3.1.27 \(\int \frac {(a+b x^2)^2}{x^9} \, dx\)

Optimal. Leaf size=30 \[ -\frac {a^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {b^2}{4 x^4} \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} -\frac {a^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {b^2}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^9,x]

[Out]

-a^2/(8*x^8) - (a*b)/(3*x^6) - b^2/(4*x^4)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^9} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {(a+b x)^2}{x^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a^2}{x^5}+\frac {2 a b}{x^4}+\frac {b^2}{x^3}\right ) \, dx,x,x^2\right )\\ &=-\frac {a^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {b^2}{4 x^4}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} -\frac {a^2}{8 x^8}-\frac {a b}{3 x^6}-\frac {b^2}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^9,x]

[Out]

-1/8*a^2/x^8 - (a*b)/(3*x^6) - b^2/(4*x^4)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a+b x^2\right )^2}{x^9} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^2/x^9,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^2/x^9, x]

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fricas [A]  time = 1.13, size = 26, normalized size = 0.87 \begin {gather*} -\frac {6 \, b^{2} x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^9,x, algorithm="fricas")

[Out]

-1/24*(6*b^2*x^4 + 8*a*b*x^2 + 3*a^2)/x^8

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giac [A]  time = 1.09, size = 26, normalized size = 0.87 \begin {gather*} -\frac {6 \, b^{2} x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^9,x, algorithm="giac")

[Out]

-1/24*(6*b^2*x^4 + 8*a*b*x^2 + 3*a^2)/x^8

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maple [A]  time = 0.01, size = 25, normalized size = 0.83 \begin {gather*} -\frac {b^{2}}{4 x^{4}}-\frac {a b}{3 x^{6}}-\frac {a^{2}}{8 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^9,x)

[Out]

-1/8*a^2/x^8-1/3*a*b/x^6-1/4*b^2/x^4

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maxima [A]  time = 1.38, size = 26, normalized size = 0.87 \begin {gather*} -\frac {6 \, b^{2} x^{4} + 8 \, a b x^{2} + 3 \, a^{2}}{24 \, x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^9,x, algorithm="maxima")

[Out]

-1/24*(6*b^2*x^4 + 8*a*b*x^2 + 3*a^2)/x^8

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mupad [B]  time = 0.04, size = 26, normalized size = 0.87 \begin {gather*} -\frac {\frac {a^2}{8}+\frac {a\,b\,x^2}{3}+\frac {b^2\,x^4}{4}}{x^8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^9,x)

[Out]

-(a^2/8 + (b^2*x^4)/4 + (a*b*x^2)/3)/x^8

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sympy [A]  time = 0.21, size = 27, normalized size = 0.90 \begin {gather*} \frac {- 3 a^{2} - 8 a b x^{2} - 6 b^{2} x^{4}}{24 x^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**9,x)

[Out]

(-3*a**2 - 8*a*b*x**2 - 6*b**2*x**4)/(24*x**8)

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