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

Optimal. Leaf size=160 \[ \frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a x^{n+1} (e x)^m (a A d+a B c+2 A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n+1} (e x)^m (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{b^2 B d x^{4 n+1} (e x)^m}{m+4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.423623, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{a^2 A c (e x)^{m+1}}{e (m+1)}+\frac{a x^{n+1} (e x)^m (a A d+a B c+2 A b c)}{m+n+1}+\frac{x^{2 n+1} (e x)^m (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n+1} (e x)^m (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{b^2 B d x^{4 n+1} (e x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 70.8568, size = 194, normalized size = 1.21 \[ \frac{A a^{2} c \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B b^{2} d x^{4 n} \left (e x\right )^{- 4 n} \left (e x\right )^{m + 4 n + 1}}{e \left (m + 4 n + 1\right )} + \frac{a x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (A a d + 2 A b c + B a c\right )}{e \left (m + n + 1\right )} + \frac{b x^{3 n} \left (e x\right )^{- 3 n} \left (e x\right )^{m + 3 n + 1} \left (A b d + 2 B a d + B b c\right )}{e \left (m + 3 n + 1\right )} + \frac{x^{- m} x^{m + 2 n + 1} \left (e x\right )^{m} \left (A b^{2} c + a \left (B a d + 2 b \left (A d + B c\right )\right )\right )}{m + 2 n + 1} \]

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),x)

[Out]

A*a**2*c*(e*x)**(m + 1)/(e*(m + 1)) + B*b**2*d*x**(4*n)*(e*x)**(-4*n)*(e*x)**(m
+ 4*n + 1)/(e*(m + 4*n + 1)) + a*x**n*(e*x)**(-n)*(e*x)**(m + n + 1)*(A*a*d + 2*
A*b*c + B*a*c)/(e*(m + n + 1)) + b*x**(3*n)*(e*x)**(-3*n)*(e*x)**(m + 3*n + 1)*(
A*b*d + 2*B*a*d + B*b*c)/(e*(m + 3*n + 1)) + x**(-m)*x**(m + 2*n + 1)*(e*x)**m*(
A*b**2*c + a*(B*a*d + 2*b*(A*d + B*c)))/(m + 2*n + 1)

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Mathematica [A]  time = 1.03958, size = 129, normalized size = 0.81 \[ x (e x)^m \left (\frac{a^2 A c}{m+1}+\frac{x^{2 n} (A b (2 a d+b c)+a B (a d+2 b c))}{m+2 n+1}+\frac{b x^{3 n} (2 a B d+A b d+b B c)}{m+3 n+1}+\frac{a x^n (a A d+a B c+2 A b c)}{m+n+1}+\frac{b^2 B d x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.114, size = 2410, normalized size = 15.1 \[ \text{result too large to display} \]

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),x)

[Out]

x*(11*B*b^2*d*n^2*(x^n)^4+A*a^2*d*m^4*x^n+4*A*b^2*c*m^3*(x^n)^2+12*A*b^2*c*n^3*(
x^n)^2+6*A*b^2*d*m^2*(x^n)^3+14*A*b^2*d*n^2*(x^n)^3+24*A*b^2*c*m^2*n*(x^n)^2+38*
A*b^2*c*m*n^2*(x^n)^2+14*A*b^2*d*m^2*n^2*(x^n)^3+8*A*b^2*d*m*n^3*(x^n)^3+11*B*b^
2*d*m^2*n^2*(x^n)^4+6*B*b^2*d*m*n^3*(x^n)^4+7*A*b^2*d*m^3*n*(x^n)^3+26*B*a^2*c*m
^2*n^2*x^n+24*B*a^2*c*m*n^3*x^n+8*A*a*b*d*m^3*(x^n)^2+24*A*a*b*d*n^3*(x^n)^2+104
*A*a*b*c*m*n^2*x^n+48*A*a*b*d*m*n*(x^n)^2+56*B*a*b*d*m*n^2*(x^n)^3+18*A*a*b*c*m^
3*n*x^n+28*B*b^2*c*m*n^2*(x^n)^3+52*A*a*b*c*n^2*x^n+B*a^2*c*m^4*x^n+4*B*a^2*d*m^
3*(x^n)^2+12*B*a^2*d*n^3*(x^n)^2+6*B*b^2*c*m^2*(x^n)^3+14*B*b^2*c*n^2*(x^n)^3+4*
m*b^2*B*d*(x^n)^4+6*b^2*B*d*(x^n)^4*n+4*A*a^2*d*m^3*x^n+24*A*a^2*d*n^3*x^n+6*A*b
^2*c*m^2*(x^n)^2+19*A*b^2*c*n^2*(x^n)^2+4*A*b^2*d*(x^n)^3*m+7*A*b^2*d*(x^n)^3*n+
4*B*a^2*c*m^3*x^n+24*B*a^2*c*n^3*x^n+6*B*a^2*d*m^2*(x^n)^2+19*B*a^2*d*n^2*(x^n)^
2+4*A*a^2*d*x^n*m+9*A*a^2*d*x^n*n+b^2*B*d*(x^n)^4+A*b^2*d*(x^n)^3+B*b^2*c*(x^n)^
3+A*b^2*c*(x^n)^2+48*A*a*b*c*m*n^3*x^n+48*A*a*b*d*m^2*n*(x^n)^2+52*A*a*b*c*m^2*n
^2*x^n+48*B*a*b*c*m*n*(x^n)^2+54*A*a*b*c*m*n*x^n+2*B*a*b*c*(x^n)^2+2*A*a*b*d*(x^
n)^2+50*A*a^2*c*n^3+6*A*a^2*c*m^2+35*A*a^2*c*n^2+8*A*a*b*c*m^3*x^n+48*A*a*b*c*n^
3*x^n+21*A*b^2*d*m*n*(x^n)^3+8*B*a^2*d*m^3*n*(x^n)^2+19*B*a^2*d*m^2*n^2*(x^n)^2+
12*B*a^2*d*m*n^3*(x^n)^2+38*B*a*b*c*n^2*(x^n)^2+8*B*a*b*d*(x^n)^3*m+24*B*a^2*d*m
*n*(x^n)^2+12*B*a*b*c*m^2*(x^n)^2+21*B*b^2*c*m*n*(x^n)^3+2*A*a*b*c*m^4*x^n+18*A*
a*b*c*x^n*n+27*B*a^2*c*m*n*x^n+38*A*a*b*d*n^2*(x^n)^2+24*A*b^2*c*m*n*(x^n)^2+38*
B*a^2*d*m*n^2*(x^n)^2+8*B*a*b*c*m^3*(x^n)^2+12*A*b^2*c*m*n^3*(x^n)^2+21*A*b^2*d*
m^2*n*(x^n)^3+28*A*b^2*d*m*n^2*(x^n)^3+2*B*a*b*d*m^4*(x^n)^3+7*B*b^2*c*m^3*n*(x^
n)^3+14*B*b^2*c*m^2*n^2*(x^n)^3+16*B*a*b*c*(x^n)^2*n+8*A*a*b*c*x^n*m+6*B*b^2*d*m
^3*n*(x^n)^4+24*B*a^2*d*m^2*n*(x^n)^2+9*A*a^2*d*m^3*n*x^n+76*A*a*b*d*m*n^2*(x^n)
^2+48*B*a*b*c*m^2*n*(x^n)^2+76*B*a*b*c*m*n^2*(x^n)^2+42*B*a*b*d*m*n*(x^n)^3+54*A
*a*b*c*m^2*n*x^n+4*B*b^2*c*(x^n)^3*m+7*B*b^2*c*(x^n)^3*n+6*A*a^2*d*m^2*x^n+26*A*
a^2*d*n^2*x^n+4*A*b^2*c*(x^n)^2*m+8*A*b^2*c*(x^n)^2*n+B*a^2*d*(x^n)^2+A*a^2*d*x^
n+B*a^2*c*x^n+24*A*a^2*c*n^4+A*a^2*c*m^4+4*A*a^2*c*m^3+A*a^2*c+8*B*a*b*c*(x^n)^2
*m+27*B*a^2*c*m^2*n*x^n+52*B*a^2*c*m*n^2*x^n+24*A*a^2*d*m*n^3*x^n+10*A*a^2*c*m^3
*n+35*A*a^2*c*m^2*n^2+50*A*a^2*c*m*n^3+6*B*a^2*c*m^2*x^n+27*A*a^2*d*m*n*x^n+12*A
*a*b*c*m^2*x^n+18*B*b^2*d*m*n*(x^n)^4+12*A*a*b*d*m^2*(x^n)^2+8*A*a*b*d*(x^n)^2*m
+16*A*a*b*d*(x^n)^2*n+27*A*a^2*d*m^2*n*x^n+9*B*a^2*c*m^3*n*x^n+42*B*a*b*d*m^2*n*
(x^n)^3+26*A*a^2*d*m^2*n^2*x^n+24*B*a*b*c*n^3*(x^n)^2+12*B*a*b*d*m^2*(x^n)^3+28*
B*a*b*d*n^2*(x^n)^3+4*A*b^2*d*m^3*(x^n)^3+8*A*b^2*d*n^3*(x^n)^3+B*a^2*d*m^4*(x^n
)^2+4*B*b^2*c*m^3*(x^n)^3+8*B*b^2*c*n^3*(x^n)^3+6*B*b^2*d*m^2*(x^n)^4+14*B*a*b*d
*m^3*n*(x^n)^3+28*B*a*b*d*m^2*n^2*(x^n)^3+16*B*a*b*d*m*n^3*(x^n)^3+16*A*a*b*d*m^
3*n*(x^n)^2+30*A*a^2*c*m^2*n+70*A*a^2*c*m*n^2+30*A*a^2*c*m*n+B*b^2*d*m^4*(x^n)^4
+A*b^2*d*m^4*(x^n)^3+B*b^2*c*m^4*(x^n)^3+4*B*b^2*d*m^3*(x^n)^4+6*B*b^2*d*n^3*(x^
n)^4+A*b^2*c*m^4*(x^n)^2+26*B*a^2*c*n^2*x^n+2*B*a*b*d*(x^n)^3+2*a*b*A*c*x^n+4*B*
a^2*d*(x^n)^2*m+8*B*a^2*d*(x^n)^2*n+9*B*a^2*c*x^n*n+38*A*a*b*d*m^2*n^2*(x^n)^2+2
4*A*a*b*d*m*n^3*(x^n)^2+16*B*a*b*c*m^3*n*(x^n)^2+38*B*a*b*c*m^2*n^2*(x^n)^2+24*B
*a*b*c*m*n^3*(x^n)^2+4*B*a^2*c*x^n*m+14*B*a*b*d*(x^n)^3*n+52*A*a^2*d*m*n^2*x^n+2
*B*a*b*c*m^4*(x^n)^2+8*B*a*b*d*m^3*(x^n)^3+16*B*a*b*d*n^3*(x^n)^3+21*B*b^2*c*m^2
*n*(x^n)^3+4*A*a^2*c*m+10*A*a^2*c*n+8*B*b^2*c*m*n^3*(x^n)^3+18*B*b^2*d*m^2*n*(x^
n)^4+22*B*b^2*d*m*n^2*(x^n)^4+2*A*a*b*d*m^4*(x^n)^2+8*A*b^2*c*m^3*n*(x^n)^2+19*A
*b^2*c*m^2*n^2*(x^n)^2)/(1+m)/(1+m+n)/(1+m+2*n)/(1+m+3*n)/(1+m+4*n)*exp(1/2*m*(-
I*Pi*csgn(I*e*x)^3+I*Pi*csgn(I*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*P
i*csgn(I*e*x)*csgn(I*e)*csgn(I*x)+2*ln(e)+2*ln(x)))

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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)*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.255673, size = 2057, normalized size = 12.86 \[ \text{result too large to display} \]

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)*(e*x)^m,x, algorithm="fricas")

[Out]

((B*b^2*d*m^4 + 4*B*b^2*d*m^3 + 6*B*b^2*d*m^2 + 4*B*b^2*d*m + B*b^2*d + 6*(B*b^2
*d*m + B*b^2*d)*n^3 + 11*(B*b^2*d*m^2 + 2*B*b^2*d*m + B*b^2*d)*n^2 + 6*(B*b^2*d*
m^3 + 3*B*b^2*d*m^2 + 3*B*b^2*d*m + B*b^2*d)*n)*x*x^(4*n)*e^(m*log(e) + m*log(x)
) + ((B*b^2*c + (2*B*a*b + A*b^2)*d)*m^4 + B*b^2*c + 4*(B*b^2*c + (2*B*a*b + A*b
^2)*d)*m^3 + 8*(B*b^2*c + (2*B*a*b + A*b^2)*d + (B*b^2*c + (2*B*a*b + A*b^2)*d)*
m)*n^3 + 6*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 + 14*(B*b^2*c + (B*b^2*c + (2*B*a
*b + A*b^2)*d)*m^2 + (2*B*a*b + A*b^2)*d + 2*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m)*
n^2 + (2*B*a*b + A*b^2)*d + 4*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m + 7*(B*b^2*c + (
B*b^2*c + (2*B*a*b + A*b^2)*d)*m^3 + 3*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m^2 + (2*
B*a*b + A*b^2)*d + 3*(B*b^2*c + (2*B*a*b + A*b^2)*d)*m)*n)*x*x^(3*n)*e^(m*log(e)
+ m*log(x)) + (((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^4 + 4*((2*B*a*b +
A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^3 + 12*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b
)*d + ((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m)*n^3 + 6*((2*B*a*b + A*b^2)*
c + (B*a^2 + 2*A*a*b)*d)*m^2 + 19*(((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m
^2 + (2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 2*((2*B*a*b + A*b^2)*c + (B*a^2
+ 2*A*a*b)*d)*m)*n^2 + (2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d + 4*((2*B*a*b
+ A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m + 8*(((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b
)*d)*m^3 + 3*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m^2 + (2*B*a*b + A*b^2)
*c + (B*a^2 + 2*A*a*b)*d + 3*((2*B*a*b + A*b^2)*c + (B*a^2 + 2*A*a*b)*d)*m)*n)*x
*x^(2*n)*e^(m*log(e) + m*log(x)) + ((A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^4 + A*a^2*
d + 4*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^3 + 24*(A*a^2*d + (B*a^2 + 2*A*a*b)*c +
(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n^3 + 6*(A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 +
26*(A*a^2*d + (A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^2 + (B*a^2 + 2*A*a*b)*c + 2*(A*
a^2*d + (B*a^2 + 2*A*a*b)*c)*m)*n^2 + (B*a^2 + 2*A*a*b)*c + 4*(A*a^2*d + (B*a^2
+ 2*A*a*b)*c)*m + 9*(A*a^2*d + (A*a^2*d + (B*a^2 + 2*A*a*b)*c)*m^3 + 3*(A*a^2*d
+ (B*a^2 + 2*A*a*b)*c)*m^2 + (B*a^2 + 2*A*a*b)*c + 3*(A*a^2*d + (B*a^2 + 2*A*a*b
)*c)*m)*n)*x*x^n*e^(m*log(e) + m*log(x)) + (A*a^2*c*m^4 + 24*A*a^2*c*n^4 + 4*A*a
^2*c*m^3 + 6*A*a^2*c*m^2 + 4*A*a^2*c*m + A*a^2*c + 50*(A*a^2*c*m + A*a^2*c)*n^3
+ 35*(A*a^2*c*m^2 + 2*A*a^2*c*m + A*a^2*c)*n^2 + 10*(A*a^2*c*m^3 + 3*A*a^2*c*m^2
+ 3*A*a^2*c*m + A*a^2*c)*n)*x*e^(m*log(e) + m*log(x)))/(m^5 + 24*(m + 1)*n^4 +
5*m^4 + 50*(m^2 + 2*m + 1)*n^3 + 10*m^3 + 35*(m^3 + 3*m^2 + 3*m + 1)*n^2 + 10*m^
2 + 10*(m^4 + 4*m^3 + 6*m^2 + 4*m + 1)*n + 5*m + 1)

_______________________________________________________________________________________

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),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222339, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

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)*(e*x)^m,x, algorithm="giac")

[Out]

Done