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

Optimal. Leaf size=43 \[ -\frac {a^3}{9 x^9}-\frac {3 a^2 b}{7 x^7}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{3 x^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 43, 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} \begin {gather*} -\frac {3 a^2 b}{7 x^7}-\frac {a^3}{9 x^9}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^10,x]

[Out]

-a^3/(9*x^9) - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(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 )^3}{x^{10}} \, dx &=\int \left (\frac {a^3}{x^{10}}+\frac {3 a^2 b}{x^8}+\frac {3 a b^2}{x^6}+\frac {b^3}{x^4}\right ) \, dx\\ &=-\frac {a^3}{9 x^9}-\frac {3 a^2 b}{7 x^7}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{3 x^3}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 43, normalized size = 1.00 \begin {gather*} -\frac {a^3}{9 x^9}-\frac {3 a^2 b}{7 x^7}-\frac {3 a b^2}{5 x^5}-\frac {b^3}{3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^10,x]

[Out]

-1/9*a^3/x^9 - (3*a^2*b)/(7*x^7) - (3*a*b^2)/(5*x^5) - b^3/(3*x^3)

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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 )^3}{x^{10}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^3/x^10,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^3/x^10, x]

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fricas [A]  time = 1.20, size = 37, normalized size = 0.86 \begin {gather*} -\frac {105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(105*b^3*x^6 + 189*a*b^2*x^4 + 135*a^2*b*x^2 + 35*a^3)/x^9

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giac [A]  time = 1.20, size = 37, normalized size = 0.86 \begin {gather*} -\frac {105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(105*b^3*x^6 + 189*a*b^2*x^4 + 135*a^2*b*x^2 + 35*a^3)/x^9

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} -\frac {b^{3}}{3 x^{3}}-\frac {3 a \,b^{2}}{5 x^{5}}-\frac {3 a^{2} b}{7 x^{7}}-\frac {a^{3}}{9 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*a^3/x^9-3/7*a^2*b/x^7-3/5*a*b^2/x^5-1/3*b^3/x^3

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maxima [A]  time = 1.49, size = 37, normalized size = 0.86 \begin {gather*} -\frac {105 \, b^{3} x^{6} + 189 \, a b^{2} x^{4} + 135 \, a^{2} b x^{2} + 35 \, a^{3}}{315 \, x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/315*(105*b^3*x^6 + 189*a*b^2*x^4 + 135*a^2*b*x^2 + 35*a^3)/x^9

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mupad [B]  time = 0.03, size = 37, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^3}{9}+\frac {3\,a^2\,b\,x^2}{7}+\frac {3\,a\,b^2\,x^4}{5}+\frac {b^3\,x^6}{3}}{x^9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^10,x)

[Out]

-(a^3/9 + (b^3*x^6)/3 + (3*a^2*b*x^2)/7 + (3*a*b^2*x^4)/5)/x^9

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sympy [A]  time = 0.27, size = 39, normalized size = 0.91 \begin {gather*} \frac {- 35 a^{3} - 135 a^{2} b x^{2} - 189 a b^{2} x^{4} - 105 b^{3} x^{6}}{315 x^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-35*a**3 - 135*a**2*b*x**2 - 189*a*b**2*x**4 - 105*b**3*x**6)/(315*x**9)

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