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

Optimal. Leaf size=25 \[ \frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \]

[Out]

1/3*a*x^3+1/5*b*x^5+1/7*c*x^7

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Rubi [A]  time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {14} \[ \frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[x*(a*x + b*x^3 + c*x^5),x]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

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 \left (a x+b x^3+c x^5\right ) \, dx &=\int \left (a x^2+b x^4+c x^6\right ) \, dx\\ &=\frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 25, normalized size = 1.00 \[ \frac {a x^3}{3}+\frac {b x^5}{5}+\frac {c x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a*x + b*x^3 + c*x^5),x]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

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fricas [A]  time = 0.52, size = 19, normalized size = 0.76 \[ \frac {1}{7} x^{7} c + \frac {1}{5} x^{5} b + \frac {1}{3} x^{3} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c + 1/5*x^5*b + 1/3*x^3*a

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giac [A]  time = 0.39, size = 19, normalized size = 0.76 \[ \frac {1}{7} \, c x^{7} + \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \[ \frac {1}{7} c \,x^{7}+\frac {1}{5} b \,x^{5}+\frac {1}{3} a \,x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3+1/5*b*x^5+1/7*c*x^7

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maxima [A]  time = 0.43, size = 19, normalized size = 0.76 \[ \frac {1}{7} \, c x^{7} + \frac {1}{5} \, b x^{5} + \frac {1}{3} \, a x^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3

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mupad [B]  time = 0.03, size = 19, normalized size = 0.76 \[ \frac {c\,x^7}{7}+\frac {b\,x^5}{5}+\frac {a\,x^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a*x + b*x^3 + c*x^5),x)

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

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sympy [A]  time = 0.06, size = 19, normalized size = 0.76 \[ \frac {a x^{3}}{3} + \frac {b x^{5}}{5} + \frac {c x^{7}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**5/5 + c*x**7/7

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