3.100 \(\int \frac{\left (a+b x^2\right )^8}{x^{17}} \, dx\)

Optimal. Leaf size=100 \[ -\frac{a^8}{16 x^{16}}-\frac{4 a^7 b}{7 x^{14}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a b^7}{x^2}+b^8 \log (x) \]

[Out]

-a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^1
0) - (35*a^4*b^4)/(4*x^8) - (28*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x
^2 + b^8*Log[x]

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Rubi [A]  time = 0.135663, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^8}{16 x^{16}}-\frac{4 a^7 b}{7 x^{14}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a b^7}{x^2}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^8/x^17,x]

[Out]

-a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^1
0) - (35*a^4*b^4)/(4*x^8) - (28*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x
^2 + b^8*Log[x]

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Rubi in Sympy [A]  time = 23.8249, size = 105, normalized size = 1.05 \[ - \frac{a^{8}}{16 x^{16}} - \frac{4 a^{7} b}{7 x^{14}} - \frac{7 a^{6} b^{2}}{3 x^{12}} - \frac{28 a^{5} b^{3}}{5 x^{10}} - \frac{35 a^{4} b^{4}}{4 x^{8}} - \frac{28 a^{3} b^{5}}{3 x^{6}} - \frac{7 a^{2} b^{6}}{x^{4}} - \frac{4 a b^{7}}{x^{2}} + \frac{b^{8} \log{\left (x^{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**8/x**17,x)

[Out]

-a**8/(16*x**16) - 4*a**7*b/(7*x**14) - 7*a**6*b**2/(3*x**12) - 28*a**5*b**3/(5*
x**10) - 35*a**4*b**4/(4*x**8) - 28*a**3*b**5/(3*x**6) - 7*a**2*b**6/x**4 - 4*a*
b**7/x**2 + b**8*log(x**2)/2

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Mathematica [A]  time = 0.00877329, size = 100, normalized size = 1. \[ -\frac{a^8}{16 x^{16}}-\frac{4 a^7 b}{7 x^{14}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{35 a^4 b^4}{4 x^8}-\frac{28 a^3 b^5}{3 x^6}-\frac{7 a^2 b^6}{x^4}-\frac{4 a b^7}{x^2}+b^8 \log (x) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^8/x^17,x]

[Out]

-a^8/(16*x^16) - (4*a^7*b)/(7*x^14) - (7*a^6*b^2)/(3*x^12) - (28*a^5*b^3)/(5*x^1
0) - (35*a^4*b^4)/(4*x^8) - (28*a^3*b^5)/(3*x^6) - (7*a^2*b^6)/x^4 - (4*a*b^7)/x
^2 + b^8*Log[x]

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Maple [A]  time = 0.011, size = 89, normalized size = 0.9 \[ -{\frac{{a}^{8}}{16\,{x}^{16}}}-{\frac{4\,{a}^{7}b}{7\,{x}^{14}}}-{\frac{7\,{a}^{6}{b}^{2}}{3\,{x}^{12}}}-{\frac{28\,{a}^{5}{b}^{3}}{5\,{x}^{10}}}-{\frac{35\,{a}^{4}{b}^{4}}{4\,{x}^{8}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,{x}^{6}}}-7\,{\frac{{a}^{2}{b}^{6}}{{x}^{4}}}-4\,{\frac{a{b}^{7}}{{x}^{2}}}+{b}^{8}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^8/x^17,x)

[Out]

-1/16*a^8/x^16-4/7*a^7*b/x^14-7/3*a^6*b^2/x^12-28/5*a^5*b^3/x^10-35/4*a^4*b^4/x^
8-28/3*a^3*b^5/x^6-7*a^2*b^6/x^4-4*a*b^7/x^2+b^8*ln(x)

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Maxima [A]  time = 1.35059, size = 127, normalized size = 1.27 \[ \frac{1}{2} \, b^{8} \log \left (x^{2}\right ) - \frac{6720 \, a b^{7} x^{14} + 11760 \, a^{2} b^{6} x^{12} + 15680 \, a^{3} b^{5} x^{10} + 14700 \, a^{4} b^{4} x^{8} + 9408 \, a^{5} b^{3} x^{6} + 3920 \, a^{6} b^{2} x^{4} + 960 \, a^{7} b x^{2} + 105 \, a^{8}}{1680 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^17,x, algorithm="maxima")

[Out]

1/2*b^8*log(x^2) - 1/1680*(6720*a*b^7*x^14 + 11760*a^2*b^6*x^12 + 15680*a^3*b^5*
x^10 + 14700*a^4*b^4*x^8 + 9408*a^5*b^3*x^6 + 3920*a^6*b^2*x^4 + 960*a^7*b*x^2 +
105*a^8)/x^16

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Fricas [A]  time = 0.199327, size = 127, normalized size = 1.27 \[ \frac{1680 \, b^{8} x^{16} \log \left (x\right ) - 6720 \, a b^{7} x^{14} - 11760 \, a^{2} b^{6} x^{12} - 15680 \, a^{3} b^{5} x^{10} - 14700 \, a^{4} b^{4} x^{8} - 9408 \, a^{5} b^{3} x^{6} - 3920 \, a^{6} b^{2} x^{4} - 960 \, a^{7} b x^{2} - 105 \, a^{8}}{1680 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^17,x, algorithm="fricas")

[Out]

1/1680*(1680*b^8*x^16*log(x) - 6720*a*b^7*x^14 - 11760*a^2*b^6*x^12 - 15680*a^3*
b^5*x^10 - 14700*a^4*b^4*x^8 - 9408*a^5*b^3*x^6 - 3920*a^6*b^2*x^4 - 960*a^7*b*x
^2 - 105*a^8)/x^16

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Sympy [A]  time = 3.7118, size = 95, normalized size = 0.95 \[ b^{8} \log{\left (x \right )} - \frac{105 a^{8} + 960 a^{7} b x^{2} + 3920 a^{6} b^{2} x^{4} + 9408 a^{5} b^{3} x^{6} + 14700 a^{4} b^{4} x^{8} + 15680 a^{3} b^{5} x^{10} + 11760 a^{2} b^{6} x^{12} + 6720 a b^{7} x^{14}}{1680 x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**8/x**17,x)

[Out]

b**8*log(x) - (105*a**8 + 960*a**7*b*x**2 + 3920*a**6*b**2*x**4 + 9408*a**5*b**3
*x**6 + 14700*a**4*b**4*x**8 + 15680*a**3*b**5*x**10 + 11760*a**2*b**6*x**12 + 6
720*a*b**7*x**14)/(1680*x**16)

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GIAC/XCAS [A]  time = 0.208714, size = 138, normalized size = 1.38 \[ \frac{1}{2} \, b^{8}{\rm ln}\left (x^{2}\right ) - \frac{2283 \, b^{8} x^{16} + 6720 \, a b^{7} x^{14} + 11760 \, a^{2} b^{6} x^{12} + 15680 \, a^{3} b^{5} x^{10} + 14700 \, a^{4} b^{4} x^{8} + 9408 \, a^{5} b^{3} x^{6} + 3920 \, a^{6} b^{2} x^{4} + 960 \, a^{7} b x^{2} + 105 \, a^{8}}{1680 \, x^{16}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8/x^17,x, algorithm="giac")

[Out]

1/2*b^8*ln(x^2) - 1/1680*(2283*b^8*x^16 + 6720*a*b^7*x^14 + 11760*a^2*b^6*x^12 +
15680*a^3*b^5*x^10 + 14700*a^4*b^4*x^8 + 9408*a^5*b^3*x^6 + 3920*a^6*b^2*x^4 +
960*a^7*b*x^2 + 105*a^8)/x^16