3.9 \(\int (e x)^m \left (a+b x^n\right )^2 \left (A+B x^n\right ) \left (c+d x^n\right )^2 \, dx\)
Optimal. Leaf size=237 \[ \frac{x^{2 n+1} (e x)^m \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac{x^{3 n+1} (e x)^m \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac{a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{a c x^{n+1} (e x)^m (2 A (a d+b c)+a B c)}{m+n+1}+\frac{b d x^{4 n+1} (e x)^m (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac{b^2 B d^2 x^{5 n+1} (e x)^m}{m+5 n+1} \]
[Out]
(a*c*(a*B*c + 2*A*(b*c + a*d))*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((2*a*B*c*(b*c +
a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) +
((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^(1 + 3*n)*(e*x)^m)/
(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4
*n) + (b^2*B*d^2*x^(1 + 5*n)*(e*x)^m)/(1 + m + 5*n) + (a^2*A*c^2*(e*x)^(1 + m))/
(e*(1 + m))
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Rubi [A] time = 0.764609, antiderivative size = 237, normalized size of antiderivative = 1.,
number of steps used = 12, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097
\[ \frac{x^{2 n+1} (e x)^m \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac{x^{3 n+1} (e x)^m \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac{a^2 A c^2 (e x)^{m+1}}{e (m+1)}+\frac{a c x^{n+1} (e x)^m (2 A (a d+b c)+a B c)}{m+n+1}+\frac{b d x^{4 n+1} (e x)^m (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac{b^2 B d^2 x^{5 n+1} (e x)^m}{m+5 n+1} \]
Antiderivative was successfully verified.
[In] Int[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
(a*c*(a*B*c + 2*A*(b*c + a*d))*x^(1 + n)*(e*x)^m)/(1 + m + n) + ((2*a*B*c*(b*c +
a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^(1 + 2*n)*(e*x)^m)/(1 + m + 2*n) +
((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^(1 + 3*n)*(e*x)^m)/
(1 + m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(1 + 4*n)*(e*x)^m)/(1 + m + 4
*n) + (b^2*B*d^2*x^(1 + 5*n)*(e*x)^m)/(1 + m + 5*n) + (a^2*A*c^2*(e*x)^(1 + m))/
(e*(1 + m))
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Rubi in Sympy [A] time = 117.925, size = 275, normalized size = 1.16 \[ \frac{A a^{2} c^{2} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B b^{2} d^{2} x^{5 n} \left (e x\right )^{- 5 n} \left (e x\right )^{m + 5 n + 1}}{e \left (m + 5 n + 1\right )} + \frac{a c x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (2 A \left (a d + b c\right ) + B a c\right )}{e \left (m + n + 1\right )} + \frac{b d x^{4 n} \left (e x\right )^{- 4 n} \left (e x\right )^{m + 4 n + 1} \left (A b d + 2 B \left (a d + b c\right )\right )}{e \left (m + 4 n + 1\right )} + \frac{x^{- m} x^{m + 2 n + 1} \left (e x\right )^{m} \left (A a^{2} d^{2} + c \left (A b^{2} c + 2 a \left (B b c + d \left (2 A b + B a\right )\right )\right )\right )}{m + 2 n + 1} + \frac{x^{3 n} \left (e x\right )^{- 3 n} \left (e x\right )^{m + 3 n + 1} \left (B a^{2} d^{2} + b \left (B b c^{2} + 2 d \left (A a d + c \left (A b + 2 B a\right )\right )\right )\right )}{e \left (m + 3 n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)*(c+d*x**n)**2,x)
[Out]
A*a**2*c**2*(e*x)**(m + 1)/(e*(m + 1)) + B*b**2*d**2*x**(5*n)*(e*x)**(-5*n)*(e*x
)**(m + 5*n + 1)/(e*(m + 5*n + 1)) + a*c*x**n*(e*x)**(-n)*(e*x)**(m + n + 1)*(2*
A*(a*d + b*c) + B*a*c)/(e*(m + n + 1)) + b*d*x**(4*n)*(e*x)**(-4*n)*(e*x)**(m +
4*n + 1)*(A*b*d + 2*B*(a*d + b*c))/(e*(m + 4*n + 1)) + x**(-m)*x**(m + 2*n + 1)*
(e*x)**m*(A*a**2*d**2 + c*(A*b**2*c + 2*a*(B*b*c + d*(2*A*b + B*a))))/(m + 2*n +
1) + x**(3*n)*(e*x)**(-3*n)*(e*x)**(m + 3*n + 1)*(B*a**2*d**2 + b*(B*b*c**2 + 2
*d*(A*a*d + c*(A*b + 2*B*a))))/(e*(m + 3*n + 1))
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Mathematica [A] time = 2.1008, size = 199, normalized size = 0.84 \[ x (e x)^m \left (\frac{x^{2 n} \left (A \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a B c (a d+b c)\right )}{m+2 n+1}+\frac{x^{3 n} \left (a^2 B d^2+2 a b d (A d+2 B c)+b^2 c (2 A d+B c)\right )}{m+3 n+1}+\frac{a^2 A c^2}{m+1}+\frac{b d x^{4 n} (2 a B d+A b d+2 b B c)}{m+4 n+1}+\frac{a c x^n (2 A (a d+b c)+a B c)}{m+n+1}+\frac{b^2 B d^2 x^{5 n}}{m+5 n+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(e*x)^m*(a + b*x^n)^2*(A + B*x^n)*(c + d*x^n)^2,x]
[Out]
x*(e*x)^m*((a^2*A*c^2)/(1 + m) + (a*c*(a*B*c + 2*A*(b*c + a*d))*x^n)/(1 + m + n)
+ ((2*a*B*c*(b*c + a*d) + A*(b^2*c^2 + 4*a*b*c*d + a^2*d^2))*x^(2*n))/(1 + m +
2*n) + ((a^2*B*d^2 + 2*a*b*d*(2*B*c + A*d) + b^2*c*(B*c + 2*A*d))*x^(3*n))/(1 +
m + 3*n) + (b*d*(2*b*B*c + A*b*d + 2*a*B*d)*x^(4*n))/(1 + m + 4*n) + (b^2*B*d^2*
x^(5*n))/(1 + m + 5*n))
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Maple [C] time = 0.158, size = 5908, normalized size = 24.9 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^m*(a+b*x^n)^2*(A+B*x^n)*(c+d*x^n)^2,x)
[Out]
result too large to display
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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^n + A)*(b*x^n + a)^2*(d*x^n + c)^2*(e*x)^m,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.274784, size = 4745, normalized size = 20.02 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(b*x^n + a)^2*(d*x^n + c)^2*(e*x)^m,x, algorithm="fricas")
[Out]
((B*b^2*d^2*m^5 + 5*B*b^2*d^2*m^4 + 10*B*b^2*d^2*m^3 + 10*B*b^2*d^2*m^2 + 5*B*b^
2*d^2*m + B*b^2*d^2 + 24*(B*b^2*d^2*m + B*b^2*d^2)*n^4 + 50*(B*b^2*d^2*m^2 + 2*B
*b^2*d^2*m + B*b^2*d^2)*n^3 + 35*(B*b^2*d^2*m^3 + 3*B*b^2*d^2*m^2 + 3*B*b^2*d^2*
m + B*b^2*d^2)*n^2 + 10*(B*b^2*d^2*m^4 + 4*B*b^2*d^2*m^3 + 6*B*b^2*d^2*m^2 + 4*B
*b^2*d^2*m + B*b^2*d^2)*n)*x*x^(5*n)*e^(m*log(e) + m*log(x)) + ((2*B*b^2*c*d + (
2*B*a*b + A*b^2)*d^2)*m^5 + 2*B*b^2*c*d + 5*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2
)*m^4 + 30*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2 + (2*B*b^2*c*d + (2*B*a*b + A*b^
2)*d^2)*m)*n^4 + 10*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + 61*(2*B*b^2*c*d
+ (2*B*a*b + A*b^2)*d^2 + (2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2 + 2*(2*B*b^2
*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n^3 + (2*B*a*b + A*b^2)*d^2 + 10*(2*B*b^2*c*d +
(2*B*a*b + A*b^2)*d^2)*m^2 + 41*(2*B*b^2*c*d + (2*B*b^2*c*d + (2*B*a*b + A*b^2)
*d^2)*m^3 + (2*B*a*b + A*b^2)*d^2 + 3*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^2
+ 3*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n^2 + 5*(2*B*b^2*c*d + (2*B*a*b + A
*b^2)*d^2)*m + 11*(2*B*b^2*c*d + (2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^4 + 4*(
2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m^3 + (2*B*a*b + A*b^2)*d^2 + 6*(2*B*b^2*c*
d + (2*B*a*b + A*b^2)*d^2)*m^2 + 4*(2*B*b^2*c*d + (2*B*a*b + A*b^2)*d^2)*m)*n)*x
*x^(4*n)*e^(m*log(e) + m*log(x)) + ((B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^
2 + 2*A*a*b)*d^2)*m^5 + B*b^2*c^2 + 5*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*
a^2 + 2*A*a*b)*d^2)*m^4 + 40*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A
*a*b)*d^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^4
+ 10*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^3 + 78*(B*
b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + (B*b^2*c^2 + 2*(2*B*
a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 2*(B*b^2*c^2 + 2*(2*B*a*b + A*b^
2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^3 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*
a*b)*d^2 + 10*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2
+ 49*(B*b^2*c^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*
m^3 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + 3*(B*b^2*c^2 + 2*(2*B*a*
b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 3*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)
*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n^2 + 5*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d +
(B*a^2 + 2*A*a*b)*d^2)*m + 12*(B*b^2*c^2 + (B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d
+ (B*a^2 + 2*A*a*b)*d^2)*m^4 + 4*(B*b^2*c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 +
2*A*a*b)*d^2)*m^3 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2 + 6*(B*b^2*
c^2 + 2*(2*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m^2 + 4*(B*b^2*c^2 + 2*(2
*B*a*b + A*b^2)*c*d + (B*a^2 + 2*A*a*b)*d^2)*m)*n)*x*x^(3*n)*e^(m*log(e) + m*log
(x)) + ((A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^5 + A*a^
2*d^2 + 5*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^4 + 60
*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + (A*a^2*d^2 + (2*
B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m)*n^4 + 10*(A*a^2*d^2 + (2*B*a*b
+ A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + 107*(A*a^2*d^2 + (2*B*a*b + A*b^2)
*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 +
2*A*a*b)*c*d)*m^2 + 2*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*
c*d)*m)*n^3 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + 10*(A*a^2*d^2 +
(2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 59*(A*a^2*d^2 + (A*a^2*d^
2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + (2*B*a*b + A*b^2)*c^2
+ 2*(B*a^2 + 2*A*a*b)*c*d + 3*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2
*A*a*b)*c*d)*m^2 + 3*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*
d)*m)*n^2 + 5*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m +
13*(A*a^2*d^2 + (A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^
4 + 4*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d)*m^3 + (2*B*a
*b + A*b^2)*c^2 + 2*(B*a^2 + 2*A*a*b)*c*d + 6*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2
+ 2*(B*a^2 + 2*A*a*b)*c*d)*m^2 + 4*(A*a^2*d^2 + (2*B*a*b + A*b^2)*c^2 + 2*(B*a^
2 + 2*A*a*b)*c*d)*m)*n)*x*x^(2*n)*e^(m*log(e) + m*log(x)) + ((2*A*a^2*c*d + (B*a
^2 + 2*A*a*b)*c^2)*m^5 + 2*A*a^2*c*d + 5*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m
^4 + 120*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2 + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)
*c^2)*m)*n^4 + 10*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + 154*(2*A*a^2*c*d +
(B*a^2 + 2*A*a*b)*c^2 + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 + 2*(2*A*a^2*
c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n^3 + (B*a^2 + 2*A*a*b)*c^2 + 10*(2*A*a^2*c*d +
(B*a^2 + 2*A*a*b)*c^2)*m^2 + 71*(2*A*a^2*c*d + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*
c^2)*m^3 + (B*a^2 + 2*A*a*b)*c^2 + 3*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^2 +
3*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n^2 + 5*(2*A*a^2*c*d + (B*a^2 + 2*A*
a*b)*c^2)*m + 14*(2*A*a^2*c*d + (2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^4 + 4*(2
*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m^3 + (B*a^2 + 2*A*a*b)*c^2 + 6*(2*A*a^2*c*d
+ (B*a^2 + 2*A*a*b)*c^2)*m^2 + 4*(2*A*a^2*c*d + (B*a^2 + 2*A*a*b)*c^2)*m)*n)*x*
x^n*e^(m*log(e) + m*log(x)) + (A*a^2*c^2*m^5 + 120*A*a^2*c^2*n^5 + 5*A*a^2*c^2*m
^4 + 10*A*a^2*c^2*m^3 + 10*A*a^2*c^2*m^2 + 5*A*a^2*c^2*m + A*a^2*c^2 + 274*(A*a^
2*c^2*m + A*a^2*c^2)*n^4 + 225*(A*a^2*c^2*m^2 + 2*A*a^2*c^2*m + A*a^2*c^2)*n^3 +
85*(A*a^2*c^2*m^3 + 3*A*a^2*c^2*m^2 + 3*A*a^2*c^2*m + A*a^2*c^2)*n^2 + 15*(A*a^
2*c^2*m^4 + 4*A*a^2*c^2*m^3 + 6*A*a^2*c^2*m^2 + 4*A*a^2*c^2*m + A*a^2*c^2)*n)*x*
e^(m*log(e) + m*log(x)))/(m^6 + 120*(m + 1)*n^5 + 6*m^5 + 274*(m^2 + 2*m + 1)*n^
4 + 15*m^4 + 225*(m^3 + 3*m^2 + 3*m + 1)*n^3 + 20*m^3 + 85*(m^4 + 4*m^3 + 6*m^2
+ 4*m + 1)*n^2 + 15*m^2 + 15*(m^5 + 5*m^4 + 10*m^3 + 10*m^2 + 5*m + 1)*n + 6*m +
1)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**m*(a+b*x**n)**2*(A+B*x**n)*(c+d*x**n)**2,x)
[Out]
Timed out
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GIAC/XCAS [A] time = 0.239614, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^n + A)*(b*x^n + a)^2*(d*x^n + c)^2*(e*x)^m,x, algorithm="giac")
[Out]
Done