3.26 \(\int \frac {(a+b x^2)^2}{x^8} \, dx\)

Optimal. Leaf size=30 \[ -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2}{3 x^3} \]

[Out]

-1/7*a^2/x^7-2/5*a*b/x^5-1/3*b^2/x^3

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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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^2/x^8,x]

[Out]

-a^2/(7*x^7) - (2*a*b)/(5*x^5) - b^2/(3*x^3)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^2\right )^2}{x^8} \, dx &=\int \left (\frac {a^2}{x^8}+\frac {2 a b}{x^6}+\frac {b^2}{x^4}\right ) \, dx\\ &=-\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.00 \[ -\frac {a^2}{7 x^7}-\frac {2 a b}{5 x^5}-\frac {b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^2/x^8,x]

[Out]

-1/7*a^2/x^7 - (2*a*b)/(5*x^5) - b^2/(3*x^3)

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fricas [A]  time = 0.79, size = 26, normalized size = 0.87 \[ -\frac {35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^8,x, algorithm="fricas")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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giac [A]  time = 1.06, size = 26, normalized size = 0.87 \[ -\frac {35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^8,x, algorithm="giac")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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maple [A]  time = 0.00, size = 25, normalized size = 0.83 \[ -\frac {b^{2}}{3 x^{3}}-\frac {2 a b}{5 x^{5}}-\frac {a^{2}}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2/x^8,x)

[Out]

-1/7*a^2/x^7-2/5*a*b/x^5-1/3*b^2/x^3

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maxima [A]  time = 1.38, size = 26, normalized size = 0.87 \[ -\frac {35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^2/x^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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mupad [B]  time = 0.04, size = 26, normalized size = 0.87 \[ -\frac {\frac {a^2}{7}+\frac {2\,a\,b\,x^2}{5}+\frac {b^2\,x^4}{3}}{x^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^2/x^8,x)

[Out]

-(a^2/7 + (b^2*x^4)/3 + (2*a*b*x^2)/5)/x^7

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sympy [A]  time = 0.20, size = 27, normalized size = 0.90 \[ \frac {- 15 a^{2} - 42 a b x^{2} - 35 b^{2} x^{4}}{105 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2/x**8,x)

[Out]

(-15*a**2 - 42*a*b*x**2 - 35*b**2*x**4)/(105*x**7)

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