3.2661 \(\int x^m \left (a+b x^n\right )^2 \, dx\)

Optimal. Leaf size=51 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+n+1}}{m+n+1}+\frac{b^2 x^{m+2 n+1}}{m+2 n+1} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(1 + m + n))/(1 + m + n) + (b^2*x^(1 + m + 2*
n))/(1 + m + 2*n)

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Rubi [A]  time = 0.0517128, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{m+n+1}}{m+n+1}+\frac{b^2 x^{m+2 n+1}}{m+2 n+1} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^n)^2,x]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(1 + m + n))/(1 + m + n) + (b^2*x^(1 + m + 2*
n))/(1 + m + 2*n)

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Rubi in Sympy [A]  time = 8.99589, size = 46, normalized size = 0.9 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{m + n + 1}}{m + n + 1} + \frac{b^{2} x^{m + 2 n + 1}}{m + 2 n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**(m + 1)/(m + 1) + 2*a*b*x**(m + n + 1)/(m + n + 1) + b**2*x**(m + 2*n +
1)/(m + 2*n + 1)

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Mathematica [A]  time = 0.048814, size = 46, normalized size = 0.9 \[ x^{m+1} \left (\frac{a^2}{m+1}+\frac{2 a b x^n}{m+n+1}+\frac{b^2 x^{2 n}}{m+2 n+1}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^m*(a + b*x^n)^2,x]

[Out]

x^(1 + m)*(a^2/(1 + m) + (2*a*b*x^n)/(1 + m + n) + (b^2*x^(2*n))/(1 + m + 2*n))

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Maple [A]  time = 0.023, size = 63, normalized size = 1.2 \[{\frac{x{a}^{2}{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{{b}^{2}x{{\rm e}^{m\ln \left ( x \right ) }} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{1+m+2\,n}}+2\,{\frac{abx{{\rm e}^{m\ln \left ( x \right ) }}{{\rm e}^{n\ln \left ( x \right ) }}}{1+m+n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^n)^2,x)

[Out]

a^2/(1+m)*x*exp(m*ln(x))+b^2/(1+m+2*n)*x*exp(m*ln(x))*exp(n*ln(x))^2+2*a*b/(1+m+
n)*x*exp(m*ln(x))*exp(n*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)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.238254, size = 204, normalized size = 4. \[ \frac{{\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2} +{\left (b^{2} m + b^{2}\right )} n\right )} x x^{m} x^{2 \, n} + 2 \,{\left (a b m^{2} + 2 \, a b m + a b + 2 \,{\left (a b m + a b\right )} n\right )} x x^{m} x^{n} +{\left (a^{2} m^{2} + 2 \, a^{2} n^{2} + 2 \, a^{2} m + a^{2} + 3 \,{\left (a^{2} m + a^{2}\right )} n\right )} x x^{m}}{m^{3} + 2 \,{\left (m + 1\right )} n^{2} + 3 \, m^{2} + 3 \,{\left (m^{2} + 2 \, m + 1\right )} n + 3 \, m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

((b^2*m^2 + 2*b^2*m + b^2 + (b^2*m + b^2)*n)*x*x^m*x^(2*n) + 2*(a*b*m^2 + 2*a*b*
m + a*b + 2*(a*b*m + a*b)*n)*x*x^m*x^n + (a^2*m^2 + 2*a^2*n^2 + 2*a^2*m + a^2 +
3*(a^2*m + a^2)*n)*x*x^m)/(m^3 + 2*(m + 1)*n^2 + 3*m^2 + 3*(m^2 + 2*m + 1)*n + 3
*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(x**m*(a+b*x**n)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.222618, size = 385, normalized size = 7.55 \[ \frac{b^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + b^{2} m n x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a b m^{2} x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + 4 \, a b m n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + a^{2} m^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a^{2} m n x e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, a^{2} n^{2} x e^{\left (m{\rm ln}\left (x\right )\right )} + 2 \, b^{2} m x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + b^{2} n x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + 4 \, a b m x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + 4 \, a b n x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + 2 \, a^{2} m x e^{\left (m{\rm ln}\left (x\right )\right )} + 3 \, a^{2} n x e^{\left (m{\rm ln}\left (x\right )\right )} + b^{2} x e^{\left (m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right )\right )} + 2 \, a b x e^{\left (m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right )\right )} + a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{m^{3} + 3 \, m^{2} n + 2 \, m n^{2} + 3 \, m^{2} + 6 \, m n + 2 \, n^{2} + 3 \, m + 3 \, n + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b^2*m^2*x*e^(m*ln(x) + 2*n*ln(x)) + b^2*m*n*x*e^(m*ln(x) + 2*n*ln(x)) + 2*a*b*m
^2*x*e^(m*ln(x) + n*ln(x)) + 4*a*b*m*n*x*e^(m*ln(x) + n*ln(x)) + a^2*m^2*x*e^(m*
ln(x)) + 3*a^2*m*n*x*e^(m*ln(x)) + 2*a^2*n^2*x*e^(m*ln(x)) + 2*b^2*m*x*e^(m*ln(x
) + 2*n*ln(x)) + b^2*n*x*e^(m*ln(x) + 2*n*ln(x)) + 4*a*b*m*x*e^(m*ln(x) + n*ln(x
)) + 4*a*b*n*x*e^(m*ln(x) + n*ln(x)) + 2*a^2*m*x*e^(m*ln(x)) + 3*a^2*n*x*e^(m*ln
(x)) + b^2*x*e^(m*ln(x) + 2*n*ln(x)) + 2*a*b*x*e^(m*ln(x) + n*ln(x)) + a^2*x*e^(
m*ln(x)))/(m^3 + 3*m^2*n + 2*m*n^2 + 3*m^2 + 6*m*n + 2*n^2 + 3*m + 3*n + 1)