\(\int (e x)^{1-2 p} (a+b x^2)^p (c+d x^2)^3 \, dx\) [1607]

Optimal result

Integrand size = 28, antiderivative size = 268 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\frac {c^3 (e x)^{2-2 p} \left (a+b x^2\right )^{1+p}}{2 a e (1-p)}+\frac {d^2 (12 b c-a d (3-p)) (e x)^{4-2 p} \left (a+b x^2\right )^{1+p}}{24 b^2 e^3}+\frac {d^3 (e x)^{6-2 p} \left (a+b x^2\right )^{1+p}}{8 b e^5}-\frac {\left (24 b^3 c^3-36 a b^2 c^2 d (1-p)+12 a^2 b c d^2 \left (2-3 p+p^2\right )-a^3 d^3 \left (6-11 p+6 p^2-p^3\right )\right ) (e x)^{4-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )}{24 a b^2 e^3 (1-p) (2-p)} \] Output:

1/2*c^3*(e*x)^(2-2*p)*(b*x^2+a)^(p+1)/a/e/(-p+1)+1/24*d^2*(12*b*c-a*d*(3-p 
))*(e*x)^(4-2*p)*(b*x^2+a)^(p+1)/b^2/e^3+1/8*d^3*(e*x)^(6-2*p)*(b*x^2+a)^( 
p+1)/b/e^5-1/24*(24*b^3*c^3-36*a*b^2*c^2*d*(-p+1)+12*a^2*b*c*d^2*(p^2-3*p+ 
2)-a^3*d^3*(-p^3+6*p^2-11*p+6))*(e*x)^(4-2*p)*(b*x^2+a)^p*hypergeom([-p, 2 
-p],[3-p],-b*x^2/a)/a/b^2/e^3/(-p+1)/(2-p)/((1+b*x^2/a)^p)
 

Mathematica [A] (verified)

Time = 5.47 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.69 \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\frac {1}{2} e x^2 (e x)^{-2 p} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {c^3 \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{1-p}+d x^2 \left (-\frac {3 c^2 \operatorname {Hypergeometric2F1}\left (2-p,-p,3-p,-\frac {b x^2}{a}\right )}{-2+p}+d x^2 \left (-\frac {3 c \operatorname {Hypergeometric2F1}\left (3-p,-p,4-p,-\frac {b x^2}{a}\right )}{-3+p}+\frac {d x^2 \operatorname {Hypergeometric2F1}\left (4-p,-p,5-p,-\frac {b x^2}{a}\right )}{4-p}\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {379, 27, 443, 27, 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.

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

\(\Big \downarrow \) 379

\(\displaystyle \frac {\int 2 (e x)^{1-2 p} \left (b x^2+a\right )^p \left (d x^2+c\right ) \left (d (6 b c-a d (3-p)) x^2+c (4 b c-a d (1-p))\right )dx}{8 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (e x)^{1-2 p} \left (b x^2+a\right )^p \left (d x^2+c\right ) \left (d (6 b c-a d (3-p)) x^2+c (4 b c-a d (1-p))\right )dx}{4 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {\frac {\int 2 (e x)^{1-2 p} \left (b x^2+a\right )^p \left (d \left (18 b^2 c^2-2 a b d (9-5 p) c+a^2 d^2 \left (p^2-5 p+6\right )\right ) x^2+c \left (12 b^2 c^2-9 a b d (1-p) c+a^2 d^2 \left (p^2-4 p+3\right )\right )\right )dx}{6 b}+\frac {d \left (c+d x^2\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (3-p))}{6 b e}}{4 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int (e x)^{1-2 p} \left (b x^2+a\right )^p \left (d \left (18 b^2 c^2-2 a b d (9-5 p) c+a^2 d^2 \left (p^2-5 p+6\right )\right ) x^2+c \left (12 b^2 c^2-9 a b d (1-p) c+a^2 d^2 \left (p^2-4 p+3\right )\right )\right )dx}{3 b}+\frac {d \left (c+d x^2\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (3-p))}{6 b e}}{4 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {\frac {\frac {\left (-a^3 d^3 \left (-p^3+6 p^2-11 p+6\right )+12 a^2 b c d^2 \left (p^2-3 p+2\right )-36 a b^2 c^2 d (1-p)+24 b^3 c^3\right ) \int (e x)^{1-2 p} \left (b x^2+a\right )^pdx}{2 b}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (a^2 d^2 \left (p^2-5 p+6\right )-2 a b c d (9-5 p)+18 b^2 c^2\right )}{4 b e}}{3 b}+\frac {d \left (c+d x^2\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (3-p))}{6 b e}}{4 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {\frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^3 d^3 \left (-p^3+6 p^2-11 p+6\right )+12 a^2 b c d^2 \left (p^2-3 p+2\right )-36 a b^2 c^2 d (1-p)+24 b^3 c^3\right ) \int (e x)^{1-2 p} \left (\frac {b x^2}{a}+1\right )^pdx}{2 b}+\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (a^2 d^2 \left (p^2-5 p+6\right )-2 a b c d (9-5 p)+18 b^2 c^2\right )}{4 b e}}{3 b}+\frac {d \left (c+d x^2\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (3-p))}{6 b e}}{4 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {\frac {\frac {d (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} \left (a^2 d^2 \left (p^2-5 p+6\right )-2 a b c d (9-5 p)+18 b^2 c^2\right )}{4 b e}+\frac {(e x)^{2-2 p} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-a^3 d^3 \left (-p^3+6 p^2-11 p+6\right )+12 a^2 b c d^2 \left (p^2-3 p+2\right )-36 a b^2 c^2 d (1-p)+24 b^3 c^3\right ) \operatorname {Hypergeometric2F1}\left (1-p,-p,2-p,-\frac {b x^2}{a}\right )}{4 b e (1-p)}}{3 b}+\frac {d \left (c+d x^2\right ) (e x)^{2-2 p} \left (a+b x^2\right )^{p+1} (6 b c-a d (3-p))}{6 b e}}{4 b}+\frac {d \left (c+d x^2\right )^2 (e x)^{2-2 p} \left (a+b x^2\right )^{p+1}}{8 b e}\)

Input:

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

Output:

(d*(e*x)^(2 - 2*p)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(8*b*e) + ((d*(6*b*c 
 - a*d*(3 - p))*(e*x)^(2 - 2*p)*(a + b*x^2)^(1 + p)*(c + d*x^2))/(6*b*e) + 
 ((d*(18*b^2*c^2 - 2*a*b*c*d*(9 - 5*p) + a^2*d^2*(6 - 5*p + p^2))*(e*x)^(2 
 - 2*p)*(a + b*x^2)^(1 + p))/(4*b*e) + ((24*b^3*c^3 - 36*a*b^2*c^2*d*(1 - 
p) + 12*a^2*b*c*d^2*(2 - 3*p + p^2) - a^3*d^3*(6 - 11*p + 6*p^2 - p^3))*(e 
*x)^(2 - 2*p)*(a + b*x^2)^p*Hypergeometric2F1[1 - p, -p, 2 - p, -((b*x^2)/ 
a)])/(4*b*e*(1 - p)*(1 + (b*x^2)/a)^p))/(3*b))/(4*b)
 

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 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]

\[\int \left (e x \right )^{1-2 p} \left (b \,x^{2}+a \right )^{p} \left (x^{2} d +c \right )^{3}d x\]

Input:

int((e*x)^(1-2*p)*(b*x^2+a)^p*(d*x^2+c)^3,x)
 

Output:

int((e*x)^(1-2*p)*(b*x^2+a)^p*(d*x^2+c)^3,x)
 

Fricas [F]

\[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int { {\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p + 1} \,d x } \] Input:

integrate((e*x)^(1-2*p)*(b*x^2+a)^p*(d*x^2+c)^3,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 + 1), x)
 

Sympy [F(-1)]

Timed out. \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\text {Timed out} \] Input:

integrate((e*x)**(1-2*p)*(b*x**2+a)**p*(d*x**2+c)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int { {\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p + 1} \,d x } \] Input:

integrate((e*x)^(1-2*p)*(b*x^2+a)^p*(d*x^2+c)^3,x, algorithm="maxima")
 

Output:

integrate((d*x^2 + c)^3*(b*x^2 + a)^p*(e*x)^(-2*p + 1), x)
 

Giac [F]

\[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int { {\left (d x^{2} + c\right )}^{3} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{-2 \, p + 1} \,d x } \] Input:

integrate((e*x)^(1-2*p)*(b*x^2+a)^p*(d*x^2+c)^3,x, algorithm="giac")
 

Output:

integrate((d*x^2 + c)^3*(b*x^2 + a)^p*(e*x)^(-2*p + 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx=\int {\left (e\,x\right )}^{1-2\,p}\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^3 \,d x \] Input:

int((e*x)^(1 - 2*p)*(a + b*x^2)^p*(c + d*x^2)^3,x)
 

Output:

int((e*x)^(1 - 2*p)*(a + b*x^2)^p*(c + d*x^2)^3, x)
 

Reduce [F]

\[ \int (e x)^{1-2 p} \left (a+b x^2\right )^p \left (c+d x^2\right )^3 \, dx =\text {Too large to display} \] Input:

int((e*x)^(1-2*p)*(b*x^2+a)^p*(d*x^2+c)^3,x)
                                                                                    
                                                                                    
 

Output:

(e*((a + b*x**2)**p*a**3*d**3*p**3*x**2 - 5*(a + b*x**2)**p*a**3*d**3*p**2 
*x**2 + 6*(a + b*x**2)**p*a**3*d**3*p*x**2 + 12*(a + b*x**2)**p*a**2*b*c*d 
**2*p**2*x**2 - 24*(a + b*x**2)**p*a**2*b*c*d**2*p*x**2 + (a + b*x**2)**p* 
a**2*b*d**3*p**2*x**4 - 3*(a + b*x**2)**p*a**2*b*d**3*p*x**4 + 36*(a + b*x 
**2)**p*a*b**2*c**2*d*p*x**2 + 12*(a + b*x**2)**p*a*b**2*c*d**2*p*x**4 + 2 
*(a + b*x**2)**p*a*b**2*d**3*p*x**6 + 24*(a + b*x**2)**p*b**3*c**3*x**2 + 
36*(a + b*x**2)**p*b**3*c**2*d*x**4 + 24*(a + b*x**2)**p*b**3*c*d**2*x**6 
+ 6*(a + b*x**2)**p*b**3*d**3*x**8 + 2*x**(2*p)*int(((a + b*x**2)**p*x)/(x 
**(2*p)*a + x**(2*p)*b*x**2),x)*a**4*d**3*p**4 - 12*x**(2*p)*int(((a + b*x 
**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**4*d**3*p**3 + 22*x**(2*p)* 
int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**4*d**3*p**2 - 
 12*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a** 
4*d**3*p + 24*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x* 
*2),x)*a**3*b*c*d**2*p**3 - 72*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)* 
a + x**(2*p)*b*x**2),x)*a**3*b*c*d**2*p**2 + 48*x**(2*p)*int(((a + b*x**2) 
**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**3*b*c*d**2*p + 72*x**(2*p)*int 
(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x)*a**2*b**2*c**2*d*p* 
*2 - 72*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x**(2*p)*b*x**2),x) 
*a**2*b**2*c**2*d*p + 48*x**(2*p)*int(((a + b*x**2)**p*x)/(x**(2*p)*a + x* 
*(2*p)*b*x**2),x)*a*b**3*c**3*p))/(48*x**(2*p)*e**(2*p)*b**3)