3.1.84 \(\int \frac {(a+b x^2)^5}{x^{20}} \, dx\)

Optimal. Leaf size=69 \[ -\frac {a^5}{19 x^{19}}-\frac {5 a^4 b}{17 x^{17}}-\frac {2 a^3 b^2}{3 x^{15}}-\frac {10 a^2 b^3}{13 x^{13}}-\frac {5 a b^4}{11 x^{11}}-\frac {b^5}{9 x^9} \]

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Rubi [A]  time = 0.02, antiderivative size = 69, 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 {2 a^3 b^2}{3 x^{15}}-\frac {10 a^2 b^3}{13 x^{13}}-\frac {5 a^4 b}{17 x^{17}}-\frac {a^5}{19 x^{19}}-\frac {5 a b^4}{11 x^{11}}-\frac {b^5}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^20,x]

[Out]

-a^5/(19*x^19) - (5*a^4*b)/(17*x^17) - (2*a^3*b^2)/(3*x^15) - (10*a^2*b^3)/(13*x^13) - (5*a*b^4)/(11*x^11) - b
^5/(9*x^9)

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^{20}} \, dx &=\int \left (\frac {a^5}{x^{20}}+\frac {5 a^4 b}{x^{18}}+\frac {10 a^3 b^2}{x^{16}}+\frac {10 a^2 b^3}{x^{14}}+\frac {5 a b^4}{x^{12}}+\frac {b^5}{x^{10}}\right ) \, dx\\ &=-\frac {a^5}{19 x^{19}}-\frac {5 a^4 b}{17 x^{17}}-\frac {2 a^3 b^2}{3 x^{15}}-\frac {10 a^2 b^3}{13 x^{13}}-\frac {5 a b^4}{11 x^{11}}-\frac {b^5}{9 x^9}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 69, normalized size = 1.00 \begin {gather*} -\frac {a^5}{19 x^{19}}-\frac {5 a^4 b}{17 x^{17}}-\frac {2 a^3 b^2}{3 x^{15}}-\frac {10 a^2 b^3}{13 x^{13}}-\frac {5 a b^4}{11 x^{11}}-\frac {b^5}{9 x^9} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^20,x]

[Out]

-1/19*a^5/x^19 - (5*a^4*b)/(17*x^17) - (2*a^3*b^2)/(3*x^15) - (10*a^2*b^3)/(13*x^13) - (5*a*b^4)/(11*x^11) - b
^5/(9*x^9)

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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^{20}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^20,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^20, x]

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fricas [A]  time = 1.08, size = 59, normalized size = 0.86 \begin {gather*} -\frac {46189 \, b^{5} x^{10} + 188955 \, a b^{4} x^{8} + 319770 \, a^{2} b^{3} x^{6} + 277134 \, a^{3} b^{2} x^{4} + 122265 \, a^{4} b x^{2} + 21879 \, a^{5}}{415701 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^20,x, algorithm="fricas")

[Out]

-1/415701*(46189*b^5*x^10 + 188955*a*b^4*x^8 + 319770*a^2*b^3*x^6 + 277134*a^3*b^2*x^4 + 122265*a^4*b*x^2 + 21
879*a^5)/x^19

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giac [A]  time = 1.13, size = 59, normalized size = 0.86 \begin {gather*} -\frac {46189 \, b^{5} x^{10} + 188955 \, a b^{4} x^{8} + 319770 \, a^{2} b^{3} x^{6} + 277134 \, a^{3} b^{2} x^{4} + 122265 \, a^{4} b x^{2} + 21879 \, a^{5}}{415701 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^20,x, algorithm="giac")

[Out]

-1/415701*(46189*b^5*x^10 + 188955*a*b^4*x^8 + 319770*a^2*b^3*x^6 + 277134*a^3*b^2*x^4 + 122265*a^4*b*x^2 + 21
879*a^5)/x^19

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maple [A]  time = 0.01, size = 58, normalized size = 0.84 \begin {gather*} -\frac {b^{5}}{9 x^{9}}-\frac {5 a \,b^{4}}{11 x^{11}}-\frac {10 a^{2} b^{3}}{13 x^{13}}-\frac {2 a^{3} b^{2}}{3 x^{15}}-\frac {5 a^{4} b}{17 x^{17}}-\frac {a^{5}}{19 x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^20,x)

[Out]

-1/19*a^5/x^19-5/17*a^4*b/x^17-2/3*a^3*b^2/x^15-10/13*a^2*b^3/x^13-5/11*a*b^4/x^11-1/9*b^5/x^9

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maxima [A]  time = 1.39, size = 59, normalized size = 0.86 \begin {gather*} -\frac {46189 \, b^{5} x^{10} + 188955 \, a b^{4} x^{8} + 319770 \, a^{2} b^{3} x^{6} + 277134 \, a^{3} b^{2} x^{4} + 122265 \, a^{4} b x^{2} + 21879 \, a^{5}}{415701 \, x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^20,x, algorithm="maxima")

[Out]

-1/415701*(46189*b^5*x^10 + 188955*a*b^4*x^8 + 319770*a^2*b^3*x^6 + 277134*a^3*b^2*x^4 + 122265*a^4*b*x^2 + 21
879*a^5)/x^19

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mupad [B]  time = 0.04, size = 59, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^5}{19}+\frac {5\,a^4\,b\,x^2}{17}+\frac {2\,a^3\,b^2\,x^4}{3}+\frac {10\,a^2\,b^3\,x^6}{13}+\frac {5\,a\,b^4\,x^8}{11}+\frac {b^5\,x^{10}}{9}}{x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^20,x)

[Out]

-(a^5/19 + (b^5*x^10)/9 + (5*a^4*b*x^2)/17 + (5*a*b^4*x^8)/11 + (2*a^3*b^2*x^4)/3 + (10*a^2*b^3*x^6)/13)/x^19

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sympy [A]  time = 0.57, size = 63, normalized size = 0.91 \begin {gather*} \frac {- 21879 a^{5} - 122265 a^{4} b x^{2} - 277134 a^{3} b^{2} x^{4} - 319770 a^{2} b^{3} x^{6} - 188955 a b^{4} x^{8} - 46189 b^{5} x^{10}}{415701 x^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**20,x)

[Out]

(-21879*a**5 - 122265*a**4*b*x**2 - 277134*a**3*b**2*x**4 - 319770*a**2*b**3*x**6 - 188955*a*b**4*x**8 - 46189
*b**5*x**10)/(415701*x**19)

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