3.19.6 \(\int (a+\frac {b}{x^2}) x^4 \, dx\) [1806]

Optimal. Leaf size=17 \[ \frac {b x^3}{3}+\frac {a x^5}{5} \]

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 17, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {a x^5}{5}+\frac {b x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*x^4,x]

[Out]

(b*x^3)/3 + (a*x^5)/5

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

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Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^3}{3}+\frac {a x^5}{5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*x^4,x]

[Out]

(b*x^3)/3 + (a*x^5)/5

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Maple [A]
time = 0.06, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {1}{3} b \,x^{3}+\frac {1}{5} a \,x^{5}\)	\(14\)
risch	\(\frac {1}{3} b \,x^{3}+\frac {1}{5} a \,x^{5}\)	\(14\)
gosper	\(\frac {x^{3} \left (3 a \,x^{2}+5 b \right )}{15}\)	\(16\)
norman	\(\frac {\frac {1}{3} b \,x^{4}+\frac {1}{5} x^{6} a}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/x^2+a)*x^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, a x^{5} + \frac {1}{3} \, b x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^4,x, algorithm="maxima")

[Out]

1/5*a*x^5 + 1/3*b*x^3

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Fricas [A]
time = 0.38, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, a x^{5} + \frac {1}{3} \, b x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^4,x, algorithm="fricas")

[Out]

1/5*a*x^5 + 1/3*b*x^3

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Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{5}}{5} + \frac {b x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**4,x)

[Out]

a*x**5/5 + b*x**3/3

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Giac [A]
time = 0.53, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{5} \, a x^{5} + \frac {1}{3} \, b x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^4,x, algorithm="giac")

[Out]

1/5*a*x^5 + 1/3*b*x^3

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {a\,x^5}{5}+\frac {b\,x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b/x^2),x)

[Out]

(a*x^5)/5 + (b*x^3)/3

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