3.1.83 \(\int \frac {(a+b x^2)^5}{x^{18}} \, dx\)

Optimal. Leaf size=69 \[ -\frac {a^5}{17 x^{17}}-\frac {a^4 b}{3 x^{15}}-\frac {10 a^3 b^2}{13 x^{13}}-\frac {10 a^2 b^3}{11 x^{11}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{7 x^7} \]

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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 {10 a^3 b^2}{13 x^{13}}-\frac {10 a^2 b^3}{11 x^{11}}-\frac {a^4 b}{3 x^{15}}-\frac {a^5}{17 x^{17}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^5/x^18,x]

[Out]

-a^5/(17*x^17) - (a^4*b)/(3*x^15) - (10*a^3*b^2)/(13*x^13) - (10*a^2*b^3)/(11*x^11) - (5*a*b^4)/(9*x^9) - b^5/
(7*x^7)

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^{18}} \, dx &=\int \left (\frac {a^5}{x^{18}}+\frac {5 a^4 b}{x^{16}}+\frac {10 a^3 b^2}{x^{14}}+\frac {10 a^2 b^3}{x^{12}}+\frac {5 a b^4}{x^{10}}+\frac {b^5}{x^8}\right ) \, dx\\ &=-\frac {a^5}{17 x^{17}}-\frac {a^4 b}{3 x^{15}}-\frac {10 a^3 b^2}{13 x^{13}}-\frac {10 a^2 b^3}{11 x^{11}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{7 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \begin {gather*} -\frac {a^5}{17 x^{17}}-\frac {a^4 b}{3 x^{15}}-\frac {10 a^3 b^2}{13 x^{13}}-\frac {10 a^2 b^3}{11 x^{11}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^5/x^18,x]

[Out]

-1/17*a^5/x^17 - (a^4*b)/(3*x^15) - (10*a^3*b^2)/(13*x^13) - (10*a^2*b^3)/(11*x^11) - (5*a*b^4)/(9*x^9) - b^5/
(7*x^7)

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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^{18}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(a + b*x^2)^5/x^18,x]

[Out]

IntegrateAlgebraic[(a + b*x^2)^5/x^18, x]

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fricas [A]  time = 1.10, size = 59, normalized size = 0.86 \begin {gather*} -\frac {21879 \, b^{5} x^{10} + 85085 \, a b^{4} x^{8} + 139230 \, a^{2} b^{3} x^{6} + 117810 \, a^{3} b^{2} x^{4} + 51051 \, a^{4} b x^{2} + 9009 \, a^{5}}{153153 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^18,x, algorithm="fricas")

[Out]

-1/153153*(21879*b^5*x^10 + 85085*a*b^4*x^8 + 139230*a^2*b^3*x^6 + 117810*a^3*b^2*x^4 + 51051*a^4*b*x^2 + 9009
*a^5)/x^17

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giac [A]  time = 1.10, size = 59, normalized size = 0.86 \begin {gather*} -\frac {21879 \, b^{5} x^{10} + 85085 \, a b^{4} x^{8} + 139230 \, a^{2} b^{3} x^{6} + 117810 \, a^{3} b^{2} x^{4} + 51051 \, a^{4} b x^{2} + 9009 \, a^{5}}{153153 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^18,x, algorithm="giac")

[Out]

-1/153153*(21879*b^5*x^10 + 85085*a*b^4*x^8 + 139230*a^2*b^3*x^6 + 117810*a^3*b^2*x^4 + 51051*a^4*b*x^2 + 9009
*a^5)/x^17

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maple [A]  time = 0.00, size = 58, normalized size = 0.84 \begin {gather*} -\frac {b^{5}}{7 x^{7}}-\frac {5 a \,b^{4}}{9 x^{9}}-\frac {10 a^{2} b^{3}}{11 x^{11}}-\frac {10 a^{3} b^{2}}{13 x^{13}}-\frac {a^{4} b}{3 x^{15}}-\frac {a^{5}}{17 x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5/x^18,x)

[Out]

-1/17*a^5/x^17-1/3*a^4*b/x^15-10/13*a^3*b^2/x^13-10/11*a^2*b^3/x^11-5/9*a*b^4/x^9-1/7*b^5/x^7

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maxima [A]  time = 1.32, size = 59, normalized size = 0.86 \begin {gather*} -\frac {21879 \, b^{5} x^{10} + 85085 \, a b^{4} x^{8} + 139230 \, a^{2} b^{3} x^{6} + 117810 \, a^{3} b^{2} x^{4} + 51051 \, a^{4} b x^{2} + 9009 \, a^{5}}{153153 \, x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5/x^18,x, algorithm="maxima")

[Out]

-1/153153*(21879*b^5*x^10 + 85085*a*b^4*x^8 + 139230*a^2*b^3*x^6 + 117810*a^3*b^2*x^4 + 51051*a^4*b*x^2 + 9009
*a^5)/x^17

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mupad [B]  time = 0.04, size = 59, normalized size = 0.86 \begin {gather*} -\frac {\frac {a^5}{17}+\frac {a^4\,b\,x^2}{3}+\frac {10\,a^3\,b^2\,x^4}{13}+\frac {10\,a^2\,b^3\,x^6}{11}+\frac {5\,a\,b^4\,x^8}{9}+\frac {b^5\,x^{10}}{7}}{x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^5/x^18,x)

[Out]

-(a^5/17 + (b^5*x^10)/7 + (a^4*b*x^2)/3 + (5*a*b^4*x^8)/9 + (10*a^3*b^2*x^4)/13 + (10*a^2*b^3*x^6)/11)/x^17

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sympy [A]  time = 0.54, size = 63, normalized size = 0.91 \begin {gather*} \frac {- 9009 a^{5} - 51051 a^{4} b x^{2} - 117810 a^{3} b^{2} x^{4} - 139230 a^{2} b^{3} x^{6} - 85085 a b^{4} x^{8} - 21879 b^{5} x^{10}}{153153 x^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5/x**18,x)

[Out]

(-9009*a**5 - 51051*a**4*b*x**2 - 117810*a**3*b**2*x**4 - 139230*a**2*b**3*x**6 - 85085*a*b**4*x**8 - 21879*b*
*5*x**10)/(153153*x**17)

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