3.19.7 \(\int (a+\frac {b}{x^2}) x^3 \, dx\) [1807]

Optimal. Leaf size=17 \[ \frac {b x^2}{2}+\frac {a x^4}{4} \]

[Out]

1/2*b*x^2+1/4*a*x^4

________________________________________________________________________________________

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^4}{4}+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x^2)*x^3,x]

[Out]

(b*x^2)/2 + (a*x^4)/4

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

________________________________________________________________________________________

Mathematica [A]
time = 0.00, size = 17, normalized size = 1.00 \begin {gather*} \frac {b x^2}{2}+\frac {a x^4}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^2)*x^3,x]

[Out]

(b*x^2)/2 + (a*x^4)/4

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 15, normalized size = 0.88

method	result	size
gosper	\(\frac {x^{2} \left (a \,x^{2}+2 b \right )}{4}\)	\(15\)
default	\(\frac {\left (a \,x^{2}+b \right )^{2}}{4 a}\)	\(15\)
norman	\(\frac {\frac {1}{4} a \,x^{5}+\frac {1}{2} b \,x^{3}}{x}\)	\(18\)
risch	\(\frac {a \,x^{4}}{4}+\frac {b \,x^{2}}{2}+\frac {b^{2}}{4 a}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/x^2+a)*x^3,x,method=_RETURNVERBOSE)

[Out]

1/4*(a*x^2+b)^2/a

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^3,x, algorithm="maxima")

[Out]

1/4*a*x^4 + 1/2*b*x^2

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^3,x, algorithm="fricas")

[Out]

1/4*a*x^4 + 1/2*b*x^2

________________________________________________________________________________________

Sympy [A]
time = 0.01, size = 12, normalized size = 0.71 \begin {gather*} \frac {a x^{4}}{4} + \frac {b x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x**2)*x**3,x)

[Out]

a*x**4/4 + b*x**2/2

________________________________________________________________________________________

Giac [A]
time = 0.50, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x^2)*x^3,x, algorithm="giac")

[Out]

1/4*a*x^4 + 1/2*b*x^2

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 13, normalized size = 0.76 \begin {gather*} \frac {a\,x^4}{4}+\frac {b\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b/x^2),x)

[Out]

(a*x^4)/4 + (b*x^2)/2

________________________________________________________________________________________