3.65 \(\int x^m (a x+b x^3+c x^5) \, dx\)

Optimal. Leaf size=37 \[ \frac{a x^{m+2}}{m+2}+\frac{b x^{m+4}}{m+4}+\frac{c x^{m+6}}{m+6} \]

[Out]

(a*x^(2 + m))/(2 + m) + (b*x^(4 + m))/(4 + m) + (c*x^(6 + m))/(6 + m)

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Rubi [A]  time = 0.0129212, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {14} \[ \frac{a x^{m+2}}{m+2}+\frac{b x^{m+4}}{m+4}+\frac{c x^{m+6}}{m+6} \]

Antiderivative was successfully verified.

[In]

Int[x^m*(a*x + b*x^3 + c*x^5),x]

[Out]

(a*x^(2 + m))/(2 + m) + (b*x^(4 + m))/(4 + m) + (c*x^(6 + m))/(6 + 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 x+b x^3+c x^5\right ) \, dx &=\int \left (a x^{1+m}+b x^{3+m}+c x^{5+m}\right ) \, dx\\ &=\frac{a x^{2+m}}{2+m}+\frac{b x^{4+m}}{4+m}+\frac{c x^{6+m}}{6+m}\\ \end{align*}

Mathematica [A]  time = 0.0288071, size = 34, normalized size = 0.92 \[ x^{m+2} \left (\frac{a}{m+2}+\frac{b x^2}{m+4}+\frac{c x^4}{m+6}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x^m*(a*x + b*x^3 + c*x^5),x]

[Out]

x^(2 + m)*(a/(2 + m) + (b*x^2)/(4 + m) + (c*x^4)/(6 + m))

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Maple [B]  time = 0.003, size = 77, normalized size = 2.1 \begin{align*}{\frac{{x}^{2+m} \left ( c{m}^{2}{x}^{4}+6\,cm{x}^{4}+b{m}^{2}{x}^{2}+8\,c{x}^{4}+8\,bm{x}^{2}+a{m}^{2}+12\,b{x}^{2}+10\,am+24\,a \right ) }{ \left ( 6+m \right ) \left ( 4+m \right ) \left ( 2+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(c*x^5+b*x^3+a*x),x)

[Out]

x^(2+m)*(c*m^2*x^4+6*c*m*x^4+b*m^2*x^2+8*c*x^4+8*b*m*x^2+a*m^2+12*b*x^2+10*a*m+24*a)/(6+m)/(4+m)/(2+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*(c*x^5+b*x^3+a*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.31314, size = 161, normalized size = 4.35 \begin{align*} \frac{{\left ({\left (c m^{2} + 6 \, c m + 8 \, c\right )} x^{6} +{\left (b m^{2} + 8 \, b m + 12 \, b\right )} x^{4} +{\left (a m^{2} + 10 \, a m + 24 \, a\right )} x^{2}\right )} x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^5+b*x^3+a*x),x, algorithm="fricas")

[Out]

((c*m^2 + 6*c*m + 8*c)*x^6 + (b*m^2 + 8*b*m + 12*b)*x^4 + (a*m^2 + 10*a*m + 24*a)*x^2)*x^m/(m^3 + 12*m^2 + 44*
m + 48)

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Sympy [A]  time = 1.09027, size = 280, normalized size = 7.57 \begin{align*} \begin{cases} - \frac{a}{4 x^{4}} - \frac{b}{2 x^{2}} + c \log{\left (x \right )} & \text{for}\: m = -6 \\- \frac{a}{2 x^{2}} + b \log{\left (x \right )} + \frac{c x^{2}}{2} & \text{for}\: m = -4 \\a \log{\left (x \right )} + \frac{b x^{2}}{2} + \frac{c x^{4}}{4} & \text{for}\: m = -2 \\\frac{a m^{2} x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{10 a m x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{24 a x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{8 b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{12 b x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{c m^{2} x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{6 c m x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac{8 c x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(c*x**5+b*x**3+a*x),x)

[Out]

Piecewise((-a/(4*x**4) - b/(2*x**2) + c*log(x), Eq(m, -6)), (-a/(2*x**2) + b*log(x) + c*x**2/2, Eq(m, -4)), (a
*log(x) + b*x**2/2 + c*x**4/4, Eq(m, -2)), (a*m**2*x**2*x**m/(m**3 + 12*m**2 + 44*m + 48) + 10*a*m*x**2*x**m/(
m**3 + 12*m**2 + 44*m + 48) + 24*a*x**2*x**m/(m**3 + 12*m**2 + 44*m + 48) + b*m**2*x**4*x**m/(m**3 + 12*m**2 +
44*m + 48) + 8*b*m*x**4*x**m/(m**3 + 12*m**2 + 44*m + 48) + 12*b*x**4*x**m/(m**3 + 12*m**2 + 44*m + 48) + c*m
**2*x**6*x**m/(m**3 + 12*m**2 + 44*m + 48) + 6*c*m*x**6*x**m/(m**3 + 12*m**2 + 44*m + 48) + 8*c*x**6*x**m/(m**
3 + 12*m**2 + 44*m + 48), True))

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Giac [B]  time = 1.11225, size = 144, normalized size = 3.89 \begin{align*} \frac{c m^{2} x^{6} x^{m} + 6 \, c m x^{6} x^{m} + b m^{2} x^{4} x^{m} + 8 \, c x^{6} x^{m} + 8 \, b m x^{4} x^{m} + a m^{2} x^{2} x^{m} + 12 \, b x^{4} x^{m} + 10 \, a m x^{2} x^{m} + 24 \, a x^{2} x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(c*x^5+b*x^3+a*x),x, algorithm="giac")

[Out]

(c*m^2*x^6*x^m + 6*c*m*x^6*x^m + b*m^2*x^4*x^m + 8*c*x^6*x^m + 8*b*m*x^4*x^m + a*m^2*x^2*x^m + 12*b*x^4*x^m +
10*a*m*x^2*x^m + 24*a*x^2*x^m)/(m^3 + 12*m^2 + 44*m + 48)