\(\int \sqrt {e x} (c+d x)^2 (a+b x^2)^p \, dx\) [65]

Optimal result

Integrand size = 24, antiderivative size = 171 \[ \int \sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {2 d^2 (e x)^{3/2} \left (a+b x^2\right )^{1+p}}{b e (7+4 p)}+\frac {2 \left (c^2-\frac {3 a d^2}{7 b+4 b p}\right ) (e x)^{3/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^2}{a}\right )}{3 e}+\frac {4 c d (e x)^{5/2} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^2}{a}\right )}{5 e^2} \] Output:

2*d^2*(e*x)^(3/2)*(b*x^2+a)^(p+1)/b/e/(7+4*p)+2/3*(c^2-3*a*d^2/(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)+4/5*c*d*(e*x)^(5/2)*(b*x^2+a)^p*hypergeom([5/4, -p],[9/4],-b*x^2/a)/ 
e^2/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.64 \[ \int \sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {2}{105} x \sqrt {e x} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (35 c^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^2}{a}\right )+3 d x \left (14 c \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^2}{a}\right )+5 d x \operatorname {Hypergeometric2F1}\left (\frac {7}{4},-p,\frac {11}{4},-\frac {b x^2}{a}\right )\right )\right ) \] Input:

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

Output:

(2*x*Sqrt[e*x]*(a + b*x^2)^p*(35*c^2*Hypergeometric2F1[3/4, -p, 7/4, -((b* 
x^2)/a)] + 3*d*x*(14*c*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^2)/a)] + 5*d 
*x*Hypergeometric2F1[7/4, -p, 11/4, -((b*x^2)/a)])))/(105*(1 + (b*x^2)/a)^ 
p)
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.10, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {559, 27, 557, 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.

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

\(\Big \downarrow \) 559

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 557

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

\(\Big \downarrow \) 279

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

\(\Big \downarrow \) 278

\(\displaystyle \frac {2 d^2 (e x)^{3/2} \left (a+b x^2\right )^{p+1}}{b e (4 p+7)}-\frac {\frac {2 (e x)^{3/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 a d^2-b c^2 (4 p+7)\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},-p,\frac {7}{4},-\frac {b x^2}{a}\right )}{3 e}-\frac {4 b c d (4 p+7) (e x)^{5/2} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},-p,\frac {9}{4},-\frac {b x^2}{a}\right )}{5 e^2}}{b (4 p+7)}\)

Input:

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

Output:

(2*d^2*(e*x)^(3/2)*(a + b*x^2)^(1 + p))/(b*e*(7 + 4*p)) - ((2*(3*a*d^2 - b 
*c^2*(7 + 4*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) - (4*b*c*d*(7 + 4*p)*(e*x)^(5/2)*(a 
 + b*x^2)^p*Hypergeometric2F1[5/4, -p, 9/4, -((b*x^2)/a)])/(5*e^2*(1 + (b* 
x^2)/a)^p))/(b*(7 + 4*p))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 557

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

rule 559

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 50.55 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.81 \[ \int \sqrt {e x} (c+d x)^2 \left (a+b x^2\right )^p \, dx=\frac {a^{p} c^{2} \sqrt {e} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, - p \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} + \frac {a^{p} c d \sqrt {e} x^{\frac {5}{2}} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, - p \\ \frac {9}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{\Gamma \left (\frac {9}{4}\right )} + \frac {a^{p} d^{2} \sqrt {e} x^{\frac {7}{2}} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {7}{4}, - p \\ \frac {11}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {11}{4}\right )} \] Input:

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

Output:

a**p*c**2*sqrt(e)*x**(3/2)*gamma(3/4)*hyper((3/4, -p), (7/4,), b*x**2*exp_ 
polar(I*pi)/a)/(2*gamma(7/4)) + a**p*c*d*sqrt(e)*x**(5/2)*gamma(5/4)*hyper 
((5/4, -p), (9/4,), b*x**2*exp_polar(I*pi)/a)/gamma(9/4) + a**p*d**2*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]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

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

Output:

(2*sqrt(e)*(128*sqrt(x)*(a + b*x**2)**p*a*c*d*p**3 + 320*sqrt(x)*(a + b*x* 
*2)**p*a*c*d*p**2 + 168*sqrt(x)*(a + b*x**2)**p*a*c*d*p + 64*sqrt(x)*(a + 
b*x**2)**p*a*d**2*p**3*x + 96*sqrt(x)*(a + b*x**2)**p*a*d**2*p**2*x + 20*s 
qrt(x)*(a + b*x**2)**p*a*d**2*p*x + 64*sqrt(x)*(a + b*x**2)**p*b*c**2*p**3 
*x + 208*sqrt(x)*(a + b*x**2)**p*b*c**2*p**2*x + 188*sqrt(x)*(a + b*x**2)* 
*p*b*c**2*p*x + 35*sqrt(x)*(a + b*x**2)**p*b*c**2*x + 128*sqrt(x)*(a + b*x 
**2)**p*b*c*d*p**3*x**2 + 352*sqrt(x)*(a + b*x**2)**p*b*c*d*p**2*x**2 + 24 
8*sqrt(x)*(a + b*x**2)**p*b*c*d*p*x**2 + 42*sqrt(x)*(a + b*x**2)**p*b*c*d* 
x**2 + 64*sqrt(x)*(a + b*x**2)**p*b*d**2*p**3*x**3 + 144*sqrt(x)*(a + b*x* 
*2)**p*b*d**2*p**2*x**3 + 92*sqrt(x)*(a + b*x**2)**p*b*d**2*p*x**3 + 15*sq 
rt(x)*(a + b*x**2)**p*b*d**2*x**3 - 16384*int((sqrt(x)*(a + b*x**2)**p)/(2 
56*a*p**4*x + 1024*a*p**3*x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256*b* 
p**4*x**3 + 1024*b*p**3*x**3 + 1376*b*p**2*x**3 + 704*b*p*x**3 + 105*b*x** 
3),x)*a**2*c*d*p**7 - 106496*int((sqrt(x)*(a + b*x**2)**p)/(256*a*p**4*x + 
 1024*a*p**3*x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256*b*p**4*x**3 + 1 
024*b*p**3*x**3 + 1376*b*p**2*x**3 + 704*b*p*x**3 + 105*b*x**3),x)*a**2*c* 
d*p**6 - 273408*int((sqrt(x)*(a + b*x**2)**p)/(256*a*p**4*x + 1024*a*p**3* 
x + 1376*a*p**2*x + 704*a*p*x + 105*a*x + 256*b*p**4*x**3 + 1024*b*p**3*x* 
*3 + 1376*b*p**2*x**3 + 704*b*p*x**3 + 105*b*x**3),x)*a**2*c*d*p**5 - 3512 
32*int((sqrt(x)*(a + b*x**2)**p)/(256*a*p**4*x + 1024*a*p**3*x + 1376*a...