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 = 0.01, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.66, size = 71, normalized size = 1.92 \[ \frac {{\left ({\left (c m^{2} + 6 \, c m + 8 \, c\right )} x^{6} + {\left (b m^{2} + 8 \, b m + 12 \, b\right )} x^{4} + {\left (a m^{2} + 10 \, a m + 24 \, a\right )} x^{2}\right )} x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} \]

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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giac [B]  time = 0.48, size = 107, normalized size = 2.89 \[ \frac {c m^{2} x^{6} x^{m} + 6 \, c m x^{6} x^{m} + b m^{2} x^{4} x^{m} + 8 \, c x^{6} x^{m} + 8 \, b m x^{4} x^{m} + a m^{2} x^{2} x^{m} + 12 \, b x^{4} x^{m} + 10 \, a m x^{2} x^{m} + 24 \, a x^{2} x^{m}}{m^{3} + 12 \, m^{2} + 44 \, m + 48} \]

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)

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maple [B]  time = 0.00, size = 77, normalized size = 2.08 \[ \frac {\left (c \,m^{2} x^{4}+6 c m \,x^{4}+b \,m^{2} x^{2}+8 c \,x^{4}+8 b m \,x^{2}+a \,m^{2}+12 b \,x^{2}+10 a m +24 a \right ) x^{m +2}}{\left (m +6\right ) \left (m +4\right ) \left (m +2\right )} \]

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^(m+2)*(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)/(m+6)/(m+4)/(m+2)

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maxima [A]  time = 0.45, size = 37, normalized size = 1.00 \[ \frac {c x^{m + 6}}{m + 6} + \frac {b x^{m + 4}}{m + 4} + \frac {a x^{m + 2}}{m + 2} \]

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]

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

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mupad [B]  time = 2.08, size = 89, normalized size = 2.41 \[ x^m\,\left (\frac {a\,x^2\,\left (m^2+10\,m+24\right )}{m^3+12\,m^2+44\,m+48}+\frac {b\,x^4\,\left (m^2+8\,m+12\right )}{m^3+12\,m^2+44\,m+48}+\frac {c\,x^6\,\left (m^2+6\,m+8\right )}{m^3+12\,m^2+44\,m+48}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*(a*x + b*x^3 + c*x^5),x)

[Out]

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

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sympy [A]  time = 1.18, size = 280, normalized size = 7.57 \[ \begin {cases} - \frac {a}{4 x^{4}} - \frac {b}{2 x^{2}} + c \log {\relax (x )} & \text {for}\: m = -6 \\- \frac {a}{2 x^{2}} + b \log {\relax (x )} + \frac {c x^{2}}{2} & \text {for}\: m = -4 \\a \log {\relax (x )} + \frac {b x^{2}}{2} + \frac {c x^{4}}{4} & \text {for}\: m = -2 \\\frac {a m^{2} x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {10 a m x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {24 a x^{2} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {b m^{2} x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {8 b m x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {12 b x^{4} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {c m^{2} x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {6 c m x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} + \frac {8 c x^{6} x^{m}}{m^{3} + 12 m^{2} + 44 m + 48} & \text {otherwise} \end {cases} \]

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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