3.2734 \(\int x^m \left (a+b x^{2+2 m}\right )^2 \, dx\)

Optimal. Leaf size=52 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{3 (m+1)}}{3 (m+1)}+\frac{b^2 x^{5 (m+1)}}{5 (m+1)} \]

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(3*(1 + m)))/(3*(1 + m)) + (b^2*x^(5*(1 + m))
)/(5*(1 + m))

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Rubi [A]  time = 0.058339, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059 \[ \frac{a^2 x^{m+1}}{m+1}+\frac{2 a b x^{3 (m+1)}}{3 (m+1)}+\frac{b^2 x^{5 (m+1)}}{5 (m+1)} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^(1 + m))/(1 + m) + (2*a*b*x^(3*(1 + m)))/(3*(1 + m)) + (b^2*x^(5*(1 + m))
)/(5*(1 + m))

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Rubi in Sympy [A]  time = 9.57496, size = 42, normalized size = 0.81 \[ \frac{a^{2} x^{m + 1}}{m + 1} + \frac{2 a b x^{3 m + 3}}{3 \left (m + 1\right )} + \frac{b^{2} x^{5 m + 5}}{5 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**(m + 1)/(m + 1) + 2*a*b*x**(3*m + 3)/(3*(m + 1)) + b**2*x**(5*m + 5)/(5*
(m + 1))

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Mathematica [A]  time = 0.0324639, size = 42, normalized size = 0.81 \[ \frac{15 a^2 x^{m+1}+10 a b x^{3 m+3}+3 b^2 x^{5 m+5}}{15 m+15} \]

Antiderivative was successfully verified.

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

[Out]

(15*a^2*x^(1 + m) + 10*a*b*x^(3 + 3*m) + 3*b^2*x^(5 + 5*m))/(15 + 15*m)

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Maple [A]  time = 0.029, size = 50, normalized size = 1. \[{\frac{{b}^{2}{x}^{5} \left ({x}^{m} \right ) ^{5}}{5+5\,m}}+{\frac{2\,ab{x}^{3} \left ({x}^{m} \right ) ^{3}}{3+3\,m}}+{\frac{x{a}^{2}{x}^{m}}{1+m}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^(2+2*m))^2,x)

[Out]

1/5*b^2*x^5/(1+m)*(x^m)^5+2/3*a*b*x^3/(1+m)*(x^m)^3+a^2/(1+m)*x*x^m

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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^(2*m + 2) + a)^2*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232124, size = 57, normalized size = 1.1 \[ \frac{3 \, b^{2} x^{5} x^{5 \, m} + 10 \, a b x^{3} x^{3 \, m} + 15 \, a^{2} x x^{m}}{15 \,{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*b^2*x^5*x^(5*m) + 10*a*b*x^3*x^(3*m) + 15*a^2*x*x^m)/(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**(2+2*m))**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220197, size = 62, normalized size = 1.19 \[ \frac{3 \, b^{2} x^{5} e^{\left (5 \, m{\rm ln}\left (x\right )\right )} + 10 \, a b x^{3} e^{\left (3 \, m{\rm ln}\left (x\right )\right )} + 15 \, a^{2} x e^{\left (m{\rm ln}\left (x\right )\right )}}{15 \,{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*b^2*x^5*e^(5*m*ln(x)) + 10*a*b*x^3*e^(3*m*ln(x)) + 15*a^2*x*e^(m*ln(x)))
/(m + 1)