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

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

[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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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} \[ -\frac {10 a^3 b^2}{11 x^{11}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a^4 b}{13 x^{13}}-\frac {a^5}{15 x^{15}}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{5 x^5} \]

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*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \[ -\frac {a^5}{15 x^{15}}-\frac {5 a^4 b}{13 x^{13}}-\frac {10 a^3 b^2}{11 x^{11}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a b^4}{7 x^7}-\frac {b^5}{5 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^16,x]

[Out]

-1/15*a^5/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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fricas [A]  time = 0.70, size = 59, normalized size = 0.86 \[ -\frac {9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \]

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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giac [A]  time = 1.08, size = 59, normalized size = 0.86 \[ -\frac {9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \]

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

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maple [A]  time = 0.01, size = 58, normalized size = 0.84 \[ -\frac {b^{5}}{5 x^{5}}-\frac {5 a \,b^{4}}{7 x^{7}}-\frac {10 a^{2} b^{3}}{9 x^{9}}-\frac {10 a^{3} b^{2}}{11 x^{11}}-\frac {5 a^{4} b}{13 x^{13}}-\frac {a^{5}}{15 x^{15}} \]

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 = 1.37, size = 59, normalized size = 0.86 \[ -\frac {9009 \, b^{5} x^{10} + 32175 \, a b^{4} x^{8} + 50050 \, a^{2} b^{3} x^{6} + 40950 \, a^{3} b^{2} x^{4} + 17325 \, a^{4} b x^{2} + 3003 \, a^{5}}{45045 \, x^{15}} \]

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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mupad [B]  time = 4.75, size = 59, normalized size = 0.86 \[ -\frac {\frac {a^5}{15}+\frac {5\,a^4\,b\,x^2}{13}+\frac {10\,a^3\,b^2\,x^4}{11}+\frac {10\,a^2\,b^3\,x^6}{9}+\frac {5\,a\,b^4\,x^8}{7}+\frac {b^5\,x^{10}}{5}}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^16,x)

[Out]

-(a^5/15 + (b^5*x^10)/5 + (5*a^4*b*x^2)/13 + (5*a*b^4*x^8)/7 + (10*a^3*b^2*x^4)/11 + (10*a^2*b^3*x^6)/9)/x^15

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sympy [A]  time = 0.51, size = 63, normalized size = 0.91 \[ \frac {- 3003 a^{5} - 17325 a^{4} b x^{2} - 40950 a^{3} b^{2} x^{4} - 50050 a^{2} b^{3} x^{6} - 32175 a b^{4} x^{8} - 9009 b^{5} x^{10}}{45045 x^{15}} \]

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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