\(\int (e x)^{-2 p} (a+b x^2)^p (c+d x^2)^3 \, dx\) [1614]

Optimal result

Integrand size = 26, antiderivative size = 296 \[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\frac {d \left (105 b^2 c^2-21 a b c d (3-2 p)+a^2 d^2 \left (15-16 p+4 p^2\right )\right ) (e x)^{1-2 p} \left (a+b x^2\right )^{1+p}}{105 b^3 e}+\frac {d^2 (21 b c-a d (5-2 p)) (e x)^{3-2 p} \left (a+b x^2\right )^{1+p}}{35 b^2 e^3}+\frac {d^3 (e x)^{5-2 p} \left (a+b x^2\right )^{1+p}}{7 b e^5}+\frac {\left (105 b^3 c^3-105 a b^2 c^2 d (1-2 p)+21 a^2 b c d^2 \left (3-8 p+4 p^2\right )-a^3 d^3 \left (15-46 p+36 p^2-8 p^3\right )\right ) (e x)^{1-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{105 b^3 e (1-2 p)} \] Output:

1/105*d*(105*b^2*c^2-21*a*b*c*d*(3-2*p)+a^2*d^2*(4*p^2-16*p+15))*(e*x)^(1- 
2*p)*(b*x^2+a)^(p+1)/b^3/e+1/35*d^2*(21*b*c-a*d*(5-2*p))*(e*x)^(3-2*p)*(b* 
x^2+a)^(p+1)/b^2/e^3+1/7*d^3*(e*x)^(5-2*p)*(b*x^2+a)^(p+1)/b/e^5+1/105*(10 
5*b^3*c^3-105*a*b^2*c^2*d*(1-2*p)+21*a^2*b*c*d^2*(4*p^2-8*p+3)-a^3*d^3*(-8 
*p^3+36*p^2-46*p+15))*(e*x)^(1-2*p)*(b*x^2+a)^p*hypergeom([-p, 1/2-p],[3/2 
-p],-b*x^2/a)/b^3/e/(1-2*p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 6.31 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.67 \[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=x (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {c^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,-\frac {b x^2}{a}\right )}{1-2 p}+d x^2 \left (\frac {3 c^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,-p,\frac {5}{2}-p,-\frac {b x^2}{a}\right )}{3-2 p}+d x^2 \left (\frac {3 c \operatorname {Hypergeometric2F1}\left (\frac {5}{2}-p,-p,\frac {7}{2}-p,-\frac {b x^2}{a}\right )}{5-2 p}+\frac {d x^2 \operatorname {Hypergeometric2F1}\left (\frac {7}{2}-p,-p,\frac {9}{2}-p,-\frac {b x^2}{a}\right )}{7-2 p}\right )\right )\right ) \] Input:

Integrate[((a + b*x^2)^p*(c + d*x^2)^3)/(e*x)^(2*p),x]
 

Output:

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

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {379, 443, 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.

\(\displaystyle \int \left (c+d x^2\right )^3 (e x)^{-2 p} \left (a+b x^2\right )^p \, dx\)

\(\Big \downarrow \) 379

\(\displaystyle \frac {\int (e x)^{-2 p} \left (b x^2+a\right )^p \left (d x^2+c\right ) \left (d (11 b c-a d (5-2 p)) x^2+c (7 b c-a d (1-2 p))\right )dx}{7 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{7 b e}\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {\frac {\int (e x)^{-2 p} \left (b x^2+a\right )^p \left (d \left (57 b^2 c^2-12 a b d (4-3 p) c+a^2 d^2 \left (4 p^2-16 p+15\right )\right ) x^2+c \left (35 b^2 c^2-16 a b d (1-2 p) c+a^2 d^2 \left (4 p^2-12 p+5\right )\right )\right )dx}{5 b}+\frac {d \left (c+d x^2\right ) (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} (11 b c-a d (5-2 p))}{5 b e}}{7 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{7 b e}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {\frac {\frac {\left (-a^3 d^3 \left (-8 p^3+36 p^2-46 p+15\right )+21 a^2 b c d^2 \left (4 p^2-8 p+3\right )-105 a b^2 c^2 d (1-2 p)+105 b^3 c^3\right ) \int (e x)^{-2 p} \left (b x^2+a\right )^pdx}{3 b}+\frac {d (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} \left (a^2 d^2 \left (4 p^2-16 p+15\right )-12 a b c d (4-3 p)+57 b^2 c^2\right )}{3 b e}}{5 b}+\frac {d \left (c+d x^2\right ) (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} (11 b c-a d (5-2 p))}{5 b e}}{7 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{7 b e}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^3 d^3 \left (-8 p^3+36 p^2-46 p+15\right )+21 a^2 b c d^2 \left (4 p^2-8 p+3\right )-105 a b^2 c^2 d (1-2 p)+105 b^3 c^3\right ) \int (e x)^{-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{3 b}+\frac {d (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} \left (a^2 d^2 \left (4 p^2-16 p+15\right )-12 a b c d (4-3 p)+57 b^2 c^2\right )}{3 b e}}{5 b}+\frac {d \left (c+d x^2\right ) (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} (11 b c-a d (5-2 p))}{5 b e}}{7 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{7 b e}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {\frac {d (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} \left (a^2 d^2 \left (4 p^2-16 p+15\right )-12 a b c d (4-3 p)+57 b^2 c^2\right )}{3 b e}+\frac {(e x)^{1-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^3 d^3 \left (-8 p^3+36 p^2-46 p+15\right )+21 a^2 b c d^2 \left (4 p^2-8 p+3\right )-105 a b^2 c^2 d (1-2 p)+105 b^3 c^3\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),-p,\frac {1}{2} (3-2 p),-\frac {b x^2}{a}\right )}{3 b e (1-2 p)}}{5 b}+\frac {d \left (c+d x^2\right ) (e x)^{1-2 p} \left (a+b x^2\right )^{p+1} (11 b c-a d (5-2 p))}{5 b e}}{7 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{1-2 p} \left (a+b x^2\right )^{p+1}}{7 b e}\)

Input:

Int[((a + b*x^2)^p*(c + d*x^2)^3)/(e*x)^(2*p),x]
 

Output:

(d*(e*x)^(1 - 2*p)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(7*b*e) + ((d*(11*b* 
c - a*d*(5 - 2*p))*(e*x)^(1 - 2*p)*(a + b*x^2)^(1 + p)*(c + d*x^2))/(5*b*e 
) + ((d*(57*b^2*c^2 - 12*a*b*c*d*(4 - 3*p) + a^2*d^2*(15 - 16*p + 4*p^2))* 
(e*x)^(1 - 2*p)*(a + b*x^2)^(1 + p))/(3*b*e) + ((105*b^3*c^3 - 105*a*b^2*c 
^2*d*(1 - 2*p) + 21*a^2*b*c*d^2*(3 - 8*p + 4*p^2) - a^3*d^3*(15 - 46*p + 3 
6*p^2 - 8*p^3))*(e*x)^(1 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[(1 - 2*p)/ 
2, -p, (3 - 2*p)/2, -((b*x^2)/a)])/(3*b*e*(1 - 2*p)*(1 + (b*x^2)/a)^p))/(5 
*b))/(7*b)
 

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]
 

rule 379

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

rule 443

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^2)^(p 
 + 1)*((c + d*x^2)^q/(b*g*(m + 2*(p + q + 1) + 1))), x] + Simp[1/(b*(m + 2* 
(p + q + 1) + 1))   Int[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(( 
b*e - a*f)*(m + 1) + b*e*2*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*2*q*(b 
*c - a*d) + b*e*d*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f 
, g, m, p}, x] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^ 
2])
 

Maple [F]

\[\int \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )^{3} \left (e x \right )^{-2 p}d x\]

Input:

int((b*x^2+a)^p*(d*x^2+c)^3/((e*x)^(2*p)),x)
 

Output:

int((b*x^2+a)^p*(d*x^2+c)^3/((e*x)^(2*p)),x)
 

Fricas [F]

\[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p}}{\left (e x\right )^{2 \, p}} \,d x } \] Input:

integrate((b*x^2+a)^p*(d*x^2+c)^3/((e*x)^(2*p)),x, algorithm="fricas")
 

Output:

integral((d^3*x^6 + 3*c*d^2*x^4 + 3*c^2*d*x^2 + c^3)*(b*x^2 + a)^p/(e*x)^( 
2*p), x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 159.97 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.73 \[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\frac {3 a^{p} c^{2} d e^{- 2 p} x^{3 - 2 p} \Gamma \left (\frac {3}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {3}{2} - p \\ \frac {5}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {5}{2} - p\right )} + \frac {3 a^{p} c d^{2} e^{- 2 p} x^{5 - 2 p} \Gamma \left (\frac {5}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {5}{2} - p \\ \frac {7}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {7}{2} - p\right )} + \frac {a^{p} d^{3} e^{- 2 p} x^{7 - 2 p} \Gamma \left (\frac {7}{2} - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {7}{2} - p \\ \frac {9}{2} - p \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {9}{2} - p\right )} + \sqrt {b} b^{p - \frac {1}{2}} c^{3} e^{- 2 p} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )} \] Input:

integrate((b*x**2+a)**p*(d*x**2+c)**3/((e*x)**(2*p)),x)
 

Output:

3*a**p*c**2*d*x**(3 - 2*p)*gamma(3/2 - p)*hyper((-p, 3/2 - p), (5/2 - p,), 
 b*x**2*exp_polar(I*pi)/a)/(2*e**(2*p)*gamma(5/2 - p)) + 3*a**p*c*d**2*x** 
(5 - 2*p)*gamma(5/2 - p)*hyper((-p, 5/2 - p), (7/2 - p,), b*x**2*exp_polar 
(I*pi)/a)/(2*e**(2*p)*gamma(7/2 - p)) + a**p*d**3*x**(7 - 2*p)*gamma(7/2 - 
 p)*hyper((-p, 7/2 - p), (9/2 - p,), b*x**2*exp_polar(I*pi)/a)/(2*e**(2*p) 
*gamma(9/2 - p)) + sqrt(b)*b**(p - 1/2)*c**3*x*hyper((-1/2, -p), (1/2,), a 
*exp_polar(I*pi)/(b*x**2))/e**(2*p)
 

Maxima [F]

\[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p}}{\left (e x\right )^{2 \, p}} \,d x } \] Input:

integrate((b*x^2+a)^p*(d*x^2+c)^3/((e*x)^(2*p)),x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^3*(b*x^2 + a)^p/(e*x)^(2*p), x)
 

Giac [F]

\[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int { \frac {{\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p}}{\left (e x\right )^{2 \, p}} \,d x } \] Input:

integrate((b*x^2+a)^p*(d*x^2+c)^3/((e*x)^(2*p)),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate((d*x^2 + c)^3*(b*x^2 + a)^p/(e*x)^(2*p), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^3}{{\left (e\,x\right )}^{2\,p}} \,d x \] Input:

int(((a + b*x^2)^p*(c + d*x^2)^3)/(e*x)^(2*p),x)
 

Output:

int(((a + b*x^2)^p*(c + d*x^2)^3)/(e*x)^(2*p), x)
 

Reduce [F]

\[ \int (e x)^{-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((b*x^2+a)^p*(d*x^2+c)^3/((e*x)^(2*p)),x)
 

Output:

(8*(a + b*x**2)**p*a**3*d**3*p**3*x - 32*(a + b*x**2)**p*a**3*d**3*p**2*x 
+ 30*(a + b*x**2)**p*a**3*d**3*p*x + 84*(a + b*x**2)**p*a**2*b*c*d**2*p**2 
*x - 126*(a + b*x**2)**p*a**2*b*c*d**2*p*x + 4*(a + b*x**2)**p*a**2*b*d**3 
*p**2*x**3 - 10*(a + b*x**2)**p*a**2*b*d**3*p*x**3 + 210*(a + b*x**2)**p*a 
*b**2*c**2*d*p*x + 42*(a + b*x**2)**p*a*b**2*c*d**2*p*x**3 + 6*(a + b*x**2 
)**p*a*b**2*d**3*p*x**5 + 105*(a + b*x**2)**p*b**3*c**3*x + 105*(a + b*x** 
2)**p*b**3*c**2*d*x**3 + 63*(a + b*x**2)**p*b**3*c*d**2*x**5 + 15*(a + b*x 
**2)**p*b**3*d**3*x**7 + 16*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x** 
(2*p)*b*x**2),x)*a**4*d**3*p**4 - 72*x**(2*p)*int((a + b*x**2)**p/(x**(2*p 
)*a + x**(2*p)*b*x**2),x)*a**4*d**3*p**3 + 92*x**(2*p)*int((a + b*x**2)**p 
/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**4*d**3*p**2 - 30*x**(2*p)*int((a + b 
*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**4*d**3*p + 168*x**(2*p)*int 
((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*b*c*d**2*p**3 - 33 
6*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*b*c* 
d**2*p**2 + 126*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2 
),x)*a**3*b*c*d**2*p + 420*x**(2*p)*int((a + b*x**2)**p/(x**(2*p)*a + x**( 
2*p)*b*x**2),x)*a**2*b**2*c**2*d*p**2 - 210*x**(2*p)*int((a + b*x**2)**p/( 
x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*b**2*c**2*d*p + 210*x**(2*p)*int((a 
+ b*x**2)**p/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a*b**3*c**3*p)/(105*x**(2*p 
)*e**(2*p)*b**3)