3.19.5 \(\int (a+\frac {b}{x^2}) x^5 \, dx\) [1805]

Optimal. Leaf size=17 \[ \frac {b x^4}{4}+\frac {a x^6}{6} \]

[Out]

1/4*b*x^4+1/6*a*x^6

________________________________________________________________________________________

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^6}{6}+\frac {b x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*x^5,x]

[Out]

(b*x^4)/4 + (a*x^6)/6

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^4}{4}+\frac {a x^6}{6} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*x^5,x]

[Out]

(b*x^4)/4 + (a*x^6)/6

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 14, normalized size = 0.82

method	result	size
default	\(\frac {1}{4} b \,x^{4}+\frac {1}{6} x^{6} a\)	\(14\)
risch	\(\frac {1}{4} b \,x^{4}+\frac {1}{6} x^{6} a\)	\(14\)
gosper	\(\frac {x^{4} \left (2 a \,x^{2}+3 b \right )}{12}\)	\(16\)
norman	\(\frac {\frac {1}{6} a \,x^{7}+\frac {1}{4} b \,x^{5}}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/x^2+a)*x^5,x,method=_RETURNVERBOSE)

[Out]

1/4*b*x^4+1/6*x^6*a

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, a x^{6} + \frac {1}{4} \, b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^5,x, algorithm="maxima")

[Out]

1/6*a*x^6 + 1/4*b*x^4

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, a x^{6} + \frac {1}{4} \, b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^5,x, algorithm="fricas")

[Out]

1/6*a*x^6 + 1/4*b*x^4

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{6}}{6} + \frac {b x^{4}}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**5,x)

[Out]

a*x**6/6 + b*x**4/4

________________________________________________________________________________________

Giac [A]
time = 0.56, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{6} \, a x^{6} + \frac {1}{4} \, b x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^5,x, algorithm="giac")

[Out]

1/6*a*x^6 + 1/4*b*x^4

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {a\,x^6}{6}+\frac {b\,x^4}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/x^2),x)

[Out]

(a*x^6)/6 + (b*x^4)/4

________________________________________________________________________________________