\(\int (e x)^m (a+b x^2)^p (A+B x^2) (c+d x^2)^2 \, dx\) [45]

Optimal result

Integrand size = 31, antiderivative size = 495 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 e (3+m+2 p) (5+m+2 p) (7+m+2 p)}-\frac {(a B d (5+m)-b (4 B c+A d (7+m+2 p))) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {(b c (3+m+2 p) (2 b c (2+p) (a B (1+m)-A b (7+m+2 p))+(b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p)))-a (1+m) (2 b c d (2+p) (a B (1+m)-A b (7+m+2 p))+d (b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p))+2 (b c-a d) (a B d (5+m)-b (4 B c+A d (7+m+2 p))))) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{b^3 e (1+m) (3+m+2 p) (5+m+2 p) (7+m+2 p)} \]

[Out]

(a^2*B*d^2*(m^2+8*m+15)+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)*(b*x^2+a)^(p+1)/b^3/e/(3+m+2*p)/(5+m+2*p)/(7+m+2*p)-(a*B*d*(5+m)-b*(4*B*c+A*d*(7+m+2*p)))*(e*x)^(1
+m)*(b*x^2+a)^(p+1)*(d*x^2+c)/b^2/e/(5+m+2*p)/(7+m+2*p)+B*(e*x)^(1+m)*(b*x^2+a)^(p+1)*(d*x^2+c)^2/b/e/(7+m+2*p
)-(b*c*(3+m+2*p)*(2*b*c*(2+p)*(a*B*(1+m)-A*b*(7+m+2*p))+(-a*d+b*c)*(1+m)*(a*B*(5+m)-A*b*(7+m+2*p)))-a*(1+m)*(2
*b*c*d*(2+p)*(a*B*(1+m)-A*b*(7+m+2*p))+d*(-a*d+b*c)*(1+m)*(a*B*(5+m)-A*b*(7+m+2*p))+2*(-a*d+b*c)*(a*B*d*(5+m)-
b*(4*B*c+A*d*(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)/b^3/e/(1+m)
/(3+m+2*p)/(5+m+2*p)/(7+m+2*p)/((1+b*x^2/a)^p)

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 464, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {595, 470, 372, 371} \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=-\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (\frac {a \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c \left ((b c-a d) (a B (m+5)-A b (m+2 p+7))+\frac {2 b c (p+2) (a B (m+1)-A b (m+2 p+7))}{m+1}\right )\right )}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}+\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]

[In]

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

[Out]

((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*(2
7 + m^2 + 2*p + 2*m*(6 + p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^3*e*(3 + m + 2*p)*(5 + m + 2*p)*(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^2*e
*(5 + m + 2*p)*(7 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*e*(7 + m + 2*p)) - ((c*
((2*b*c*(2 + p)*(a*B*(1 + m) - A*b*(7 + m + 2*p)))/(1 + m) + (b*c - a*d)*(a*B*(5 + m) - A*b*(7 + m + 2*p))) +
(a*(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)])/(b^2*e*(5 + m + 2*p)*(7 + m + 2*p)*(1 + (b*x^2)/a)^p)

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 372

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

Rule 470

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

Rule 595

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

Rubi steps \begin{align*} \text {integral}& = \frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}+\frac {\int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right ) \left (-c (a B (1+m)-A b (7+m+2 p))+(4 b B c-a B d (5+m)+A b d (7+m+2 p)) x^2\right ) \, dx}{b (7+m+2 p)} \\ & = \frac {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}+\frac {\int (e x)^m \left (a+b x^2\right )^p \left (-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)))-(2 b c d (2+p) (a B (1+m)-A b (7+m+2 p))+d (b c-a d) (1+m) (a B (5+m)-A b (7+m+2 p))-2 (b c-a d) (4 b B c-a B d (5+m)+A b d (7+m+2 p))) x^2\right ) \, dx}{b^2 (5+m+2 p) (7+m+2 p)} \\ & = \frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\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 {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {\left (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)))+\frac {a (1+m) \left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right )}{b (3+m+2 p)}\right ) \int (e x)^m \left (a+b x^2\right )^p \, dx}{b^2 (5+m+2 p) (7+m+2 p)} \\ & = \frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\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 {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {\left (\left (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)))+\frac {a (1+m) \left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right )}{b (3+m+2 p)}\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac {b x^2}{a}\right )^p \, dx}{b^2 (5+m+2 p) (7+m+2 p)} \\ & = \frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\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 {(4 b B c-a B d (5+m)+A b d (7+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^2 e (5+m+2 p) (7+m+2 p)}+\frac {B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b e (7+m+2 p)}-\frac {\left (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)))+\frac {a (1+m) \left (a^2 B d^2 \left (15+8 m+m^2\right )+b^2 c \left (8 B c+A d (7+m+2 p)^2\right )-a b d \left (A d (3+m) (7+m+2 p)+B c \left (27+m^2+2 p+2 m (6+p)\right )\right )\right )}{b (3+m+2 p)}\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1+m}{2},-p;\frac {3+m}{2};-\frac {b x^2}{a}\right )}{b^2 e (1+m) (5+m+2 p) (7+m+2 p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.40 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {A c^2 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+\frac {c (B c+2 A d) x^2 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+d x^4 \left (\frac {(2 B c+A d) \operatorname {Hypergeometric2F1}\left (\frac {5+m}{2},-p,\frac {7+m}{2},-\frac {b x^2}{a}\right )}{5+m}+\frac {B d x^2 \operatorname {Hypergeometric2F1}\left (\frac {7+m}{2},-p,\frac {9+m}{2},-\frac {b x^2}{a}\right )}{7+m}\right )\right ) \]

[In]

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

[Out]

(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)*Hyperg
eometric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + (B*d*x^2*Hypergeometric2F1[(7 + m)/2, -p, (9 +
m)/2, -((b*x^2)/a)])/(7 + m))))/(1 + (b*x^2)/a)^p

Maple [F]

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

[In]

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x)

[Out]

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x)

Fricas [F]

\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \]

[In]

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

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)

Sympy [F(-1)]

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

[In]

integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c)**2,x)

[Out]

Timed out

Maxima [F]

\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \]

[In]

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(d*x^2 + c)^2*(b*x^2 + a)^p*(e*x)^m, x)

Giac [F]

\[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\int { {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m} \,d x } \]

[In]

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^2,x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(d*x^2 + c)^2*(b*x^2 + a)^p*(e*x)^m, x)

Mupad [F(-1)]

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

[In]

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^2,x)

[Out]

int((A + B*x^2)*(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^2, x)