3.603 \(\int x^m (a+b x^4) \, dx\)

Optimal. Leaf size=25 \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+5}}{m+5} \]

[Out]

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

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Rubi [A]  time = 0.0075716, 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} \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+5}}{m+5} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a + b*x^4),x]

[Out]

(a*x^(1 + m))/(1 + m) + (b*x^(5 + m))/(5 + 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^4\right ) \, dx &=\int \left (a x^m+b x^{4+m}\right ) \, dx\\ &=\frac{a x^{1+m}}{1+m}+\frac{b x^{5+m}}{5+m}\\ \end{align*}

Mathematica [A]  time = 0.0126902, size = 25, normalized size = 1. \[ \frac{a x^{m+1}}{m+1}+\frac{b x^{m+5}}{m+5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a + b*x^4),x]

[Out]

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

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Maple [A]  time = 0.004, size = 35, normalized size = 1.4 \begin{align*}{\frac{{x}^{1+m} \left ( bm{x}^{4}+b{x}^{4}+am+5\,a \right ) }{ \left ( 5+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(b*x^4+a),x)

[Out]

x^(1+m)*(b*m*x^4+b*x^4+a*m+5*a)/(5+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^4+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.54975, size = 72, normalized size = 2.88 \begin{align*} \frac{{\left ({\left (b m + b\right )} x^{5} +{\left (a m + 5 \, a\right )} x\right )} x^{m}}{m^{2} + 6 \, m + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a),x, algorithm="fricas")

[Out]

((b*m + b)*x^5 + (a*m + 5*a)*x)*x^m/(m^2 + 6*m + 5)

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Sympy [A]  time = 0.898886, size = 94, normalized size = 3.76 \begin{align*} \begin{cases} - \frac{a}{4 x^{4}} + b \log{\left (x \right )} & \text{for}\: m = -5 \\a \log{\left (x \right )} + \frac{b x^{4}}{4} & \text{for}\: m = -1 \\\frac{a m x x^{m}}{m^{2} + 6 m + 5} + \frac{5 a x x^{m}}{m^{2} + 6 m + 5} + \frac{b m x^{5} x^{m}}{m^{2} + 6 m + 5} + \frac{b x^{5} x^{m}}{m^{2} + 6 m + 5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(b*x**4+a),x)

[Out]

Piecewise((-a/(4*x**4) + b*log(x), Eq(m, -5)), (a*log(x) + b*x**4/4, Eq(m, -1)), (a*m*x*x**m/(m**2 + 6*m + 5)
+ 5*a*x*x**m/(m**2 + 6*m + 5) + b*m*x**5*x**m/(m**2 + 6*m + 5) + b*x**5*x**m/(m**2 + 6*m + 5), True))

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Giac [A]  time = 1.09412, size = 58, normalized size = 2.32 \begin{align*} \frac{b m x^{5} x^{m} + b x^{5} x^{m} + a m x x^{m} + 5 \, a x x^{m}}{m^{2} + 6 \, m + 5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(b*x^4+a),x, algorithm="giac")

[Out]

(b*m*x^5*x^m + b*x^5*x^m + a*m*x*x^m + 5*a*x*x^m)/(m^2 + 6*m + 5)