3.51 \(\int \frac{(a+b x^2)^3}{x^{12}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{a^2 b}{3 x^9}-\frac{a^3}{11 x^{11}}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{5 x^5} \]

[Out]

-a^3/(11*x^11) - (a^2*b)/(3*x^9) - (3*a*b^2)/(7*x^7) - b^3/(5*x^5)

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Rubi [A]  time = 0.014571, antiderivative size = 43, normalized size of antiderivative = 1., 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} \[ -\frac{a^2 b}{3 x^9}-\frac{a^3}{11 x^{11}}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^3/x^12,x]

[Out]

-a^3/(11*x^11) - (a^2*b)/(3*x^9) - (3*a*b^2)/(7*x^7) - b^3/(5*x^5)

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^{12}} \, dx &=\int \left (\frac{a^3}{x^{12}}+\frac{3 a^2 b}{x^{10}}+\frac{3 a b^2}{x^8}+\frac{b^3}{x^6}\right ) \, dx\\ &=-\frac{a^3}{11 x^{11}}-\frac{a^2 b}{3 x^9}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0061936, size = 43, normalized size = 1. \[ -\frac{a^2 b}{3 x^9}-\frac{a^3}{11 x^{11}}-\frac{3 a b^2}{7 x^7}-\frac{b^3}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^12,x]

[Out]

-a^3/(11*x^11) - (a^2*b)/(3*x^9) - (3*a*b^2)/(7*x^7) - b^3/(5*x^5)

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Maple [A]  time = 0.003, size = 36, normalized size = 0.8 \begin{align*} -{\frac{{a}^{3}}{11\,{x}^{11}}}-{\frac{{a}^{2}b}{3\,{x}^{9}}}-{\frac{3\,a{b}^{2}}{7\,{x}^{7}}}-{\frac{{b}^{3}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3/x^12,x)

[Out]

-1/11*a^3/x^11-1/3*a^2*b/x^9-3/7*a*b^2/x^7-1/5*b^3/x^5

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Maxima [A]  time = 2.53168, size = 50, normalized size = 1.16 \begin{align*} -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^12,x, algorithm="maxima")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11

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Fricas [A]  time = 1.27293, size = 95, normalized size = 2.21 \begin{align*} -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^12,x, algorithm="fricas")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11

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Sympy [A]  time = 0.441176, size = 39, normalized size = 0.91 \begin{align*} - \frac{105 a^{3} + 385 a^{2} b x^{2} + 495 a b^{2} x^{4} + 231 b^{3} x^{6}}{1155 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3/x**12,x)

[Out]

-(105*a**3 + 385*a**2*b*x**2 + 495*a*b**2*x**4 + 231*b**3*x**6)/(1155*x**11)

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Giac [A]  time = 2.50031, size = 50, normalized size = 1.16 \begin{align*} -\frac{231 \, b^{3} x^{6} + 495 \, a b^{2} x^{4} + 385 \, a^{2} b x^{2} + 105 \, a^{3}}{1155 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3/x^12,x, algorithm="giac")

[Out]

-1/1155*(231*b^3*x^6 + 495*a*b^2*x^4 + 385*a^2*b*x^2 + 105*a^3)/x^11