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

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*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))

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Rubi [A]  time = 0.183558, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037 \[ \frac{(e x)^{m+5} (a B d+A b d+b B c)}{e^5 (m+5)}+\frac{(e x)^{m+3} (a A d+a B c+A b c)}{e^3 (m+3)}+\frac{a A c (e x)^{m+1}}{e (m+1)}+\frac{b B d (e x)^{m+7}}{e^7 (m+7)} \]

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))

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Rubi in Sympy [A]  time = 32.0537, size = 92, normalized size = 0.95 \[ \frac{A a c \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B b d \left (e x\right )^{m + 7}}{e^{7} \left (m + 7\right )} + \frac{\left (e x\right )^{m + 3} \left (A a d + A b c + B a c\right )}{e^{3} \left (m + 3\right )} + \frac{\left (e x\right )^{m + 5} \left (A b d + B a d + B b c\right )}{e^{5} \left (m + 5\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.107822, size = 73, normalized size = 0.75 \[ (e x)^m \left (\frac{x^5 (a B d+A b d+b B c)}{m+5}+\frac{x^3 (a A d+a B c+A b c)}{m+3}+\frac{a A c x}{m+1}+\frac{b B d x^7}{m+7}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 321, normalized size = 3.3 \[{\frac{ \left ( Bbd{m}^{3}{x}^{6}+9\,Bbd{m}^{2}{x}^{6}+Abd{m}^{3}{x}^{4}+Bad{m}^{3}{x}^{4}+Bbc{m}^{3}{x}^{4}+23\,Bbdm{x}^{6}+11\,Abd{m}^{2}{x}^{4}+11\,Bad{m}^{2}{x}^{4}+11\,Bbc{m}^{2}{x}^{4}+15\,Bbd{x}^{6}+Aad{m}^{3}{x}^{2}+Abc{m}^{3}{x}^{2}+31\,Abdm{x}^{4}+Bac{m}^{3}{x}^{2}+31\,Badm{x}^{4}+31\,Bbcm{x}^{4}+13\,Aad{m}^{2}{x}^{2}+13\,Abc{m}^{2}{x}^{2}+21\,Abd{x}^{4}+13\,Bac{m}^{2}{x}^{2}+21\,Bad{x}^{4}+21\,Bbc{x}^{4}+Aac{m}^{3}+47\,Aadm{x}^{2}+47\,Abcm{x}^{2}+47\,Bacm{x}^{2}+15\,Aac{m}^{2}+35\,Aad{x}^{2}+35\,Abc{x}^{2}+35\,Bac{x}^{2}+71\,Aacm+105\,Aac \right ) x \left ( ex \right ) ^{m}}{ \left ( 7+m \right ) \left ( 5+m \right ) \left ( 3+m \right ) \left ( 1+m \right ) }} \]

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+3
5*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/(7+m)/(5+m)/
(3+m)/(1+m)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25366, size = 317, normalized size = 3.27 \[ \frac{{\left ({\left (B b d m^{3} + 9 \, B b d m^{2} + 23 \, B b d m + 15 \, B b d\right )} x^{7} +{\left ({\left (B b c +{\left (B a + A b\right )} d\right )} m^{3} + 21 \, B b c + 11 \,{\left (B b c +{\left (B a + A b\right )} d\right )} m^{2} + 21 \,{\left (B a + A b\right )} d + 31 \,{\left (B b c +{\left (B a + A b\right )} d\right )} m\right )} x^{5} +{\left ({\left (A a d +{\left (B a + A b\right )} c\right )} m^{3} + 35 \, A a d + 13 \,{\left (A a d +{\left (B a + A b\right )} c\right )} m^{2} + 35 \,{\left (B a + A b\right )} c + 47 \,{\left (A a d +{\left (B a + A b\right )} c\right )} m\right )} x^{3} +{\left (A a c m^{3} + 15 \, A a c m^{2} + 71 \, A a c m + 105 \, A a c\right )} x\right )} \left (e x\right )^{m}}{m^{4} + 16 \, m^{3} + 86 \, m^{2} + 176 \, m + 105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*(d*x^2 + c)*(e*x)^m,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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Sympy [A]  time = 3.045, 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 + 1
05) + 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 + 16*m**3 + 86*m**2 + 1
76*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 + 86*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 + 1
6*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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GIAC/XCAS [A]  time = 0.223712, size = 732, normalized size = 7.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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