3.297 \(\int \frac{\left (a+b x^3\right )^8}{x^{16}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^8}{15 x^{15}}-\frac{2 a^7 b}{3 x^{12}}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{3 x^6}-\frac{70 a^4 b^4}{3 x^3}+56 a^3 b^5 \log (x)+\frac{28}{3} a^2 b^6 x^3+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]

[Out]

-a^8/(15*x^15) - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/(3*x^6
) - (70*a^4*b^4)/(3*x^3) + (28*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b^8*x^9)/9 +
56*a^3*b^5*Log[x]

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Rubi [A]  time = 0.126611, antiderivative size = 105, 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 \[ -\frac{a^8}{15 x^{15}}-\frac{2 a^7 b}{3 x^{12}}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{3 x^6}-\frac{70 a^4 b^4}{3 x^3}+56 a^3 b^5 \log (x)+\frac{28}{3} a^2 b^6 x^3+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^16,x]

[Out]

-a^8/(15*x^15) - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/(3*x^6
) - (70*a^4*b^4)/(3*x^3) + (28*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b^8*x^9)/9 +
56*a^3*b^5*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{15 x^{15}} - \frac{2 a^{7} b}{3 x^{12}} - \frac{28 a^{6} b^{2}}{9 x^{9}} - \frac{28 a^{5} b^{3}}{3 x^{6}} - \frac{70 a^{4} b^{4}}{3 x^{3}} + \frac{56 a^{3} b^{5} \log{\left (x^{3} \right )}}{3} + \frac{28 a^{2} b^{6} x^{3}}{3} + \frac{8 a b^{7} \int ^{x^{3}} x\, dx}{3} + \frac{b^{8} x^{9}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**16,x)

[Out]

-a**8/(15*x**15) - 2*a**7*b/(3*x**12) - 28*a**6*b**2/(9*x**9) - 28*a**5*b**3/(3*
x**6) - 70*a**4*b**4/(3*x**3) + 56*a**3*b**5*log(x**3)/3 + 28*a**2*b**6*x**3/3 +
8*a*b**7*Integral(x, (x, x**3))/3 + b**8*x**9/9

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Mathematica [A]  time = 0.0122083, size = 105, normalized size = 1. \[ -\frac{a^8}{15 x^{15}}-\frac{2 a^7 b}{3 x^{12}}-\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{3 x^6}-\frac{70 a^4 b^4}{3 x^3}+56 a^3 b^5 \log (x)+\frac{28}{3} a^2 b^6 x^3+\frac{4}{3} a b^7 x^6+\frac{b^8 x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^16,x]

[Out]

-a^8/(15*x^15) - (2*a^7*b)/(3*x^12) - (28*a^6*b^2)/(9*x^9) - (28*a^5*b^3)/(3*x^6
) - (70*a^4*b^4)/(3*x^3) + (28*a^2*b^6*x^3)/3 + (4*a*b^7*x^6)/3 + (b^8*x^9)/9 +
56*a^3*b^5*Log[x]

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Maple [A]  time = 0.012, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{15\,{x}^{15}}}-{\frac{2\,{a}^{7}b}{3\,{x}^{12}}}-{\frac{28\,{a}^{6}{b}^{2}}{9\,{x}^{9}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-{\frac{70\,{a}^{4}{b}^{4}}{3\,{x}^{3}}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{3}}{3}}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{9}}{9}}+56\,{a}^{3}{b}^{5}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^16,x)

[Out]

-1/15*a^8/x^15-2/3*a^7*b/x^12-28/9*a^6*b^2/x^9-28/3*a^5*b^3/x^6-70/3*a^4*b^4/x^3
+28/3*a^2*b^6*x^3+4/3*a*b^7*x^6+1/9*b^8*x^9+56*a^3*b^5*ln(x)

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Maxima [A]  time = 1.4247, size = 127, normalized size = 1.21 \[ \frac{1}{9} \, b^{8} x^{9} + \frac{4}{3} \, a b^{7} x^{6} + \frac{28}{3} \, a^{2} b^{6} x^{3} + \frac{56}{3} \, a^{3} b^{5} \log \left (x^{3}\right ) - \frac{1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^16,x, algorithm="maxima")

[Out]

1/9*b^8*x^9 + 4/3*a*b^7*x^6 + 28/3*a^2*b^6*x^3 + 56/3*a^3*b^5*log(x^3) - 1/45*(1
050*a^4*b^4*x^12 + 420*a^5*b^3*x^9 + 140*a^6*b^2*x^6 + 30*a^7*b*x^3 + 3*a^8)/x^1
5

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Fricas [A]  time = 0.213781, size = 127, normalized size = 1.21 \[ \frac{5 \, b^{8} x^{24} + 60 \, a b^{7} x^{21} + 420 \, a^{2} b^{6} x^{18} + 2520 \, a^{3} b^{5} x^{15} \log \left (x\right ) - 1050 \, a^{4} b^{4} x^{12} - 420 \, a^{5} b^{3} x^{9} - 140 \, a^{6} b^{2} x^{6} - 30 \, a^{7} b x^{3} - 3 \, a^{8}}{45 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^16,x, algorithm="fricas")

[Out]

1/45*(5*b^8*x^24 + 60*a*b^7*x^21 + 420*a^2*b^6*x^18 + 2520*a^3*b^5*x^15*log(x) -
1050*a^4*b^4*x^12 - 420*a^5*b^3*x^9 - 140*a^6*b^2*x^6 - 30*a^7*b*x^3 - 3*a^8)/x
^15

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Sympy [A]  time = 2.86569, size = 100, normalized size = 0.95 \[ 56 a^{3} b^{5} \log{\left (x \right )} + \frac{28 a^{2} b^{6} x^{3}}{3} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{9}}{9} - \frac{3 a^{8} + 30 a^{7} b x^{3} + 140 a^{6} b^{2} x^{6} + 420 a^{5} b^{3} x^{9} + 1050 a^{4} b^{4} x^{12}}{45 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**16,x)

[Out]

56*a**3*b**5*log(x) + 28*a**2*b**6*x**3/3 + 4*a*b**7*x**6/3 + b**8*x**9/9 - (3*a
**8 + 30*a**7*b*x**3 + 140*a**6*b**2*x**6 + 420*a**5*b**3*x**9 + 1050*a**4*b**4*
x**12)/(45*x**15)

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GIAC/XCAS [A]  time = 0.222698, size = 140, normalized size = 1.33 \[ \frac{1}{9} \, b^{8} x^{9} + \frac{4}{3} \, a b^{7} x^{6} + \frac{28}{3} \, a^{2} b^{6} x^{3} + 56 \, a^{3} b^{5}{\rm ln}\left ({\left | x \right |}\right ) - \frac{1918 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 420 \, a^{5} b^{3} x^{9} + 140 \, a^{6} b^{2} x^{6} + 30 \, a^{7} b x^{3} + 3 \, a^{8}}{45 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^16,x, algorithm="giac")

[Out]

1/9*b^8*x^9 + 4/3*a*b^7*x^6 + 28/3*a^2*b^6*x^3 + 56*a^3*b^5*ln(abs(x)) - 1/45*(1
918*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 420*a^5*b^3*x^9 + 140*a^6*b^2*x^6 + 30*a^
7*b*x^3 + 3*a^8)/x^15