\(\int \sqrt {e x} (a+b x^2)^p (c+d x^2) \, dx\) [520]

Optimal result

Integrand size = 24, antiderivative size = 107 \[ \int \sqrt {e x} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {2 d (e x)^{3/2} \left (a+b x^2\right )^{1+p}}{b e (7+4 p)}+\frac {2 \left (c-\frac {3 a d}{7 b+4 b p}\right ) (e x)^{3/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^2}{a}\right )}{3 e} \] Output:

2*d*(e*x)^(3/2)*(b*x^2+a)^(p+1)/b/e/(7+4*p)+2/3*(c-3*a*d/(4*b*p+7*b))*(e*x 
)^(3/2)*(b*x^2+a)^p*hypergeom([3/4, -p],[7/4],-b*x^2/a)/e/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.78 \[ \int \sqrt {e x} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {2}{21} x \sqrt {e x} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (7 c \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^2}{a}\right )+3 d x^2 \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^2}{a}\right )\right ) \] Input:

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

Output:

(2*x*Sqrt[e*x]*(a + b*x^2)^p*(7*c*Hypergeometric2F1[3/4, -p, 7/4, -((b*x^2 
)/a)] + 3*d*x^2*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^2)/a)]))/(21*(1 + 
(b*x^2)/a)^p)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {363, 279, 278}

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.

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

\(\Big \downarrow \) 363

\(\displaystyle \left (c-\frac {3 a d}{4 b p+7 b}\right ) \int \sqrt {e x} \left (b x^2+a\right )^pdx+\frac {2 d (e x)^{3/2} \left (a+b x^2\right )^{p+1}}{b e (4 p+7)}\)

\(\Big \downarrow \) 279

\(\displaystyle \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c-\frac {3 a d}{4 b p+7 b}\right ) \int \sqrt {e x} \left (\frac {b x^2}{a}+1\right )^pdx+\frac {2 d (e x)^{3/2} \left (a+b x^2\right )^{p+1}}{b e (4 p+7)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {2 (e x)^{3/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c-\frac {3 a d}{4 b p+7 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^2}{a}\right )}{3 e}+\frac {2 d (e x)^{3/2} \left (a+b x^2\right )^{p+1}}{b e (4 p+7)}\)

Input:

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

Output:

(2*d*(e*x)^(3/2)*(a + b*x^2)^(1 + p))/(b*e*(7 + 4*p)) + (2*(c - (3*a*d)/(7 
*b + 4*b*p))*(e*x)^(3/2)*(a + b*x^2)^p*Hypergeometric2F1[3/4, -p, 7/4, -(( 
b*x^2)/a)])/(3*e*(1 + (b*x^2)/a)^p)
 

Defintions of rubi rules used

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((d*x^2 + c)*sqrt(e*x)*(b*x^2 + a)^p, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 35.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.84 \[ \int \sqrt {e x} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {a^{p} c \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{p} d \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, - p \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \] Input:

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

Output:

a**p*c*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**2*exp_pol 
ar(I*pi)/a)/(2*gamma(7/4)) + a**p*d*sqrt(e)*x**(7/2)*gamma(7/4)*hyper((7/4 
, -p), (11/4,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(11/4))
                                                                                    
                                                                                    
 

Maxima [F]

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

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

Output:

integrate((d*x^2 + c)*sqrt(e*x)*(b*x^2 + a)^p, x)
 

Giac [F]

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

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

Output:

integrate((d*x^2 + c)*sqrt(e*x)*(b*x^2 + a)^p, x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {e x} \left (a+b x^2\right )^p \left (c+d x^2\right ) \, dx=\frac {2 \sqrt {e}\, \left (4 \sqrt {x}\, \left (b \,x^{2}+a \right )^{p} a d p x +4 \sqrt {x}\, \left (b \,x^{2}+a \right )^{p} b c p x +7 \sqrt {x}\, \left (b \,x^{2}+a \right )^{p} b c x +4 \sqrt {x}\, \left (b \,x^{2}+a \right )^{p} b d p \,x^{3}+3 \sqrt {x}\, \left (b \,x^{2}+a \right )^{p} b d \,x^{3}-96 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a^{2} d \,p^{3}-240 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a^{2} d \,p^{2}-126 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a^{2} d p +128 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a b c \,p^{4}+544 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a b c \,p^{3}+728 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a b c \,p^{2}+294 \left (\int \frac {\sqrt {x}\, \left (b \,x^{2}+a \right )^{p}}{16 b \,p^{2} x^{2}+40 b p \,x^{2}+16 a \,p^{2}+21 b \,x^{2}+40 a p +21 a}d x \right ) a b c p \right )}{b \left (16 p^{2}+40 p +21\right )} \] Input:

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

Output:

(2*sqrt(e)*(4*sqrt(x)*(a + b*x**2)**p*a*d*p*x + 4*sqrt(x)*(a + b*x**2)**p* 
b*c*p*x + 7*sqrt(x)*(a + b*x**2)**p*b*c*x + 4*sqrt(x)*(a + b*x**2)**p*b*d* 
p*x**3 + 3*sqrt(x)*(a + b*x**2)**p*b*d*x**3 - 96*int((sqrt(x)*(a + b*x**2) 
**p)/(16*a*p**2 + 40*a*p + 21*a + 16*b*p**2*x**2 + 40*b*p*x**2 + 21*b*x**2 
),x)*a**2*d*p**3 - 240*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2 + 40*a*p + 
 21*a + 16*b*p**2*x**2 + 40*b*p*x**2 + 21*b*x**2),x)*a**2*d*p**2 - 126*int 
((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2 + 40*a*p + 21*a + 16*b*p**2*x**2 + 4 
0*b*p*x**2 + 21*b*x**2),x)*a**2*d*p + 128*int((sqrt(x)*(a + b*x**2)**p)/(1 
6*a*p**2 + 40*a*p + 21*a + 16*b*p**2*x**2 + 40*b*p*x**2 + 21*b*x**2),x)*a* 
b*c*p**4 + 544*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p**2 + 40*a*p + 21*a + 
16*b*p**2*x**2 + 40*b*p*x**2 + 21*b*x**2),x)*a*b*c*p**3 + 728*int((sqrt(x) 
*(a + b*x**2)**p)/(16*a*p**2 + 40*a*p + 21*a + 16*b*p**2*x**2 + 40*b*p*x** 
2 + 21*b*x**2),x)*a*b*c*p**2 + 294*int((sqrt(x)*(a + b*x**2)**p)/(16*a*p** 
2 + 40*a*p + 21*a + 16*b*p**2*x**2 + 40*b*p*x**2 + 21*b*x**2),x)*a*b*c*p)) 
/(b*(16*p**2 + 40*p + 21))