3.82 \(\int \frac{(a+b x^2)^5}{x^{16}} \, dx\)

Optimal. Leaf size=69 \[ -\frac{10 a^3 b^2}{11 x^{11}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a^4 b}{13 x^{13}}-\frac{a^5}{15 x^{15}}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{5 x^5} \]

[Out]

-a^5/(15*x^15) - (5*a^4*b)/(13*x^13) - (10*a^3*b^2)/(11*x^11) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(7*x^7) - b^5
/(5*x^5)

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Rubi [A]  time = 0.0231295, 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} \[ -\frac{10 a^3 b^2}{11 x^{11}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a^4 b}{13 x^{13}}-\frac{a^5}{15 x^{15}}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^16,x]

[Out]

-a^5/(15*x^15) - (5*a^4*b)/(13*x^13) - (10*a^3*b^2)/(11*x^11) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(7*x^7) - b^5
/(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 )^5}{x^{16}} \, dx &=\int \left (\frac{a^5}{x^{16}}+\frac{5 a^4 b}{x^{14}}+\frac{10 a^3 b^2}{x^{12}}+\frac{10 a^2 b^3}{x^{10}}+\frac{5 a b^4}{x^8}+\frac{b^5}{x^6}\right ) \, dx\\ &=-\frac{a^5}{15 x^{15}}-\frac{5 a^4 b}{13 x^{13}}-\frac{10 a^3 b^2}{11 x^{11}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{5 x^5}\\ \end{align*}

Mathematica [A]  time = 0.0040832, size = 69, normalized size = 1. \[ -\frac{10 a^3 b^2}{11 x^{11}}-\frac{10 a^2 b^3}{9 x^9}-\frac{5 a^4 b}{13 x^{13}}-\frac{a^5}{15 x^{15}}-\frac{5 a b^4}{7 x^7}-\frac{b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^16,x]

[Out]

-a^5/(15*x^15) - (5*a^4*b)/(13*x^13) - (10*a^3*b^2)/(11*x^11) - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(7*x^7) - b^5
/(5*x^5)

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Maple [A]  time = 0.004, size = 58, normalized size = 0.8 \begin{align*} -{\frac{{a}^{5}}{15\,{x}^{15}}}-{\frac{5\,{a}^{4}b}{13\,{x}^{13}}}-{\frac{10\,{a}^{3}{b}^{2}}{11\,{x}^{11}}}-{\frac{10\,{a}^{2}{b}^{3}}{9\,{x}^{9}}}-{\frac{5\,a{b}^{4}}{7\,{x}^{7}}}-{\frac{{b}^{5}}{5\,{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^16,x)

[Out]

-1/15*a^5/x^15-5/13*a^4*b/x^13-10/11*a^3*b^2/x^11-10/9*a^2*b^3/x^9-5/7*a*b^4/x^7-1/5*b^5/x^5

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Maxima [A]  time = 2.30062, size = 80, normalized size = 1.16 \begin{align*} -\frac{9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^16,x, algorithm="maxima")

[Out]

-1/45045*(9009*b^5*x^10 + 32175*a*b^4*x^8 + 50050*a^2*b^3*x^6 + 40950*a^3*b^2*x^4 + 17325*a^4*b*x^2 + 3003*a^5
)/x^15

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Fricas [A]  time = 1.23189, size = 159, normalized size = 2.3 \begin{align*} -\frac{9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^16,x, algorithm="fricas")

[Out]

-1/45045*(9009*b^5*x^10 + 32175*a*b^4*x^8 + 50050*a^2*b^3*x^6 + 40950*a^3*b^2*x^4 + 17325*a^4*b*x^2 + 3003*a^5
)/x^15

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Sympy [A]  time = 0.729849, size = 63, normalized size = 0.91 \begin{align*} - \frac{3003 a^{5} + 17325 a^{4} b x^{2} + 40950 a^{3} b^{2} x^{4} + 50050 a^{2} b^{3} x^{6} + 32175 a b^{4} x^{8} + 9009 b^{5} x^{10}}{45045 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**16,x)

[Out]

-(3003*a**5 + 17325*a**4*b*x**2 + 40950*a**3*b**2*x**4 + 50050*a**2*b**3*x**6 + 32175*a*b**4*x**8 + 9009*b**5*
x**10)/(45045*x**15)

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Giac [A]  time = 2.1227, size = 80, normalized size = 1.16 \begin{align*} -\frac{9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^16,x, algorithm="giac")

[Out]

-1/45045*(9009*b^5*x^10 + 32175*a*b^4*x^8 + 50050*a^2*b^3*x^6 + 40950*a^3*b^2*x^4 + 17325*a^4*b*x^2 + 3003*a^5
)/x^15