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

Optimal. Leaf size=495 \[ \frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right ) (b c (m+2 p+3) ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))-a (m+1) (2 b c d (p+2) (a B (m+1)-A b (m+2 p+7))+d (m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 (b c-a d) (a B d (m+5)-b (A d (m+2 p+7)+4 B c))))}{b^3 e (m+1) (m+2 p+3) (m+2 p+5) (m+2 p+7)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+5)-b (A d (m+2 p+7)+4 B c))}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]

[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)^(1+p)/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)^(1+p)*(d*x^2+c)/b^2/e/(5+m+2*p)/(7+m+2*p)+B*(e*x)^(1+m)*(b*x^2+a)^(1+p)*(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)

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Rubi [A]  time = 0.75, antiderivative size = 464, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {581, 459, 365, 364} \[ -\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right ) \left (\frac {a \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c \left ((b c-a d) (a B (m+5)-A b (m+2 p+7))+\frac {2 b c (p+2) (a B (m+1)-A b (m+2 p+7))}{m+1}\right )\right )}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b^3 e (m+2 p+3) (m+2 p+5) (m+2 p+7)}+\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b^2 e (m+2 p+5) (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)} \]

Antiderivative was successfully verified.

[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 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 365

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 459

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 581

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*} \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 {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*}

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Mathematica [A]  time = 0.27, size = 198, normalized size = 0.40 \[ x (e x)^m \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (\frac {c x^2 (2 A d+B c) \, _2F_1\left (\frac {m+3}{2},-p;\frac {m+5}{2};-\frac {b x^2}{a}\right )}{m+3}+d x^4 \left (\frac {(A d+2 B c) \, _2F_1\left (\frac {m+5}{2},-p;\frac {m+7}{2};-\frac {b x^2}{a}\right )}{m+5}+\frac {B d x^2 \, _2F_1\left (\frac {m+7}{2},-p;\frac {m+9}{2};-\frac {b x^2}{a}\right )}{m+7}\right )+\frac {A c^2 \, _2F_1\left (\frac {m+1}{2},-p;\frac {m+3}{2};-\frac {b x^2}{a}\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[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

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fricas [F]  time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B d^{2} x^{6} + {\left (2 \, B c d + A d^{2}\right )} x^{4} + A c^{2} + {\left (B c^{2} + 2 \, A c d\right )} x^{2}\right )} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [F]  time = 0.07, size = 0, normalized size = 0.00 \[ \int \left (B \,x^{2}+A \right ) \left (d \,x^{2}+c \right )^{2} \left (e x \right )^{m} \left (b \,x^{2}+a \right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B x^{2} + A\right )} {\left (d x^{2} + c\right )}^{2} {\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (B\,x^2+A\right )\,{\left (e\,x\right )}^m\,{\left (b\,x^2+a\right )}^p\,{\left (d\,x^2+c\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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