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

Optimal. Leaf size=97 \[ \frac {(e x)^{m+5} (a B d+A b d+b B c)}{e^5 (m+5)}+\frac {(e x)^{m+3} (a A d+a B c+A b c)}{e^3 (m+3)}+\frac {a A c (e x)^{m+1}}{e (m+1)}+\frac {b B d (e x)^{m+7}}{e^7 (m+7)} \]

[Out]

a*A*c*(e*x)^(1+m)/e/(1+m)+(A*a*d+A*b*c+B*a*c)*(e*x)^(3+m)/e^3/(3+m)+(A*b*d+B*a*d+B*b*c)*(e*x)^(5+m)/e^5/(5+m)+
b*B*d*(e*x)^(7+m)/e^7/(7+m)

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Rubi [A]  time = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {570} \[ \frac {(e x)^{m+3} (a A d+a B c+A b c)}{e^3 (m+3)}+\frac {(e x)^{m+5} (a B d+A b d+b B c)}{e^5 (m+5)}+\frac {a A c (e x)^{m+1}}{e (m+1)}+\frac {b B d (e x)^{m+7}}{e^7 (m+7)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*A*c*(e*x)^(1 + m))/(e*(1 + m)) + ((A*b*c + a*B*c + a*A*d)*(e*x)^(3 + m))/(e^3*(3 + m)) + ((b*B*c + A*b*d +
a*B*d)*(e*x)^(5 + m))/(e^5*(5 + m)) + (b*B*d*(e*x)^(7 + m))/(e^7*(7 + m))

Rule 570

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^
(r_.), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a,
b, c, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]

Rubi steps

\begin {align*} \int (e x)^m \left (a+b x^2\right ) \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx &=\int \left (a A c (e x)^m+\frac {(A b c+a B c+a A d) (e x)^{2+m}}{e^2}+\frac {(b B c+A b d+a B d) (e x)^{4+m}}{e^4}+\frac {b B d (e x)^{6+m}}{e^6}\right ) \, dx\\ &=\frac {a A c (e x)^{1+m}}{e (1+m)}+\frac {(A b c+a B c+a A d) (e x)^{3+m}}{e^3 (3+m)}+\frac {(b B c+A b d+a B d) (e x)^{5+m}}{e^5 (5+m)}+\frac {b B d (e x)^{7+m}}{e^7 (7+m)}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 73, normalized size = 0.75 \[ x (e x)^m \left (\frac {x^4 (a B d+A b d+b B c)}{m+5}+\frac {x^2 (a A d+a B c+A b c)}{m+3}+\frac {a A c}{m+1}+\frac {b B d x^6}{m+7}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

x*(e*x)^m*((a*A*c)/(1 + m) + ((A*b*c + a*B*c + a*A*d)*x^2)/(3 + m) + ((b*B*c + A*b*d + a*B*d)*x^4)/(5 + m) + (
b*B*d*x^6)/(7 + m))

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fricas [B]  time = 0.98, size = 235, normalized size = 2.42 \[ \frac {{\left ({\left (B b d m^{3} + 9 \, B b d m^{2} + 23 \, B b d m + 15 \, B b d\right )} x^{7} + {\left ({\left (B b c + {\left (B a + A b\right )} d\right )} m^{3} + 21 \, B b c + 11 \, {\left (B b c + {\left (B a + A b\right )} d\right )} m^{2} + 21 \, {\left (B a + A b\right )} d + 31 \, {\left (B b c + {\left (B a + A b\right )} d\right )} m\right )} x^{5} + {\left ({\left (A a d + {\left (B a + A b\right )} c\right )} m^{3} + 35 \, A a d + 13 \, {\left (A a d + {\left (B a + A b\right )} c\right )} m^{2} + 35 \, {\left (B a + A b\right )} c + 47 \, {\left (A a d + {\left (B a + A b\right )} c\right )} m\right )} x^{3} + {\left (A a c m^{3} + 15 \, A a c m^{2} + 71 \, A a c m + 105 \, A a c\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((B*b*d*m^3 + 9*B*b*d*m^2 + 23*B*b*d*m + 15*B*b*d)*x^7 + ((B*b*c + (B*a + A*b)*d)*m^3 + 21*B*b*c + 11*(B*b*c +
(B*a + A*b)*d)*m^2 + 21*(B*a + A*b)*d + 31*(B*b*c + (B*a + A*b)*d)*m)*x^5 + ((A*a*d + (B*a + A*b)*c)*m^3 + 35
*A*a*d + 13*(A*a*d + (B*a + A*b)*c)*m^2 + 35*(B*a + A*b)*c + 47*(A*a*d + (B*a + A*b)*c)*m)*x^3 + (A*a*c*m^3 +
15*A*a*c*m^2 + 71*A*a*c*m + 105*A*a*c)*x)*(e*x)^m/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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giac [B]  time = 0.44, size = 478, normalized size = 4.93 \[ \frac {B b d m^{3} x^{7} x^{m} e^{m} + 9 \, B b d m^{2} x^{7} x^{m} e^{m} + B b c m^{3} x^{5} x^{m} e^{m} + B a d m^{3} x^{5} x^{m} e^{m} + A b d m^{3} x^{5} x^{m} e^{m} + 23 \, B b d m x^{7} x^{m} e^{m} + 11 \, B b c m^{2} x^{5} x^{m} e^{m} + 11 \, B a d m^{2} x^{5} x^{m} e^{m} + 11 \, A b d m^{2} x^{5} x^{m} e^{m} + 15 \, B b d x^{7} x^{m} e^{m} + B a c m^{3} x^{3} x^{m} e^{m} + A b c m^{3} x^{3} x^{m} e^{m} + A a d m^{3} x^{3} x^{m} e^{m} + 31 \, B b c m x^{5} x^{m} e^{m} + 31 \, B a d m x^{5} x^{m} e^{m} + 31 \, A b d m x^{5} x^{m} e^{m} + 13 \, B a c m^{2} x^{3} x^{m} e^{m} + 13 \, A b c m^{2} x^{3} x^{m} e^{m} + 13 \, A a d m^{2} x^{3} x^{m} e^{m} + 21 \, B b c x^{5} x^{m} e^{m} + 21 \, B a d x^{5} x^{m} e^{m} + 21 \, A b d x^{5} x^{m} e^{m} + A a c m^{3} x x^{m} e^{m} + 47 \, B a c m x^{3} x^{m} e^{m} + 47 \, A b c m x^{3} x^{m} e^{m} + 47 \, A a d m x^{3} x^{m} e^{m} + 15 \, A a c m^{2} x x^{m} e^{m} + 35 \, B a c x^{3} x^{m} e^{m} + 35 \, A b c x^{3} x^{m} e^{m} + 35 \, A a d x^{3} x^{m} e^{m} + 71 \, A a c m x x^{m} e^{m} + 105 \, A a c x x^{m} e^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d*m^3*x^7*x^m*e^m + 9*B*b*d*m^2*x^7*x^m*e^m + B*b*c*m^3*x^5*x^m*e^m + B*a*d*m^3*x^5*x^m*e^m + A*b*d*m^3*x
^5*x^m*e^m + 23*B*b*d*m*x^7*x^m*e^m + 11*B*b*c*m^2*x^5*x^m*e^m + 11*B*a*d*m^2*x^5*x^m*e^m + 11*A*b*d*m^2*x^5*x
^m*e^m + 15*B*b*d*x^7*x^m*e^m + B*a*c*m^3*x^3*x^m*e^m + A*b*c*m^3*x^3*x^m*e^m + A*a*d*m^3*x^3*x^m*e^m + 31*B*b
*c*m*x^5*x^m*e^m + 31*B*a*d*m*x^5*x^m*e^m + 31*A*b*d*m*x^5*x^m*e^m + 13*B*a*c*m^2*x^3*x^m*e^m + 13*A*b*c*m^2*x
^3*x^m*e^m + 13*A*a*d*m^2*x^3*x^m*e^m + 21*B*b*c*x^5*x^m*e^m + 21*B*a*d*x^5*x^m*e^m + 21*A*b*d*x^5*x^m*e^m + A
*a*c*m^3*x*x^m*e^m + 47*B*a*c*m*x^3*x^m*e^m + 47*A*b*c*m*x^3*x^m*e^m + 47*A*a*d*m*x^3*x^m*e^m + 15*A*a*c*m^2*x
*x^m*e^m + 35*B*a*c*x^3*x^m*e^m + 35*A*b*c*x^3*x^m*e^m + 35*A*a*d*x^3*x^m*e^m + 71*A*a*c*m*x*x^m*e^m + 105*A*a
*c*x*x^m*e^m)/(m^4 + 16*m^3 + 86*m^2 + 176*m + 105)

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maple [B]  time = 0.00, size = 321, normalized size = 3.31 \[ \frac {\left (B b d \,m^{3} x^{6}+9 B b d \,m^{2} x^{6}+A b d \,m^{3} x^{4}+B a d \,m^{3} x^{4}+B b c \,m^{3} x^{4}+23 B b d m \,x^{6}+11 A b d \,m^{2} x^{4}+11 B a d \,m^{2} x^{4}+11 B b c \,m^{2} x^{4}+15 B b d \,x^{6}+A a d \,m^{3} x^{2}+A b c \,m^{3} x^{2}+31 A b d m \,x^{4}+B a c \,m^{3} x^{2}+31 B a d m \,x^{4}+31 B b c m \,x^{4}+13 A a d \,m^{2} x^{2}+13 A b c \,m^{2} x^{2}+21 A b d \,x^{4}+13 B a c \,m^{2} x^{2}+21 B a d \,x^{4}+21 B b c \,x^{4}+A a c \,m^{3}+47 A a d m \,x^{2}+47 A b c m \,x^{2}+47 B a c m \,x^{2}+15 A a c \,m^{2}+35 A a d \,x^{2}+35 A b c \,x^{2}+35 B a c \,x^{2}+71 A a c m +105 A a c \right ) x \left (e x \right )^{m}}{\left (m +7\right ) \left (m +5\right ) \left (m +3\right ) \left (m +1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(B*b*d*m^3*x^6+9*B*b*d*m^2*x^6+A*b*d*m^3*x^4+B*a*d*m^3*x^4+B*b*c*m^3*x^4+23*B*b*d*m*x^6+11*A*b*d*m^2*x^4+11*
B*a*d*m^2*x^4+11*B*b*c*m^2*x^4+15*B*b*d*x^6+A*a*d*m^3*x^2+A*b*c*m^3*x^2+31*A*b*d*m*x^4+B*a*c*m^3*x^2+31*B*a*d*
m*x^4+31*B*b*c*m*x^4+13*A*a*d*m^2*x^2+13*A*b*c*m^2*x^2+21*A*b*d*x^4+13*B*a*c*m^2*x^2+21*B*a*d*x^4+21*B*b*c*x^4
+A*a*c*m^3+47*A*a*d*m*x^2+47*A*b*c*m*x^2+47*B*a*c*m*x^2+15*A*a*c*m^2+35*A*a*d*x^2+35*A*b*c*x^2+35*B*a*c*x^2+71
*A*a*c*m+105*A*a*c)*(e*x)^m/(m+7)/(m+5)/(m+3)/(m+1)

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maxima [A]  time = 1.63, size = 146, normalized size = 1.51 \[ \frac {B b d e^{m} x^{7} x^{m}}{m + 7} + \frac {B b c e^{m} x^{5} x^{m}}{m + 5} + \frac {B a d e^{m} x^{5} x^{m}}{m + 5} + \frac {A b d e^{m} x^{5} x^{m}}{m + 5} + \frac {B a c e^{m} x^{3} x^{m}}{m + 3} + \frac {A b c e^{m} x^{3} x^{m}}{m + 3} + \frac {A a d e^{m} x^{3} x^{m}}{m + 3} + \frac {\left (e x\right )^{m + 1} A a c}{e {\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

B*b*d*e^m*x^7*x^m/(m + 7) + B*b*c*e^m*x^5*x^m/(m + 5) + B*a*d*e^m*x^5*x^m/(m + 5) + A*b*d*e^m*x^5*x^m/(m + 5)
+ B*a*c*e^m*x^3*x^m/(m + 3) + A*b*c*e^m*x^3*x^m/(m + 3) + A*a*d*e^m*x^3*x^m/(m + 3) + (e*x)^(m + 1)*A*a*c/(e*(
m + 1))

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mupad [B]  time = 1.03, size = 185, normalized size = 1.91 \[ {\left (e\,x\right )}^m\,\left (\frac {x^3\,\left (A\,a\,d+A\,b\,c+B\,a\,c\right )\,\left (m^3+13\,m^2+47\,m+35\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {x^5\,\left (A\,b\,d+B\,a\,d+B\,b\,c\right )\,\left (m^3+11\,m^2+31\,m+21\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {B\,b\,d\,x^7\,\left (m^3+9\,m^2+23\,m+15\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}+\frac {A\,a\,c\,x\,\left (m^3+15\,m^2+71\,m+105\right )}{m^4+16\,m^3+86\,m^2+176\,m+105}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e*x)^m*((x^3*(A*a*d + A*b*c + B*a*c)*(47*m + 13*m^2 + m^3 + 35))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (x^5
*(A*b*d + B*a*d + B*b*c)*(31*m + 11*m^2 + m^3 + 21))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (B*b*d*x^7*(23*m
+ 9*m^2 + m^3 + 15))/(176*m + 86*m^2 + 16*m^3 + m^4 + 105) + (A*a*c*x*(71*m + 15*m^2 + m^3 + 105))/(176*m + 86
*m^2 + 16*m^3 + m^4 + 105))

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sympy [A]  time = 2.60, size = 1515, normalized size = 15.62 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-A*a*c/(6*x**6) - A*a*d/(4*x**4) - A*b*c/(4*x**4) - A*b*d/(2*x**2) - B*a*c/(4*x**4) - B*a*d/(2*x**
2) - B*b*c/(2*x**2) + B*b*d*log(x))/e**7, Eq(m, -7)), ((-A*a*c/(4*x**4) - A*a*d/(2*x**2) - A*b*c/(2*x**2) + A*
b*d*log(x) - B*a*c/(2*x**2) + B*a*d*log(x) + B*b*c*log(x) + B*b*d*x**2/2)/e**5, Eq(m, -5)), ((-A*a*c/(2*x**2)
+ A*a*d*log(x) + A*b*c*log(x) + A*b*d*x**2/2 + B*a*c*log(x) + B*a*d*x**2/2 + B*b*c*x**2/2 + B*b*d*x**4/4)/e**3
, Eq(m, -3)), ((A*a*c*log(x) + A*a*d*x**2/2 + A*b*c*x**2/2 + A*b*d*x**4/4 + B*a*c*x**2/2 + B*a*d*x**4/4 + B*b*
c*x**4/4 + B*b*d*x**6/6)/e, Eq(m, -1)), (A*a*c*e**m*m**3*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*
A*a*c*e**m*m**2*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 71*A*a*c*e**m*m*x*x**m/(m**4 + 16*m**3 + 86*
m**2 + 176*m + 105) + 105*A*a*c*e**m*x*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + A*a*d*e**m*m**3*x**3*x*
*m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*A*a*d*e**m*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m +
105) + 47*A*a*d*e**m*m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 35*A*a*d*e**m*x**3*x**m/(m**4 + 1
6*m**3 + 86*m**2 + 176*m + 105) + A*b*c*e**m*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*A*b*
c*e**m*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*A*b*c*e**m*m*x**3*x**m/(m**4 + 16*m**3 + 8
6*m**2 + 176*m + 105) + 35*A*b*c*e**m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + A*b*d*e**m*m**3*x**
5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*A*b*d*e**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176
*m + 105) + 31*A*b*d*e**m*m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 21*A*b*d*e**m*x**5*x**m/(m**4
+ 16*m**3 + 86*m**2 + 176*m + 105) + B*a*c*e**m*m**3*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 13*
B*a*c*e**m*m**2*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 47*B*a*c*e**m*m*x**3*x**m/(m**4 + 16*m**3
+ 86*m**2 + 176*m + 105) + 35*B*a*c*e**m*x**3*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*a*d*e**m*m**3
*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 11*B*a*d*e**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 +
176*m + 105) + 31*B*a*d*e**m*m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 21*B*a*d*e**m*x**5*x**m/(
m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*b*c*e**m*m**3*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) +
11*B*b*c*e**m*m**2*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 31*B*b*c*e**m*m*x**5*x**m/(m**4 + 16*
m**3 + 86*m**2 + 176*m + 105) + 21*B*b*c*e**m*x**5*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + B*b*d*e**m*
m**3*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 9*B*b*d*e**m*m**2*x**7*x**m/(m**4 + 16*m**3 + 86*m**
2 + 176*m + 105) + 23*B*b*d*e**m*m*x**7*x**m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105) + 15*B*b*d*e**m*x**7*x**
m/(m**4 + 16*m**3 + 86*m**2 + 176*m + 105), True))

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