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

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

[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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Rubi [A]  time = 0.04, antiderivative size = 99, 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} \[ -\frac {28 a^6 b^2}{11 x^{11}}-\frac {56 a^5 b^3}{9 x^9}-\frac {10 a^4 b^4}{x^7}-\frac {56 a^3 b^5}{5 x^5}-\frac {28 a^2 b^6}{3 x^3}-\frac {8 a^7 b}{13 x^{13}}-\frac {a^8}{15 x^{15}}-\frac {8 a b^7}{x}+b^8 x \]

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*}

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Mathematica [A]  time = 0.01, size = 99, normalized size = 1.00 \[ -\frac {a^8}{15 x^{15}}-\frac {8 a^7 b}{13 x^{13}}-\frac {28 a^6 b^2}{11 x^{11}}-\frac {56 a^5 b^3}{9 x^9}-\frac {10 a^4 b^4}{x^7}-\frac {56 a^3 b^5}{5 x^5}-\frac {28 a^2 b^6}{3 x^3}-\frac {8 a b^7}{x}+b^8 x \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^8/x^16,x]

[Out]

-1/15*a^8/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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fricas [A]  time = 0.70, size = 92, normalized size = 0.93 \[ \frac {6435 \, b^{8} x^{16} - 51480 \, a b^{7} x^{14} - 60060 \, a^{2} b^{6} x^{12} - 72072 \, a^{3} b^{5} x^{10} - 64350 \, a^{4} b^{4} x^{8} - 40040 \, a^{5} b^{3} x^{6} - 16380 \, a^{6} b^{2} x^{4} - 3960 \, a^{7} b x^{2} - 429 \, a^{8}}{6435 \, x^{15}} \]

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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giac [A]  time = 0.87, size = 90, normalized size = 0.91 \[ b^{8} x - \frac {51480 \, a b^{7} x^{14} + 60060 \, a^{2} b^{6} x^{12} + 72072 \, a^{3} b^{5} x^{10} + 64350 \, a^{4} b^{4} x^{8} + 40040 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 3960 \, a^{7} b x^{2} + 429 \, a^{8}}{6435 \, x^{15}} \]

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

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maple [A]  time = 0.01, size = 88, normalized size = 0.89 \[ b^{8} x -\frac {8 a \,b^{7}}{x}-\frac {28 a^{2} b^{6}}{3 x^{3}}-\frac {56 a^{3} b^{5}}{5 x^{5}}-\frac {10 a^{4} b^{4}}{x^{7}}-\frac {56 a^{5} b^{3}}{9 x^{9}}-\frac {28 a^{6} b^{2}}{11 x^{11}}-\frac {8 a^{7} b}{13 x^{13}}-\frac {a^{8}}{15 x^{15}} \]

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 = 1.41, size = 90, normalized size = 0.91 \[ b^{8} x - \frac {51480 \, a b^{7} x^{14} + 60060 \, a^{2} b^{6} x^{12} + 72072 \, a^{3} b^{5} x^{10} + 64350 \, a^{4} b^{4} x^{8} + 40040 \, a^{5} b^{3} x^{6} + 16380 \, a^{6} b^{2} x^{4} + 3960 \, a^{7} b x^{2} + 429 \, a^{8}}{6435 \, x^{15}} \]

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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mupad [B]  time = 4.52, size = 90, normalized size = 0.91 \[ b^8\,x-\frac {\frac {a^8}{15}+\frac {8\,a^7\,b\,x^2}{13}+\frac {28\,a^6\,b^2\,x^4}{11}+\frac {56\,a^5\,b^3\,x^6}{9}+10\,a^4\,b^4\,x^8+\frac {56\,a^3\,b^5\,x^{10}}{5}+\frac {28\,a^2\,b^6\,x^{12}}{3}+8\,a\,b^7\,x^{14}}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^2)^8/x^16,x)

[Out]

b^8*x - (a^8/15 + (8*a^7*b*x^2)/13 + 8*a*b^7*x^14 + (28*a^6*b^2*x^4)/11 + (56*a^5*b^3*x^6)/9 + 10*a^4*b^4*x^8
+ (56*a^3*b^5*x^10)/5 + (28*a^2*b^6*x^12)/3)/x^15

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sympy [A]  time = 0.69, size = 95, normalized size = 0.96 \[ b^{8} x + \frac {- 429 a^{8} - 3960 a^{7} b x^{2} - 16380 a^{6} b^{2} x^{4} - 40040 a^{5} b^{3} x^{6} - 64350 a^{4} b^{4} x^{8} - 72072 a^{3} b^{5} x^{10} - 60060 a^{2} b^{6} x^{12} - 51480 a b^{7} x^{14}}{6435 x^{15}} \]

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 -
72072*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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