\) [425]

Optimal result

Integrand size = 24, antiderivative size = 123 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=-\frac {2 d \left (a+b x^n\right )^{1+p}}{b e (3-2 n (1+p)) (e x)^{3/2}}-\frac {2 \left (c-\frac {3 a d}{b (3-2 n (1+p))}\right ) \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2 n},-p,1-\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 e (e x)^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\frac {2 x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (3 d x^n \operatorname {Hypergeometric2F1}\left (1-\frac {3}{2 n},-p,2-\frac {3}{2 n},-\frac {b x^n}{a}\right )+c (3-2 n) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2 n},-p,1-\frac {3}{2 n},-\frac {b x^n}{a}\right )\right )}{3 (-3+2 n) (e x)^{5/2}} \] Input:

Integrate[((a + b*x^n)^p*(c + d*x^n))/(e*x)^(5/2),x]
 

Output:

(2*x*(a + b*x^n)^p*(3*d*x^n*Hypergeometric2F1[1 - 3/(2*n), -p, 2 - 3/(2*n) 
, -((b*x^n)/a)] + c*(3 - 2*n)*Hypergeometric2F1[-3/(2*n), -p, 1 - 3/(2*n), 
 -((b*x^n)/a)]))/(3*(-3 + 2*n)*(e*x)^(5/2)*(1 + (b*x^n)/a)^p)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.01, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

\(\displaystyle \int \frac {\left (c+d x^n\right ) \left (a+b x^n\right )^p}{(e x)^{5/2}} \, dx\)

\(\Big \downarrow \) 959

\(\displaystyle \left (c-\frac {3 a d}{b (3-2 n (p+1))}\right ) \int \frac {\left (b x^n+a\right )^p}{(e x)^{5/2}}dx-\frac {2 d \left (a+b x^n\right )^{p+1}}{b e (e x)^{3/2} (-2 n p-2 n+3)}\)

\(\Big \downarrow \) 889

\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c-\frac {3 a d}{b (3-2 n (p+1))}\right ) \int \frac {\left (\frac {b x^n}{a}+1\right )^p}{(e x)^{5/2}}dx-\frac {2 d \left (a+b x^n\right )^{p+1}}{b e (e x)^{3/2} (-2 n p-2 n+3)}\)

\(\Big \downarrow \) 888

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

Input:

Int[((a + b*x^n)^p*(c + d*x^n))/(e*x)^(5/2),x]
 

Output:

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

\[\int \frac {\left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )}{\left (e x \right )^{\frac {5}{2}}}d x\]

Input:

int((a+b*x^n)^p*(c+d*x^n)/(e*x)^(5/2),x)
 

Output:

int((a+b*x^n)^p*(c+d*x^n)/(e*x)^(5/2),x)
 

Fricas [F]

\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*x^n)^p*(c+d*x^n)/(e*x)^(5/2),x, algorithm="fricas")
 

Output:

integral((d*x^n + c)*sqrt(e*x)*(b*x^n + a)^p/(e^3*x^3), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\text {Timed out} \] Input:

integrate((a+b*x**n)**p*(c+d*x**n)/(e*x)**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\int { \frac {{\left (d x^{n} + c\right )} {\left (b x^{n} + a\right )}^{p}}{\left (e x\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate((a+b*x^n)^p*(c+d*x^n)/(e*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*x^n + c)*(b*x^n + a)^p/(e*x)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((a+b*x^n)^p*(c+d*x^n)/(e*x)^(5/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-2,[0,0,0,2,1,0,1,2]%%%}+%%%{-2,[0,0,0,2,1,0,0,2]%%%} / %% 
%{4,[0,0,
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\int \frac {{\left (a+b\,x^n\right )}^p\,\left (c+d\,x^n\right )}{{\left (e\,x\right )}^{5/2}} \,d x \] Input:

int(((a + b*x^n)^p*(c + d*x^n))/(e*x)^(5/2),x)
 

Output:

int(((a + b*x^n)^p*(c + d*x^n))/(e*x)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^n\right )^p \left (c+d x^n\right )}{(e x)^{5/2}} \, dx=\text {too large to display} \] Input:

int((a+b*x^n)^p*(c+d*x^n)/(e*x)^(5/2),x)
 

Output:

(2*sqrt(e)*(2*x**n*(x**n*b + a)**p*b*d*n*p - 3*x**n*(x**n*b + a)**p*b*d + 
2*(x**n*b + a)**p*a*d*n*p + 2*(x**n*b + a)**p*b*c*n*p + 2*(x**n*b + a)**p* 
b*c*n - 3*(x**n*b + a)**p*b*c + 12*sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/( 
4*x**n*b*n**2*p**2*x**3 + 4*x**n*b*n**2*p*x**3 - 12*x**n*b*n*p*x**3 - 6*x* 
*n*b*n*x**3 + 9*x**n*b*x**3 + 4*a*n**2*p**2*x**3 + 4*a*n**2*p*x**3 - 12*a* 
n*p*x**3 - 6*a*n*x**3 + 9*a*x**3),x)*a**2*d*n**3*p**3*x + 12*sqrt(x)*int(( 
sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2*x**3 + 4*x**n*b*n**2*p*x**3 - 
 12*x**n*b*n*p*x**3 - 6*x**n*b*n*x**3 + 9*x**n*b*x**3 + 4*a*n**2*p**2*x**3 
 + 4*a*n**2*p*x**3 - 12*a*n*p*x**3 - 6*a*n*x**3 + 9*a*x**3),x)*a**2*d*n**3 
*p**2*x - 36*sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2*x** 
3 + 4*x**n*b*n**2*p*x**3 - 12*x**n*b*n*p*x**3 - 6*x**n*b*n*x**3 + 9*x**n*b 
*x**3 + 4*a*n**2*p**2*x**3 + 4*a*n**2*p*x**3 - 12*a*n*p*x**3 - 6*a*n*x**3 
+ 9*a*x**3),x)*a**2*d*n**2*p**2*x - 18*sqrt(x)*int((sqrt(x)*(x**n*b + a)** 
p)/(4*x**n*b*n**2*p**2*x**3 + 4*x**n*b*n**2*p*x**3 - 12*x**n*b*n*p*x**3 - 
6*x**n*b*n*x**3 + 9*x**n*b*x**3 + 4*a*n**2*p**2*x**3 + 4*a*n**2*p*x**3 - 1 
2*a*n*p*x**3 - 6*a*n*x**3 + 9*a*x**3),x)*a**2*d*n**2*p*x + 27*sqrt(x)*int( 
(sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2*x**3 + 4*x**n*b*n**2*p*x**3 
- 12*x**n*b*n*p*x**3 - 6*x**n*b*n*x**3 + 9*x**n*b*x**3 + 4*a*n**2*p**2*x** 
3 + 4*a*n**2*p*x**3 - 12*a*n*p*x**3 - 6*a*n*x**3 + 9*a*x**3),x)*a**2*d*n*p 
*x + 8*sqrt(x)*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2*x**3 +...