Integrand size = 31, antiderivative size = 399 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )-a b d (2 B c+A d) (3+m) (7+m+2 p)+b^2 c (B c+2 A d) \left (35+m^2+24 p+4 p^2+4 m (3+p)\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 e (3+m+2 p) (5+m+2 p) (7+m+2 p)}-\frac {d (a B d (5+m)-b (2 B c+A d) (7+m+2 p)) (e x)^{3+m} \left (a+b x^2\right )^{1+p}}{b^2 e^3 (5+m+2 p) (7+m+2 p)}+\frac {B d^2 (e x)^{5+m} \left (a+b x^2\right )^{1+p}}{b e^5 (7+m+2 p)}+\frac {\left (\frac {A c^2}{1+m}-\frac {a \left (a^2 B d^2 \left (15+8 m+m^2\right )-a b d (2 B c+A d) (3+m) (7+m+2 p)+b^2 c (B c+2 A d) \left (35+m^2+24 p+4 p^2+4 m (3+p)\right )\right )}{b^3 (3+m+2 p) (5+m+2 p) (7+m+2 p)}\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e} \] Output:
(a^2*B*d^2*(m^2+8*m+15)-a*b*d*(A*d+2*B*c)*(3+m)*(7+m+2*p)+b^2*c*(2*A*d+B*c )*(35+m^2+24*p+4*p^2+4*m*(3+p)))*(e*x)^(1+m)*(b*x^2+a)^(p+1)/b^3/e/(3+m+2* p)/(5+m+2*p)/(7+m+2*p)-d*(a*B*d*(5+m)-b*(A*d+2*B*c)*(7+m+2*p))*(e*x)^(3+m) *(b*x^2+a)^(p+1)/b^2/e^3/(5+m+2*p)/(7+m+2*p)+B*d^2*(e*x)^(5+m)*(b*x^2+a)^( p+1)/b/e^5/(7+m+2*p)+(A*c^2/(1+m)-a*(a^2*B*d^2*(m^2+8*m+15)-a*b*d*(A*d+2*B *c)*(3+m)*(7+m+2*p)+b^2*c*(2*A*d+B*c)*(35+m^2+24*p+4*p^2+4*m*(3+p)))/b^3/( 3+m+2*p)/(5+m+2*p)/(7+m+2*p))*(e*x)^(1+m)*(b*x^2+a)^p*hypergeom([-p, 1/2+1 /2*m],[3/2+1/2*m],-b*x^2/a)/e/((1+b*x^2/a)^p)
Time = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.50 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {A c^2 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+\frac {c (B c+2 A d) x^2 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+d x^4 \left (\frac {(2 B c+A d) \operatorname {Hypergeometric2F1}\left (\frac {5+m}{2},-p,\frac {7+m}{2},-\frac {b x^2}{a}\right )}{5+m}+\frac {B d x^2 \operatorname {Hypergeometric2F1}\left (\frac {7+m}{2},-p,\frac {9+m}{2},-\frac {b x^2}{a}\right )}{7+m}\right )\right ) \] Input:
Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^2,x]
Output:
(x*(e*x)^m*(a + b*x^2)^p*((A*c^2*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/ 2, -((b*x^2)/a)])/(1 + m) + (c*(B*c + 2*A*d)*x^2*Hypergeometric2F1[(3 + m) /2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + d*x^4*(((2*B*c + A*d)*Hypergeo metric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + (B*d*x^2*Hype rgeometric2F1[(7 + m)/2, -p, (9 + m)/2, -((b*x^2)/a)])/(7 + m))))/(1 + (b* x^2)/a)^p
Time = 0.81 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {443, 25, 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.
\(\displaystyle \int \left (A+B x^2\right ) \left (c+d x^2\right )^2 (e x)^m \left (a+b x^2\right )^p \, dx\) |
\(\Big \downarrow \) 443 |
\(\displaystyle \frac {\int -(e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right ) \left (c (a B (m+1)-A b (m+2 p+7))-(4 b B c-a B d (m+5)+A b d (m+2 p+7)) x^2\right )dx}{b (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\int (e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right ) \left ((a B d (m+5)-b (4 B c+A d (m+2 p+7))) x^2+c (a B (m+1)-A b (m+2 p+7))\right )dx}{b (m+2 p+7)}\) |
\(\Big \downarrow \) 443 |
\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\int (e x)^m \left (b x^2+a\right )^p \left ((2 b c d (p+2) (a B (m+1)-A b (m+2 p+7))+d (b c-a d) (m+1) (a B (m+5)-A b (m+2 p+7))+2 (b c-a d) (a B d (m+5)-b (4 B c+A d (m+2 p+7)))) x^2+c (2 b c (p+2) (a B (m+1)-A b (m+2 p+7))+(b c-a d) (m+1) (a B (m+5)-A b (m+2 p+7)))\right )dx}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\) |
\(\Big \downarrow \) 363 |
\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\left (\frac {a (m+1) \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))\right ) \int (e x)^m \left (b x^2+a\right )^pdx-\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b e (m+2 p+3)}}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\) |
\(\Big \downarrow \) 279 |
\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (\frac {a (m+1) \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))\right ) \int (e x)^m \left (\frac {b x^2}{a}+1\right )^pdx-\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b e (m+2 p+3)}}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\) |
\(\Big \downarrow \) 278 |
\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (\frac {a (m+1) \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))\right )}{e (m+1)}-\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b e (m+2 p+3)}}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\) |
Input:
Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^2,x]
Output:
(B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*e*(7 + m + 2*p)) - (-(((4*b*B*c - a*B*d*(5 + m) + A*b*d*(7 + m + 2*p))*(e*x)^(1 + m)*(a + b*x ^2)^(1 + p)*(c + d*x^2))/(b*e*(5 + m + 2*p))) + (-(((a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b*e*(3 + m + 2*p))) + ((c*(2*b*c*(2 + p)*(a*B*(1 + m) - A*b*(7 + m + 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(5 + m) - A*b*(7 + m + 2*p))) + (a*(1 + m)*(a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b *d*(A*d*(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p)))))/(b*( 3 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p , (3 + m)/2, -((b*x^2)/a)])/(e*(1 + m)*(1 + (b*x^2)/a)^p))/(b*(5 + m + 2*p )))/(b*(7 + m + 2*p))
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])
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])
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]
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])
\[\int \left (e x \right )^{m} \left (b \,x^{2}+a \right )^{p} \left (x^{2} B +A \right ) \left (x^{2} d +c \right )^{2}d x\]
Input:
int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x)
Output:
int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x)
\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="fricas")
Output:
integral((B*d^2*x^6 + (2*B*c*d + A*d^2)*x^4 + A*c^2 + (B*c^2 + 2*A*c*d)*x^ 2)*(b*x^2 + a)^p*(e*x)^m, x)
Timed out. \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {Timed out} \] Input:
integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c)**2,x)
Output:
Timed out
\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="maxima")
Output:
integrate((B*x^2 + A)*(d*x^2 + c)^2*(b*x^2 + a)^p*(e*x)^m, x)
\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \] Input:
integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="giac")
Output:
integrate((B*x^2 + A)*(d*x^2 + c)^2*(b*x^2 + a)^p*(e*x)^m, x)
Timed out. \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^2 \,d x \] Input:
int((A + B*x^2)*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^2,x)
Output:
int((A + B*x^2)*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^2, x)
\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\text {too large to display} \] Input:
int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x)
Output:
(e**m*( - 4*x**m*(a + b*x**2)**p*a**3*d**2*m*p**2*x - 4*x**m*(a + b*x**2)* *p*a**3*d**2*m*p*x - 12*x**m*(a + b*x**2)**p*a**3*d**2*p**2*x - 12*x**m*(a + b*x**2)**p*a**3*d**2*p*x + 8*x**m*(a + b*x**2)**p*a**2*b*c*d*m*p**2*x + 8*x**m*(a + b*x**2)**p*a**2*b*c*d*m*p*x + 16*x**m*(a + b*x**2)**p*a**2*b* c*d*p**3*x + 72*x**m*(a + b*x**2)**p*a**2*b*c*d*p**2*x + 56*x**m*(a + b*x* *2)**p*a**2*b*c*d*p*x + 4*x**m*(a + b*x**2)**p*a**2*b*d**2*m*p**2*x**3 + 4 *x**m*(a + b*x**2)**p*a**2*b*d**2*m*p*x**3 + 8*x**m*(a + b*x**2)**p*a**2*b *d**2*p**3*x**3 + 12*x**m*(a + b*x**2)**p*a**2*b*d**2*p**2*x**3 + 4*x**m*( a + b*x**2)**p*a**2*b*d**2*p*x**3 + x**m*(a + b*x**2)**p*a*b**2*c**2*m**3* x + 8*x**m*(a + b*x**2)**p*a*b**2*c**2*m**2*p*x + 15*x**m*(a + b*x**2)**p* a*b**2*c**2*m**2*x + 20*x**m*(a + b*x**2)**p*a*b**2*c**2*m*p**2*x + 84*x** m*(a + b*x**2)**p*a*b**2*c**2*m*p*x + 71*x**m*(a + b*x**2)**p*a*b**2*c**2* m*x + 16*x**m*(a + b*x**2)**p*a*b**2*c**2*p**3*x + 108*x**m*(a + b*x**2)** p*a*b**2*c**2*p**2*x + 212*x**m*(a + b*x**2)**p*a*b**2*c**2*p*x + 105*x**m *(a + b*x**2)**p*a*b**2*c**2*x + 2*x**m*(a + b*x**2)**p*a*b**2*c*d*m**3*x* *3 + 16*x**m*(a + b*x**2)**p*a*b**2*c*d*m**2*p*x**3 + 26*x**m*(a + b*x**2) **p*a*b**2*c*d*m**2*x**3 + 40*x**m*(a + b*x**2)**p*a*b**2*c*d*m*p**2*x**3 + 136*x**m*(a + b*x**2)**p*a*b**2*c*d*m*p*x**3 + 94*x**m*(a + b*x**2)**p*a *b**2*c*d*m*x**3 + 32*x**m*(a + b*x**2)**p*a*b**2*c*d*p**3*x**3 + 168*x**m *(a + b*x**2)**p*a*b**2*c*d*p**2*x**3 + 216*x**m*(a + b*x**2)**p*a*b**2...