3.18 \(\int (e x)^m \left (A+B x^n\right ) \left (c+d x^n\right )^3 \, dx\)

Optimal. Leaf size=137 \[ \frac{c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 x^{4 n+1} (e x)^m}{m+4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.275137, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{c^2 x^{n+1} (e x)^m (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n+1} (e x)^m (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n+1} (e x)^m (A d+B c)}{m+2 n+1}+\frac{A c^3 (e x)^{m+1}}{e (m+1)}+\frac{B d^3 x^{4 n+1} (e x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(A + B*x^n)*(c + d*x^n)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 47.3937, size = 165, normalized size = 1.2 \[ \frac{A c^{3} \left (e x\right )^{m + 1}}{e \left (m + 1\right )} + \frac{B d^{3} x^{4 n} \left (e x\right )^{- 4 n} \left (e x\right )^{m + 4 n + 1}}{e \left (m + 4 n + 1\right )} + \frac{c^{2} x^{n} \left (e x\right )^{- n} \left (e x\right )^{m + n + 1} \left (3 A d + B c\right )}{e \left (m + n + 1\right )} + \frac{3 c d x^{2 n} \left (e x\right )^{- 2 n} \left (e x\right )^{m + 2 n + 1} \left (A d + B c\right )}{e \left (m + 2 n + 1\right )} + \frac{d^{2} x^{- m} x^{m + 3 n + 1} \left (e x\right )^{m} \left (A d + 3 B c\right )}{m + 3 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(A+B*x**n)*(c+d*x**n)**3,x)

[Out]

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

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Mathematica [A]  time = 0.227091, size = 106, normalized size = 0.77 \[ x (e x)^m \left (\frac{c^2 x^n (3 A d+B c)}{m+n+1}+\frac{d^2 x^{3 n} (A d+3 B c)}{m+3 n+1}+\frac{3 c d x^{2 n} (A d+B c)}{m+2 n+1}+\frac{A c^3}{m+1}+\frac{B d^3 x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*(A + B*x^n)*(c + d*x^n)^3,x]

[Out]

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

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Maple [C]  time = 0.105, size = 1609, normalized size = 11.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(A+B*x^n)*(c+d*x^n)^3,x)

[Out]

x*(A*c^3*m^4+81*A*c^2*d*m^2*n*x^n+156*A*c^2*d*m*n^2*x^n+72*A*c*d^2*m*n*(x^n)^2+7
2*B*c^2*d*m*n*(x^n)^2+81*A*c^2*d*m*n*x^n+84*B*c*d^2*m*n^2*(x^n)^3+27*A*c^2*d*m^3
*n*x^n+B*c^3*m^4*x^n+4*m*B*d^3*(x^n)^4+6*B*d^3*(x^n)^4*n+4*A*d^3*(x^n)^3*m+7*A*d
^3*(x^n)^3*n+27*B*c^3*m*n*x^n+12*B*c^2*d*(x^n)^2*m+24*B*c^2*d*(x^n)^2*n+12*A*c^2
*d*x^n*m+27*A*c^2*d*x^n*n+4*A*c^3*m+10*A*c^3*n+10*A*c^3*m^3*n+35*A*c^3*m^2*n^2+5
0*A*c^3*m*n^3+30*A*c^3*m^2*n+70*A*c^3*m*n^2+30*A*c^3*m*n+36*B*c^2*d*m*n^3*(x^n)^
2+63*B*c*d^2*m^2*n*(x^n)^3+114*B*c^2*d*m*n^2*(x^n)^2+63*B*c*d^2*m*n*(x^n)^3+6*B*
d^3*m^3*n*(x^n)^4+11*B*d^3*m^2*n^2*(x^n)^4+6*B*d^3*m*n^3*(x^n)^4+7*A*d^3*m^3*n*(
x^n)^3+14*A*d^3*m^2*n^2*(x^n)^3+18*B*d^3*m*n*(x^n)^4+3*A*c^2*d*m^4*x^n+12*A*c*d^
2*m^3*(x^n)^2+36*A*c*d^2*n^3*(x^n)^2+21*A*d^3*m*n*(x^n)^3+9*B*c^3*m^3*n*x^n+B*d^
3*(x^n)^4+A*d^3*(x^n)^3+B*c^3*x^n+3*B*c*d^2*(x^n)^3+3*A*c*d^2*(x^n)^2+3*B*c^2*d*
(x^n)^2+3*A*c^2*d*x^n+B*d^3*m^4*(x^n)^4+A*d^3*m^4*(x^n)^3+4*B*d^3*m^3*(x^n)^4+A*
c^3+4*B*c^3*x^n*m+9*B*c^3*x^n*n+72*A*c^2*d*m*n^3*x^n+72*A*c*d^2*m^2*n*(x^n)^2+11
4*A*c*d^2*m*n^2*(x^n)^2+72*B*c^2*d*m^2*n*(x^n)^2+12*A*c^2*d*m^3*x^n+72*A*c^2*d*n
^3*x^n+24*A*c^3*n^4+4*A*c^3*m^3+50*A*c^3*n^3+6*A*c^3*m^2+35*A*c^3*n^2+8*A*d^3*m*
n^3*(x^n)^3+3*B*c*d^2*m^4*(x^n)^3+18*B*d^3*m^2*n*(x^n)^4+22*B*d^3*m*n^2*(x^n)^4+
3*A*c*d^2*m^4*(x^n)^2+21*A*d^3*m^2*n*(x^n)^3+28*A*d^3*m*n^2*(x^n)^3+3*B*c^2*d*m^
4*(x^n)^2+12*B*c*d^2*m^3*(x^n)^3+24*B*c*d^2*n^3*(x^n)^3+24*B*c*d^2*m*n^3*(x^n)^3
+24*A*c*d^2*m^3*n*(x^n)^2+57*A*c*d^2*m^2*n^2*(x^n)^2+36*A*c*d^2*m*n^3*(x^n)^2+24
*B*c^2*d*m^3*n*(x^n)^2+57*B*c^2*d*m^2*n^2*(x^n)^2+21*B*c*d^2*m^3*n*(x^n)^3+42*B*
c*d^2*m^2*n^2*(x^n)^3+4*B*c^3*m^3*x^n+24*B*c^3*n^3*x^n+6*B*c^3*m^2*x^n+26*B*c^3*
n^2*x^n+18*A*c*d^2*m^2*(x^n)^2+57*A*c*d^2*n^2*(x^n)^2+27*B*c^3*m^2*n*x^n+52*B*c^
3*m*n^2*x^n+18*B*c^2*d*m^2*(x^n)^2+57*B*c^2*d*n^2*(x^n)^2+12*B*c*d^2*(x^n)^3*m+7
8*A*c^2*d*m^2*n^2*x^n+6*B*d^3*n^3*(x^n)^4+4*A*d^3*m^3*(x^n)^3+8*A*d^3*n^3*(x^n)^
3+6*B*d^3*m^2*(x^n)^4+11*B*d^3*n^2*(x^n)^4+6*A*d^3*m^2*(x^n)^3+14*A*d^3*n^2*(x^n
)^3+26*B*c^3*m^2*n^2*x^n+24*B*c^3*m*n^3*x^n+12*B*c^2*d*m^3*(x^n)^2+36*B*c^2*d*n^
3*(x^n)^2+18*B*c*d^2*m^2*(x^n)^3+42*B*c*d^2*n^2*(x^n)^3+21*B*c*d^2*(x^n)^3*n+18*
A*c^2*d*m^2*x^n+78*A*c^2*d*n^2*x^n+12*A*c*d^2*(x^n)^2*m+24*A*c*d^2*(x^n)^2*n)/(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*cs
gn(I*e*x)^2*csgn(I*e)+I*Pi*csgn(I*e*x)^2*csgn(I*x)-I*Pi*csgn(I*e*x)*csgn(I*e)*cs
gn(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)*(d*x^n + c)^3*(e*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261279, size = 1490, normalized size = 10.88 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(d*x^n + c)^3*(e*x)^m,x, algorithm="fricas")

[Out]

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

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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)*(c+d*x**n)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220328, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^n + A)*(d*x^n + c)^3*(e*x)^m,x, algorithm="giac")

[Out]

Done