3.1.80 \(\int \frac {(a+b x^2)^5}{x^{12}} \, dx\)

Optimal. Leaf size=65 \[ -\frac {a^5}{11 x^{11}}-\frac {5 a^4 b}{9 x^9}-\frac {10 a^3 b^2}{7 x^7}-\frac {2 a^2 b^3}{x^5}-\frac {5 a b^4}{3 x^3}-\frac {b^5}{x} \]

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Rubi [A]  time = 0.02, antiderivative size = 65, 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 {10 a^3 b^2}{7 x^7}-\frac {2 a^2 b^3}{x^5}-\frac {5 a^4 b}{9 x^9}-\frac {a^5}{11 x^{11}}-\frac {5 a b^4}{3 x^3}-\frac {b^5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^12,x]

[Out]

-a^5/(11*x^11) - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x

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 )^5}{x^{12}} \, dx &=\int \left (\frac {a^5}{x^{12}}+\frac {5 a^4 b}{x^{10}}+\frac {10 a^3 b^2}{x^8}+\frac {10 a^2 b^3}{x^6}+\frac {5 a b^4}{x^4}+\frac {b^5}{x^2}\right ) \, dx\\ &=-\frac {a^5}{11 x^{11}}-\frac {5 a^4 b}{9 x^9}-\frac {10 a^3 b^2}{7 x^7}-\frac {2 a^2 b^3}{x^5}-\frac {5 a b^4}{3 x^3}-\frac {b^5}{x}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 65, normalized size = 1.00 \begin {gather*} -\frac {a^5}{11 x^{11}}-\frac {5 a^4 b}{9 x^9}-\frac {10 a^3 b^2}{7 x^7}-\frac {2 a^2 b^3}{x^5}-\frac {5 a b^4}{3 x^3}-\frac {b^5}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^12,x]

[Out]

-1/11*a^5/x^11 - (5*a^4*b)/(9*x^9) - (10*a^3*b^2)/(7*x^7) - (2*a^2*b^3)/x^5 - (5*a*b^4)/(3*x^3) - b^5/x

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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 )^5}{x^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^12,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^12, x]

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fricas [A]  time = 0.73, size = 59, normalized size = 0.91 \begin {gather*} -\frac {693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^12,x, algorithm="fricas")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385*a^4*b*x^2 + 63*a^5)/x^11

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giac [A]  time = 1.04, size = 59, normalized size = 0.91 \begin {gather*} -\frac {693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^12,x, algorithm="giac")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385*a^4*b*x^2 + 63*a^5)/x^11

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maple [A]  time = 0.01, size = 58, normalized size = 0.89 \begin {gather*} -\frac {b^{5}}{x}-\frac {5 a \,b^{4}}{3 x^{3}}-\frac {2 a^{2} b^{3}}{x^{5}}-\frac {10 a^{3} b^{2}}{7 x^{7}}-\frac {5 a^{4} b}{9 x^{9}}-\frac {a^{5}}{11 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^12,x)

[Out]

-1/11*a^5/x^11-5/9*a^4*b/x^9-10/7*a^3*b^2/x^7-2*a^2*b^3/x^5-5/3*a*b^4/x^3-b^5/x

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maxima [A]  time = 1.36, size = 59, normalized size = 0.91 \begin {gather*} -\frac {693 \, b^{5} x^{10} + 1155 \, a b^{4} x^{8} + 1386 \, a^{2} b^{3} x^{6} + 990 \, a^{3} b^{2} x^{4} + 385 \, a^{4} b x^{2} + 63 \, a^{5}}{693 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^12,x, algorithm="maxima")

[Out]

-1/693*(693*b^5*x^10 + 1155*a*b^4*x^8 + 1386*a^2*b^3*x^6 + 990*a^3*b^2*x^4 + 385*a^4*b*x^2 + 63*a^5)/x^11

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mupad [B]  time = 0.04, size = 58, normalized size = 0.89 \begin {gather*} -\frac {\frac {a^5}{11}+\frac {5\,a^4\,b\,x^2}{9}+\frac {10\,a^3\,b^2\,x^4}{7}+2\,a^2\,b^3\,x^6+\frac {5\,a\,b^4\,x^8}{3}+b^5\,x^{10}}{x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^12,x)

[Out]

-(a^5/11 + b^5*x^10 + (5*a^4*b*x^2)/9 + (5*a*b^4*x^8)/3 + (10*a^3*b^2*x^4)/7 + 2*a^2*b^3*x^6)/x^11

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sympy [A]  time = 0.43, size = 63, normalized size = 0.97 \begin {gather*} \frac {- 63 a^{5} - 385 a^{4} b x^{2} - 990 a^{3} b^{2} x^{4} - 1386 a^{2} b^{3} x^{6} - 1155 a b^{4} x^{8} - 693 b^{5} x^{10}}{693 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**12,x)

[Out]

(-63*a**5 - 385*a**4*b*x**2 - 990*a**3*b**2*x**4 - 1386*a**2*b**3*x**6 - 1155*a*b**4*x**8 - 693*b**5*x**10)/(6
93*x**11)

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