\(\int \sqrt {e x} (a+b x^n)^p (c+d x^n) \, dx\) [422]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \sqrt {e x} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {2 d (e x)^{3/2} \left (a+b x^n\right )^{1+p}}{b e (3+2 n+2 n p)}+\frac {2 \left (c-\frac {3 a d}{b (3+2 n (1+p))}\right ) (e x)^{3/2} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{2 n},-p,1+\frac {3}{2 n},-\frac {b x^n}{a}\right )}{3 e} \] Output:

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int \sqrt {e x} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {2 x \sqrt {e x} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (3 d x^n \operatorname {Hypergeometric2F1}\left (1+\frac {3}{2 n},-p,2+\frac {3}{2 n},-\frac {b x^n}{a}\right )+c (3+2 n) \operatorname {Hypergeometric2F1}\left (\frac {3}{2 n},-p,1+\frac {3}{2 n},-\frac {b x^n}{a}\right )\right )}{3 (3+2 n)} \] Input:

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

Output:

(2*x*Sqrt[e*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)*(1 + (b*x^n)/a)^p)
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.10, 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.

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

\(\Big \downarrow \) 959

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

\(\Big \downarrow \) 889

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

\(\Big \downarrow \) 888

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

Input:

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

Output:

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

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

Input:

int((e*x)^(1/2)*(a+b*x^n)^p*(c+d*x^n),x)
 

Output:

int((e*x)^(1/2)*(a+b*x^n)^p*(c+d*x^n),x)
 

Fricas [F]

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

integrate((e*x)^(1/2)*(a+b*x^n)^p*(c+d*x^n),x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 24.28 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.15 \[ \int \sqrt {e x} \left (a+b x^n\right )^p \left (c+d x^n\right ) \, dx=\frac {a^{\frac {3}{2 n}} a^{p - \frac {3}{2 n}} c \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2 n}, - p \\ 1 + \frac {3}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {3}{2 n}\right )} + \frac {a^{1 + \frac {3}{2 n}} a^{p - 1 - \frac {3}{2 n}} d \sqrt {e} x^{n + \frac {3}{2}} \Gamma \left (1 + \frac {3}{2 n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 + \frac {3}{2 n} \\ 2 + \frac {3}{2 n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (2 + \frac {3}{2 n}\right )} \] Input:

integrate((e*x)**(1/2)*(a+b*x**n)**p*(c+d*x**n),x)
 

Output:

a**(3/(2*n))*a**(p - 3/(2*n))*c*sqrt(e)*x**(3/2)*gamma(3/(2*n))*hyper((3/( 
2*n), -p), (1 + 3/(2*n),), b*x**n*exp_polar(I*pi)/a)/(n*gamma(1 + 3/(2*n)) 
) + a**(1 + 3/(2*n))*a**(p - 1 - 3/(2*n))*d*sqrt(e)*x**(n + 3/2)*gamma(1 + 
 3/(2*n))*hyper((-p, 1 + 3/(2*n)), (2 + 3/(2*n),), b*x**n*exp_polar(I*pi)/ 
a)/(n*gamma(2 + 3/(2*n)))
                                                                                    
                                                                                    
 

Maxima [F]

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

integrate((e*x)^(1/2)*(a+b*x^n)^p*(c+d*x^n),x, algorithm="maxima")
 

Output:

integrate((d*x^n + c)*sqrt(e*x)*(b*x^n + a)^p, x)
 

Giac [F(-2)]

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

integrate((e*x)^(1/2)*(a+b*x^n)^p*(c+d*x^n),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,[0,0,0,5,2,1,2,2]%%%}+%%%{-4,[0,0,0,5,2,1,1,2]%%%}+%%%{ 
-2,[0,0,0
 

Mupad [F(-1)]

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

int((e*x)^(1/2)*(a + b*x^n)^p*(c + d*x^n),x)
 

Output:

int((e*x)^(1/2)*(a + b*x^n)^p*(c + d*x^n), x)
 

Reduce [F]

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

int((e*x)^(1/2)*(a+b*x^n)^p*(c+d*x^n),x)
 

Output:

(2*sqrt(e)*(2*x**((2*n + 1)/2)*(x**n*b + a)**p*b*d*n*p*x + 3*x**((2*n + 1) 
/2)*(x**n*b + a)**p*b*d*x + 2*sqrt(x)*(x**n*b + a)**p*a*d*n*p*x + 2*sqrt(x 
)*(x**n*b + a)**p*b*c*n*p*x + 2*sqrt(x)*(x**n*b + a)**p*b*c*n*x + 3*sqrt(x 
)*(x**n*b + a)**p*b*c*x - 12*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2* 
p**2 + 4*x**n*b*n**2*p + 12*x**n*b*n*p + 6*x**n*b*n + 9*x**n*b + 4*a*n**2* 
p**2 + 4*a*n**2*p + 12*a*n*p + 6*a*n + 9*a),x)*a**2*d*n**3*p**3 - 12*int(( 
sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2 + 4*x**n*b*n**2*p + 12*x**n*b 
*n*p + 6*x**n*b*n + 9*x**n*b + 4*a*n**2*p**2 + 4*a*n**2*p + 12*a*n*p + 6*a 
*n + 9*a),x)*a**2*d*n**3*p**2 - 36*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b 
*n**2*p**2 + 4*x**n*b*n**2*p + 12*x**n*b*n*p + 6*x**n*b*n + 9*x**n*b + 4*a 
*n**2*p**2 + 4*a*n**2*p + 12*a*n*p + 6*a*n + 9*a),x)*a**2*d*n**2*p**2 - 18 
*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2 + 4*x**n*b*n**2*p + 12* 
x**n*b*n*p + 6*x**n*b*n + 9*x**n*b + 4*a*n**2*p**2 + 4*a*n**2*p + 12*a*n*p 
 + 6*a*n + 9*a),x)*a**2*d*n**2*p - 27*int((sqrt(x)*(x**n*b + a)**p)/(4*x** 
n*b*n**2*p**2 + 4*x**n*b*n**2*p + 12*x**n*b*n*p + 6*x**n*b*n + 9*x**n*b + 
4*a*n**2*p**2 + 4*a*n**2*p + 12*a*n*p + 6*a*n + 9*a),x)*a**2*d*n*p + 8*int 
((sqrt(x)*(x**n*b + a)**p)/(4*x**n*b*n**2*p**2 + 4*x**n*b*n**2*p + 12*x**n 
*b*n*p + 6*x**n*b*n + 9*x**n*b + 4*a*n**2*p**2 + 4*a*n**2*p + 12*a*n*p + 6 
*a*n + 9*a),x)*a*b*c*n**4*p**4 + 16*int((sqrt(x)*(x**n*b + a)**p)/(4*x**n* 
b*n**2*p**2 + 4*x**n*b*n**2*p + 12*x**n*b*n*p + 6*x**n*b*n + 9*x**n*b +...