3.119 \(\int \frac{(a+b x^2)^8}{x^{12}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{11 x^{11}}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5} \]

[Out]

-a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (70*a^4*b^4)/(3*x^3) - (56*a^3*b
^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)/5

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Rubi [A]  time = 0.0398323, antiderivative size = 100, 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} \[ -\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{11 x^{11}}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^12,x]

[Out]

-a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (70*a^4*b^4)/(3*x^3) - (56*a^3*b
^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)/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 )^8}{x^{12}} \, dx &=\int \left (28 a^2 b^6+\frac{a^8}{x^{12}}+\frac{8 a^7 b}{x^{10}}+\frac{28 a^6 b^2}{x^8}+\frac{56 a^5 b^3}{x^6}+\frac{70 a^4 b^4}{x^4}+\frac{56 a^3 b^5}{x^2}+8 a b^7 x^2+b^8 x^4\right ) \, dx\\ &=-\frac{a^8}{11 x^{11}}-\frac{8 a^7 b}{9 x^9}-\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0103317, size = 100, normalized size = 1. \[ -\frac{4 a^6 b^2}{x^7}-\frac{56 a^5 b^3}{5 x^5}-\frac{70 a^4 b^4}{3 x^3}-\frac{56 a^3 b^5}{x}+28 a^2 b^6 x-\frac{8 a^7 b}{9 x^9}-\frac{a^8}{11 x^{11}}+\frac{8}{3} a b^7 x^3+\frac{b^8 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^12,x]

[Out]

-a^8/(11*x^11) - (8*a^7*b)/(9*x^9) - (4*a^6*b^2)/x^7 - (56*a^5*b^3)/(5*x^5) - (70*a^4*b^4)/(3*x^3) - (56*a^3*b
^5)/x + 28*a^2*b^6*x + (8*a*b^7*x^3)/3 + (b^8*x^5)/5

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Maple [A]  time = 0.007, size = 89, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{11\,{x}^{11}}}-{\frac{8\,{a}^{7}b}{9\,{x}^{9}}}-4\,{\frac{{a}^{6}{b}^{2}}{{x}^{7}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,{x}^{5}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}-56\,{\frac{{a}^{3}{b}^{5}}{x}}+28\,{a}^{2}{b}^{6}x+{\frac{8\,a{b}^{7}{x}^{3}}{3}}+{\frac{{b}^{8}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^12,x)

[Out]

-1/11*a^8/x^11-8/9*a^7*b/x^9-4*a^6*b^2/x^7-56/5*a^5*b^3/x^5-70/3*a^4*b^4/x^3-56*a^3*b^5/x+28*a^2*b^6*x+8/3*a*b
^7*x^3+1/5*b^8*x^5

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Maxima [A]  time = 2.00433, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{5} \, b^{8} x^{5} + \frac{8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6} x - \frac{27720 \, a^{3} b^{5} x^{10} + 11550 \, a^{4} b^{4} x^{8} + 5544 \, a^{5} b^{3} x^{6} + 1980 \, a^{6} b^{2} x^{4} + 440 \, a^{7} b x^{2} + 45 \, a^{8}}{495 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^12,x, algorithm="maxima")

[Out]

1/5*b^8*x^5 + 8/3*a*b^7*x^3 + 28*a^2*b^6*x - 1/495*(27720*a^3*b^5*x^10 + 11550*a^4*b^4*x^8 + 5544*a^5*b^3*x^6
+ 1980*a^6*b^2*x^4 + 440*a^7*b*x^2 + 45*a^8)/x^11

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Fricas [A]  time = 1.28241, size = 228, normalized size = 2.28 \begin{align*} \frac{99 \, b^{8} x^{16} + 1320 \, a b^{7} x^{14} + 13860 \, a^{2} b^{6} x^{12} - 27720 \, a^{3} b^{5} x^{10} - 11550 \, a^{4} b^{4} x^{8} - 5544 \, a^{5} b^{3} x^{6} - 1980 \, a^{6} b^{2} x^{4} - 440 \, a^{7} b x^{2} - 45 \, a^{8}}{495 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^12,x, algorithm="fricas")

[Out]

1/495*(99*b^8*x^16 + 1320*a*b^7*x^14 + 13860*a^2*b^6*x^12 - 27720*a^3*b^5*x^10 - 11550*a^4*b^4*x^8 - 5544*a^5*
b^3*x^6 - 1980*a^6*b^2*x^4 - 440*a^7*b*x^2 - 45*a^8)/x^11

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Sympy [A]  time = 0.759729, size = 97, normalized size = 0.97 \begin{align*} 28 a^{2} b^{6} x + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{5}}{5} - \frac{45 a^{8} + 440 a^{7} b x^{2} + 1980 a^{6} b^{2} x^{4} + 5544 a^{5} b^{3} x^{6} + 11550 a^{4} b^{4} x^{8} + 27720 a^{3} b^{5} x^{10}}{495 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**12,x)

[Out]

28*a**2*b**6*x + 8*a*b**7*x**3/3 + b**8*x**5/5 - (45*a**8 + 440*a**7*b*x**2 + 1980*a**6*b**2*x**4 + 5544*a**5*
b**3*x**6 + 11550*a**4*b**4*x**8 + 27720*a**3*b**5*x**10)/(495*x**11)

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Giac [A]  time = 2.58107, size = 123, normalized size = 1.23 \begin{align*} \frac{1}{5} \, b^{8} x^{5} + \frac{8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6} x - \frac{27720 \, a^{3} b^{5} x^{10} + 11550 \, a^{4} b^{4} x^{8} + 5544 \, a^{5} b^{3} x^{6} + 1980 \, a^{6} b^{2} x^{4} + 440 \, a^{7} b x^{2} + 45 \, a^{8}}{495 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^12,x, algorithm="giac")

[Out]

1/5*b^8*x^5 + 8/3*a*b^7*x^3 + 28*a^2*b^6*x - 1/495*(27720*a^3*b^5*x^10 + 11550*a^4*b^4*x^8 + 5544*a^5*b^3*x^6
+ 1980*a^6*b^2*x^4 + 440*a^7*b*x^2 + 45*a^8)/x^11