3.121 \(\int \frac{(a+b x^2)^8}{x^{16}} \, dx\)

Optimal. Leaf size=99 \[ -\frac{28 a^6 b^2}{11 x^{11}}-\frac{56 a^5 b^3}{9 x^9}-\frac{10 a^4 b^4}{x^7}-\frac{56 a^3 b^5}{5 x^5}-\frac{28 a^2 b^6}{3 x^3}-\frac{8 a^7 b}{13 x^{13}}-\frac{a^8}{15 x^{15}}-\frac{8 a b^7}{x}+b^8 x \]

[Out]

-a^8/(15*x^15) - (8*a^7*b)/(13*x^13) - (28*a^6*b^2)/(11*x^11) - (56*a^5*b^3)/(9*x^9) - (10*a^4*b^4)/x^7 - (56*
a^3*b^5)/(5*x^5) - (28*a^2*b^6)/(3*x^3) - (8*a*b^7)/x + b^8*x

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Rubi [A]  time = 0.0393084, antiderivative size = 99, 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} \[ -\frac{28 a^6 b^2}{11 x^{11}}-\frac{56 a^5 b^3}{9 x^9}-\frac{10 a^4 b^4}{x^7}-\frac{56 a^3 b^5}{5 x^5}-\frac{28 a^2 b^6}{3 x^3}-\frac{8 a^7 b}{13 x^{13}}-\frac{a^8}{15 x^{15}}-\frac{8 a b^7}{x}+b^8 x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)^8/x^16,x]

[Out]

-a^8/(15*x^15) - (8*a^7*b)/(13*x^13) - (28*a^6*b^2)/(11*x^11) - (56*a^5*b^3)/(9*x^9) - (10*a^4*b^4)/x^7 - (56*
a^3*b^5)/(5*x^5) - (28*a^2*b^6)/(3*x^3) - (8*a*b^7)/x + b^8*x

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^{16}} \, dx &=\int \left (b^8+\frac{a^8}{x^{16}}+\frac{8 a^7 b}{x^{14}}+\frac{28 a^6 b^2}{x^{12}}+\frac{56 a^5 b^3}{x^{10}}+\frac{70 a^4 b^4}{x^8}+\frac{56 a^3 b^5}{x^6}+\frac{28 a^2 b^6}{x^4}+\frac{8 a b^7}{x^2}\right ) \, dx\\ &=-\frac{a^8}{15 x^{15}}-\frac{8 a^7 b}{13 x^{13}}-\frac{28 a^6 b^2}{11 x^{11}}-\frac{56 a^5 b^3}{9 x^9}-\frac{10 a^4 b^4}{x^7}-\frac{56 a^3 b^5}{5 x^5}-\frac{28 a^2 b^6}{3 x^3}-\frac{8 a b^7}{x}+b^8 x\\ \end{align*}

Mathematica [A]  time = 0.0053595, size = 99, normalized size = 1. \[ -\frac{28 a^6 b^2}{11 x^{11}}-\frac{56 a^5 b^3}{9 x^9}-\frac{10 a^4 b^4}{x^7}-\frac{56 a^3 b^5}{5 x^5}-\frac{28 a^2 b^6}{3 x^3}-\frac{8 a^7 b}{13 x^{13}}-\frac{a^8}{15 x^{15}}-\frac{8 a b^7}{x}+b^8 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^16,x]

[Out]

-a^8/(15*x^15) - (8*a^7*b)/(13*x^13) - (28*a^6*b^2)/(11*x^11) - (56*a^5*b^3)/(9*x^9) - (10*a^4*b^4)/x^7 - (56*
a^3*b^5)/(5*x^5) - (28*a^2*b^6)/(3*x^3) - (8*a*b^7)/x + b^8*x

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Maple [A]  time = 0.007, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{a}^{8}}{15\,{x}^{15}}}-{\frac{8\,{a}^{7}b}{13\,{x}^{13}}}-{\frac{28\,{a}^{6}{b}^{2}}{11\,{x}^{11}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,{x}^{9}}}-10\,{\frac{{a}^{4}{b}^{4}}{{x}^{7}}}-{\frac{56\,{a}^{3}{b}^{5}}{5\,{x}^{5}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,{x}^{3}}}-8\,{\frac{a{b}^{7}}{x}}+{b}^{8}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^8/x^16,x)

[Out]

-1/15*a^8/x^15-8/13*a^7*b/x^13-28/11*a^6*b^2/x^11-56/9*a^5*b^3/x^9-10*a^4*b^4/x^7-56/5*a^3*b^5/x^5-28/3*a^2*b^
6/x^3-8*a*b^7/x+b^8*x

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Maxima [A]  time = 1.3335, size = 122, normalized size = 1.23 \begin{align*} b^{8} x - \frac{51480 \, a b^{7} x^{14} + 60060 \, a^{2} b^{6} x^{12} + 72072 \, a^{3} b^{5} x^{10} + 64350 \, a^{4} b^{4} x^{8} + 40040 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 3960 \, a^{7} b x^{2} + 429 \, a^{8}}{6435 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^16,x, algorithm="maxima")

[Out]

b^8*x - 1/6435*(51480*a*b^7*x^14 + 60060*a^2*b^6*x^12 + 72072*a^3*b^5*x^10 + 64350*a^4*b^4*x^8 + 40040*a^5*b^3
*x^6 + 16380*a^6*b^2*x^4 + 3960*a^7*b*x^2 + 429*a^8)/x^15

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Fricas [A]  time = 1.26084, size = 239, normalized size = 2.41 \begin{align*} \frac{6435 \, b^{8} x^{16} - 51480 \, a b^{7} x^{14} - 60060 \, a^{2} b^{6} x^{12} - 72072 \, a^{3} b^{5} x^{10} - 64350 \, a^{4} b^{4} x^{8} - 40040 \, a^{5} b^{3} x^{6} - 16380 \, a^{6} b^{2} x^{4} - 3960 \, a^{7} b x^{2} - 429 \, a^{8}}{6435 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^16,x, algorithm="fricas")

[Out]

1/6435*(6435*b^8*x^16 - 51480*a*b^7*x^14 - 60060*a^2*b^6*x^12 - 72072*a^3*b^5*x^10 - 64350*a^4*b^4*x^8 - 40040
*a^5*b^3*x^6 - 16380*a^6*b^2*x^4 - 3960*a^7*b*x^2 - 429*a^8)/x^15

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Sympy [A]  time = 0.977237, size = 94, normalized size = 0.95 \begin{align*} b^{8} x - \frac{429 a^{8} + 3960 a^{7} b x^{2} + 16380 a^{6} b^{2} x^{4} + 40040 a^{5} b^{3} x^{6} + 64350 a^{4} b^{4} x^{8} + 72072 a^{3} b^{5} x^{10} + 60060 a^{2} b^{6} x^{12} + 51480 a b^{7} x^{14}}{6435 x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**8/x**16,x)

[Out]

b**8*x - (429*a**8 + 3960*a**7*b*x**2 + 16380*a**6*b**2*x**4 + 40040*a**5*b**3*x**6 + 64350*a**4*b**4*x**8 + 7
2072*a**3*b**5*x**10 + 60060*a**2*b**6*x**12 + 51480*a*b**7*x**14)/(6435*x**15)

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Giac [A]  time = 2.15924, size = 122, normalized size = 1.23 \begin{align*} b^{8} x - \frac{51480 \, a b^{7} x^{14} + 60060 \, a^{2} b^{6} x^{12} + 72072 \, a^{3} b^{5} x^{10} + 64350 \, a^{4} b^{4} x^{8} + 40040 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 3960 \, a^{7} b x^{2} + 429 \, a^{8}}{6435 \, x^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^8/x^16,x, algorithm="giac")

[Out]

b^8*x - 1/6435*(51480*a*b^7*x^14 + 60060*a^2*b^6*x^12 + 72072*a^3*b^5*x^10 + 64350*a^4*b^4*x^8 + 40040*a^5*b^3
*x^6 + 16380*a^6*b^2*x^4 + 3960*a^7*b*x^2 + 429*a^8)/x^15