3.597 \(\int (d x)^m \left (a+b x^n+c x^{2 n}\right )^2 \, dx\)

Optimal. Leaf size=117 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{x^{2 n+1} \left (2 a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac{2 a b x^{n+1} (d x)^m}{m+n+1}+\frac{2 b c x^{3 n+1} (d x)^m}{m+3 n+1}+\frac{c^2 x^{4 n+1} (d x)^m}{m+4 n+1} \]

[Out]

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

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Rubi [A]  time = 0.146697, antiderivative size = 117, 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 \[ \frac{a^2 (d x)^{m+1}}{d (m+1)}+\frac{x^{2 n+1} \left (2 a c+b^2\right ) (d x)^m}{m+2 n+1}+\frac{2 a b x^{n+1} (d x)^m}{m+n+1}+\frac{2 b c x^{3 n+1} (d x)^m}{m+3 n+1}+\frac{c^2 x^{4 n+1} (d x)^m}{m+4 n+1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 27.872, size = 141, normalized size = 1.21 \[ \frac{a^{2} \left (d x\right )^{m + 1}}{d \left (m + 1\right )} + \frac{2 a b x^{- m} x^{m + n + 1} \left (d x\right )^{m}}{m + n + 1} + \frac{2 b c x^{3 n} \left (d x\right )^{- 3 n} \left (d x\right )^{m + 3 n + 1}}{d \left (m + 3 n + 1\right )} + \frac{c^{2} x^{4 n} \left (d x\right )^{- 4 n} \left (d x\right )^{m + 4 n + 1}}{d \left (m + 4 n + 1\right )} + \frac{x^{- m} x^{m + 2 n + 1} \left (d x\right )^{m} \left (2 a c + b^{2}\right )}{m + 2 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.139337, size = 86, normalized size = 0.74 \[ x (d x)^m \left (\frac{a^2}{m+1}+\frac{x^{2 n} \left (2 a c+b^2\right )}{m+2 n+1}+\frac{2 a b x^n}{m+n+1}+\frac{2 b c x^{3 n}}{m+3 n+1}+\frac{c^2 x^{4 n}}{m+4 n+1}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.101, size = 1065, normalized size = 9.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x)^m*(a+b*x^n+c*x^(2*n))^2,x)

[Out]

x*(12*a*c*m^2*(x^n)^2+38*a*c*n^2*(x^n)^2+24*b^2*m*n*(x^n)^2+8*m*b*c*(x^n)^3+14*b
*c*(x^n)^3*n+12*a*b*m^2*x^n+52*a*b*n^2*x^n+8*a*c*(x^n)^2*m+16*a*c*(x^n)^2*n+8*a*
b*x^n*m+18*a*b*x^n*n+a^2+48*a*c*m^2*n*(x^n)^2+76*a*c*m*n^2*(x^n)^2+42*b*c*m*n*(x
^n)^3+54*a*b*m^2*n*x^n+104*a*b*m*n^2*x^n+48*a*c*m*n*(x^n)^2+54*a*b*m*n*x^n+c^2*(
x^n)^4+a^2*m^4+4*a^2*m^3+50*a^2*n^3+6*a^2*m^2+35*a^2*n^2+24*a^2*n^4+4*a^2*m+10*a
^2*n+b^2*(x^n)^2+10*a^2*m^3*n+35*a^2*m^2*n^2+50*a^2*m*n^3+30*a^2*m^2*n+70*a^2*m*
n^2+30*a^2*m*n+8*b*c*m^3*(x^n)^3+16*b*c*n^3*(x^n)^3+18*c^2*m*n*(x^n)^4+2*a*b*m^4
*x^n+8*a*c*m^3*(x^n)^2+24*a*c*n^3*(x^n)^2+24*b^2*m^2*n*(x^n)^2+38*b^2*m*n^2*(x^n
)^2+12*b*c*m^2*(x^n)^3+28*b*c*n^2*(x^n)^3+8*a*b*m^3*x^n+48*a*b*n^3*x^n+38*a*c*m^
2*n^2*(x^n)^2+24*a*c*m*n^3*(x^n)^2+42*b*c*m^2*n*(x^n)^3+56*b*c*m*n^2*(x^n)^3+18*
a*b*m^3*n*x^n+52*a*b*m^2*n^2*x^n+48*a*b*m*n^3*x^n+2*b*c*m^4*(x^n)^3+18*c^2*m^2*n
*(x^n)^4+22*c^2*m*n^2*(x^n)^4+2*a*c*m^4*(x^n)^2+8*b^2*m^3*n*(x^n)^2+19*b^2*m^2*n
^2*(x^n)^2+12*b^2*m*n^3*(x^n)^2+11*c^2*m^2*n^2*(x^n)^4+6*c^2*m*n^3*(x^n)^4+6*c^2
*m^3*n*(x^n)^4+2*a*b*x^n+4*b^2*(x^n)^2*m+8*b^2*(x^n)^2*n+2*b*c*(x^n)^3+2*a*c*(x^
n)^2+11*c^2*n^2*(x^n)^4+4*b^2*m^3*(x^n)^2+12*b^2*n^3*(x^n)^2+4*m*c^2*(x^n)^4+6*c
^2*(x^n)^4*n+6*b^2*m^2*(x^n)^2+19*b^2*n^2*(x^n)^2+c^2*m^4*(x^n)^4+4*c^2*m^3*(x^n
)^4+6*c^2*n^3*(x^n)^4+b^2*m^4*(x^n)^2+6*c^2*m^2*(x^n)^4+14*b*c*m^3*n*(x^n)^3+28*
b*c*m^2*n^2*(x^n)^3+16*b*c*m*n^3*(x^n)^3+16*a*c*m^3*n*(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*d*x)^3-I*Pi*csgn(I*d*x)^2*csg
n(I*d)-I*Pi*csgn(I*d*x)^2*csgn(I*x)+I*Pi*csgn(I*d*x)*csgn(I*d)*csgn(I*x)-2*ln(x)
-2*ln(d)))

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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((c*x^(2*n) + b*x^n + a)^2*(d*x)^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294903, size = 953, normalized size = 8.15 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^(2*n) + b*x^n + a)^2*(d*x)^m,x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.283026, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done