3.2742 \(\int x^m (a+b x^{2+2 m}) \, dx\)

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))

________________________________________________________________________________________

Rubi [A]  time = 0.0082643, 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, Rules used = {14} \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{3 (m+1)}}{3 (m+1)} \]

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))

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^m \left (a+b x^{2+2 m}\right ) \, dx &=\int \left (a x^m+b x^{2+3 m}\right ) \, dx\\ &=\frac{a x^{1+m}}{1+m}+\frac{b x^{3 (1+m)}}{3 (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.0130887, 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)

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 33, normalized size = 1.1 \begin{align*}{\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}} \end{align*}

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.34212, size = 55, normalized size = 1.83 \begin{align*} \frac{b x^{3} x^{3 \, m} + 3 \, a x x^{m}}{3 \,{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 12.1479, size = 36, normalized size = 1.2 \begin{align*} \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} \end{align*}

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))

________________________________________________________________________________________

Giac [A]  time = 1.12278, size = 34, normalized size = 1.13 \begin{align*} \frac{b x^{3} x^{3 \, m} + 3 \, a x x^{m}}{3 \,{\left (m + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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