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

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

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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 {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} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 43, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^3/x^12,x]

[Out]

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

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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^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^3/x^12,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^3/x^12, x]

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fricas [A]  time = 1.00, size = 37, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 1.03, size = 37, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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

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maple [A]  time = 0.00, size = 36, normalized size = 0.84 \begin {gather*} -\frac {b^{3}}{5 x^{5}}-\frac {3 a \,b^{2}}{7 x^{7}}-\frac {a^{2} b}{3 x^{9}}-\frac {a^{3}}{11 x^{11}} \end {gather*}

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 = 1.42, size = 37, normalized size = 0.86 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 0.03, size = 37, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^3}{11}+\frac {a^2\,b\,x^2}{3}+\frac {3\,a\,b^2\,x^4}{7}+\frac {b^3\,x^6}{5}}{x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^3/x^12,x)

[Out]

-(a^3/11 + (b^3*x^6)/5 + (a^2*b*x^2)/3 + (3*a*b^2*x^4)/7)/x^11

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sympy [A]  time = 0.30, size = 39, normalized size = 0.91 \begin {gather*} \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 {gather*}

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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