3.232 \(\int \frac {(a+b x^3)^2}{x^{10}} \, dx\)

Optimal. Leaf size=19 \[ -\frac {\left (a+b x^3\right )^3}{9 a x^9} \]

[Out]

-1/9*(b*x^3+a)^3/a/x^9

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Rubi [A]  time = 0.00, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {264} \[ -\frac {\left (a+b x^3\right )^3}{9 a x^9} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^3)^2/x^10,x]

[Out]

-(a + b*x^3)^3/(9*a*x^9)

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^3\right )^2}{x^{10}} \, dx &=-\frac {\left (a+b x^3\right )^3}{9 a x^9}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 30, normalized size = 1.58 \[ -\frac {a^2}{9 x^9}-\frac {a b}{3 x^6}-\frac {b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^3)^2/x^10,x]

[Out]

-1/9*a^2/x^9 - (a*b)/(3*x^6) - b^2/(3*x^3)

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fricas [A]  time = 0.80, size = 24, normalized size = 1.26 \[ -\frac {3 \, b^{2} x^{6} + 3 \, a b x^{3} + a^{2}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^10,x, algorithm="fricas")

[Out]

-1/9*(3*b^2*x^6 + 3*a*b*x^3 + a^2)/x^9

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giac [A]  time = 0.15, size = 24, normalized size = 1.26 \[ -\frac {3 \, b^{2} x^{6} + 3 \, a b x^{3} + a^{2}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^10,x, algorithm="giac")

[Out]

-1/9*(3*b^2*x^6 + 3*a*b*x^3 + a^2)/x^9

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maple [A]  time = 0.00, size = 25, normalized size = 1.32 \[ -\frac {b^{2}}{3 x^{3}}-\frac {a b}{3 x^{6}}-\frac {a^{2}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^2/x^10,x)

[Out]

-1/3*b^2/x^3-1/3*a*b/x^6-1/9*a^2/x^9

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maxima [A]  time = 1.40, size = 24, normalized size = 1.26 \[ -\frac {3 \, b^{2} x^{6} + 3 \, a b x^{3} + a^{2}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^2/x^10,x, algorithm="maxima")

[Out]

-1/9*(3*b^2*x^6 + 3*a*b*x^3 + a^2)/x^9

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mupad [B]  time = 0.03, size = 26, normalized size = 1.37 \[ -\frac {\frac {a^2}{9}+\frac {a\,b\,x^3}{3}+\frac {b^2\,x^6}{3}}{x^9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^3)^2/x^10,x)

[Out]

-(a^2/9 + (b^2*x^6)/3 + (a*b*x^3)/3)/x^9

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sympy [A]  time = 0.27, size = 26, normalized size = 1.37 \[ \frac {- a^{2} - 3 a b x^{3} - 3 b^{2} x^{6}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**2/x**10,x)

[Out]

(-a**2 - 3*a*b*x**3 - 3*b**2*x**6)/(9*x**9)

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