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

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

[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)

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Rubi [A]  time = 0.0225537, antiderivative size = 69, 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} \[ -\frac{2 a^3 b^2}{3 x^{15}}-\frac{10 a^2 b^3}{13 x^{13}}-\frac{5 a^4 b}{17 x^{17}}-\frac{a^5}{19 x^{19}}-\frac{5 a b^4}{11 x^{11}}-\frac{b^5}{9 x^9} \]

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*}

Mathematica [A]  time = 0.0060701, size = 69, normalized size = 1. \[ -\frac{2 a^3 b^2}{3 x^{15}}-\frac{10 a^2 b^3}{13 x^{13}}-\frac{5 a^4 b}{17 x^{17}}-\frac{a^5}{19 x^{19}}-\frac{5 a b^4}{11 x^{11}}-\frac{b^5}{9 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[(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)

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Maple [A]  time = 0.005, size = 58, normalized size = 0.8 \begin{align*} -{\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*}

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 = 1.78504, size = 80, normalized size = 1.16 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.18763, size = 169, normalized size = 2.45 \begin{align*} -\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{align*}

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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Sympy [A]  time = 0.740657, size = 63, normalized size = 0.91 \begin{align*} - \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{align*}

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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Giac [A]  time = 2.47468, size = 80, normalized size = 1.16 \begin{align*} -\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{align*}

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