3.2735 \(\int x^m \left (a+b x^{2+2 m}\right ) \, dx\)

Optimal. Leaf size=30 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{3 (m+1)}}{3 (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0263151, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{3 (m+1)}}{3 (m+1)} \]

Antiderivative was successfully verified.

[In]  Int[x^m*(a + b*x^(2 + 2*m)),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.55048, size = 22, normalized size = 0.73 \[ \frac{a x^{m + 1}}{m + 1} + \frac{b x^{3 m + 3}}{3 \left (m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.01554, size = 26, normalized size = 0.87 \[ \frac{3 a x^{m+1}+b x^{3 m+3}}{3 m+3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.025, size = 33, normalized size = 1.1 \[{\frac{ax{{\rm e}^{m\ln \left ( x \right ) }}}{1+m}}+{\frac{b{x}^{3} \left ({{\rm e}^{m\ln \left ( x \right ) }} \right ) ^{3}}{3+3\,m}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^m*(a+b*x^(2+2*m)),x)

[Out]

a/(1+m)*x*exp(m*ln(x))+1/3*b/(1+m)*x^3*exp(m*ln(x))^3

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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)*x^m,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231383, size = 34, normalized size = 1.13 \[ \frac{b x^{3} x^{3 \, m} + 3 \, a x x^{m}}{3 \,{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 40.6921, size = 36, normalized size = 1.2 \[ \begin{cases} \frac{3 a x x^{m}}{3 m + 3} + \frac{b x^{3} x^{3 m}}{3 m + 3} & \text{for}\: m \neq -1 \\\left (a + b\right ) \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((3*a*x*x**m/(3*m + 3) + b*x**3*x**(3*m)/(3*m + 3), Ne(m, -1)), ((a + b
)*log(x), True))

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GIAC/XCAS [A]  time = 0.217384, size = 38, normalized size = 1.27 \[ \frac{b x^{3} e^{\left (3 \, m{\rm ln}\left (x\right )\right )} + 3 \, a x e^{\left (m{\rm ln}\left (x\right )\right )}}{3 \,{\left (m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b*x^3*e^(3*m*ln(x)) + 3*a*x*e^(m*ln(x)))/(m + 1)