3.343 \(\int x^m (a+b x^2) \, dx\)

Optimal. Leaf size=25 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+3}}{m+3} \]

[Out]

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

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Rubi [A]  time = 0.0073573, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+3}}{m+3} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^2),x]

[Out]

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

Mathematica [A]  time = 0.0123004, size = 25, normalized size = 1. \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+3}}{m+3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^2),x]

[Out]

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

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Maple [A]  time = 0.002, size = 35, normalized size = 1.4 \begin{align*}{\frac{{x}^{1+m} \left ( bm{x}^{2}+b{x}^{2}+am+3\,a \right ) }{ \left ( 3+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^2+a),x)

[Out]

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

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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*(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.53331, size = 72, normalized size = 2.88 \begin{align*} \frac{{\left ({\left (b m + b\right )} x^{3} +{\left (a m + 3 \, a\right )} x\right )} x^{m}}{m^{2} + 4 \, m + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*m + b)*x^3 + (a*m + 3*a)*x)*x^m/(m^2 + 4*m + 3)

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Sympy [A]  time = 0.375347, size = 94, normalized size = 3.76 \begin{align*} \begin{cases} - \frac{a}{2 x^{2}} + b \log{\left (x \right )} & \text{for}\: m = -3 \\a \log{\left (x \right )} + \frac{b x^{2}}{2} & \text{for}\: m = -1 \\\frac{a m x x^{m}}{m^{2} + 4 m + 3} + \frac{3 a x x^{m}}{m^{2} + 4 m + 3} + \frac{b m x^{3} x^{m}}{m^{2} + 4 m + 3} + \frac{b x^{3} x^{m}}{m^{2} + 4 m + 3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**2+a),x)

[Out]

Piecewise((-a/(2*x**2) + b*log(x), Eq(m, -3)), (a*log(x) + b*x**2/2, Eq(m, -1)), (a*m*x*x**m/(m**2 + 4*m + 3)
+ 3*a*x*x**m/(m**2 + 4*m + 3) + b*m*x**3*x**m/(m**2 + 4*m + 3) + b*x**3*x**m/(m**2 + 4*m + 3), True))

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Giac [A]  time = 2.58023, size = 58, normalized size = 2.32 \begin{align*} \frac{b m x^{3} x^{m} + b x^{3} x^{m} + a m x x^{m} + 3 \, a x x^{m}}{m^{2} + 4 \, m + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*m*x^3*x^m + b*x^3*x^m + a*m*x*x^m + 3*a*x*x^m)/(m^2 + 4*m + 3)