3.5.19 \(\int \frac {a^2+2 a b x^2+b^2 x^4}{x^7} \, dx\) [419]

Optimal. Leaf size=30 \[ -\frac {a^2}{6 x^6}-\frac {a b}{2 x^4}-\frac {b^2}{2 x^2} \]

[Out]

-1/6*a^2/x^6-1/2*a*b/x^4-1/2*b^2/x^2

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {14} \begin {gather*} -\frac {a^2}{6 x^6}-\frac {a b}{2 x^4}-\frac {b^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7,x]

[Out]

-1/6*a^2/x^6 - (a*b)/(2*x^4) - b^2/(2*x^2)

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

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Mathematica [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{6 x^6}-\frac {a b}{2 x^4}-\frac {b^2}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^7,x]

[Out]

-1/6*a^2/x^6 - (a*b)/(2*x^4) - b^2/(2*x^2)

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Maple [A]
time = 0.01, size = 25, normalized size = 0.83

method	result	size
gosper	\(-\frac {3 b^{2} x^{4}+3 a b \,x^{2}+a^{2}}{6 x^{6}}\)	\(25\)
default	\(-\frac {a^{2}}{6 x^{6}}-\frac {a b}{2 x^{4}}-\frac {b^{2}}{2 x^{2}}\)	\(25\)
norman	\(\frac {-\frac {1}{2} b^{2} x^{4}-\frac {1}{2} a b \,x^{2}-\frac {1}{6} a^{2}}{x^{6}}\)	\(26\)
risch	\(\frac {-\frac {1}{2} b^{2} x^{4}-\frac {1}{2} a b \,x^{2}-\frac {1}{6} a^{2}}{x^{6}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b^2*x^4+2*a*b*x^2+a^2)/x^7,x,method=_RETURNVERBOSE)

[Out]

-1/6*a^2/x^6-1/2*a*b/x^4-1/2*b^2/x^2

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Maxima [A]
time = 0.30, size = 24, normalized size = 0.80 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^7,x, algorithm="maxima")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Fricas [A]
time = 0.34, size = 24, normalized size = 0.80 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^7,x, algorithm="fricas")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Sympy [A]
time = 0.07, size = 26, normalized size = 0.87 \begin {gather*} \frac {- a^{2} - 3 a b x^{2} - 3 b^{2} x^{4}}{6 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b**2*x**4+2*a*b*x**2+a**2)/x**7,x)

[Out]

(-a**2 - 3*a*b*x**2 - 3*b**2*x**4)/(6*x**6)

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Giac [A]
time = 3.65, size = 24, normalized size = 0.80 \begin {gather*} -\frac {3 \, b^{2} x^{4} + 3 \, a b x^{2} + a^{2}}{6 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b^2*x^4+2*a*b*x^2+a^2)/x^7,x, algorithm="giac")

[Out]

-1/6*(3*b^2*x^4 + 3*a*b*x^2 + a^2)/x^6

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Mupad [B]
time = 0.03, size = 26, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^2}{6}+\frac {a\,b\,x^2}{2}+\frac {b^2\,x^4}{2}}{x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2 + b^2*x^4 + 2*a*b*x^2)/x^7,x)

[Out]

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

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