3.298 \(\int \frac{\left (a+b x^3\right )^8}{x^{19}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^8}{18 x^{18}}-\frac{8 a^7 b}{15 x^{15}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6} \]

[Out]

-a^8/(18*x^18) - (8*a^7*b)/(15*x^15) - (7*a^6*b^2)/(3*x^12) - (56*a^5*b^3)/(9*x^
9) - (35*a^4*b^4)/(3*x^6) - (56*a^3*b^5)/(3*x^3) + (8*a*b^7*x^3)/3 + (b^8*x^6)/6
+ 28*a^2*b^6*Log[x]

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Rubi [A]  time = 0.125017, 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 \[ -\frac{a^8}{18 x^{18}}-\frac{8 a^7 b}{15 x^{15}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^19,x]

[Out]

-a^8/(18*x^18) - (8*a^7*b)/(15*x^15) - (7*a^6*b^2)/(3*x^12) - (56*a^5*b^3)/(9*x^
9) - (35*a^4*b^4)/(3*x^6) - (56*a^3*b^5)/(3*x^3) + (8*a*b^7*x^3)/3 + (b^8*x^6)/6
+ 28*a^2*b^6*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{18 x^{18}} - \frac{8 a^{7} b}{15 x^{15}} - \frac{7 a^{6} b^{2}}{3 x^{12}} - \frac{56 a^{5} b^{3}}{9 x^{9}} - \frac{35 a^{4} b^{4}}{3 x^{6}} - \frac{56 a^{3} b^{5}}{3 x^{3}} + \frac{28 a^{2} b^{6} \log{\left (x^{3} \right )}}{3} + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} \int ^{x^{3}} x\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**19,x)

[Out]

-a**8/(18*x**18) - 8*a**7*b/(15*x**15) - 7*a**6*b**2/(3*x**12) - 56*a**5*b**3/(9
*x**9) - 35*a**4*b**4/(3*x**6) - 56*a**3*b**5/(3*x**3) + 28*a**2*b**6*log(x**3)/
3 + 8*a*b**7*x**3/3 + b**8*Integral(x, (x, x**3))/3

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Mathematica [A]  time = 0.00877233, size = 105, normalized size = 1. \[ -\frac{a^8}{18 x^{18}}-\frac{8 a^7 b}{15 x^{15}}-\frac{7 a^6 b^2}{3 x^{12}}-\frac{56 a^5 b^3}{9 x^9}-\frac{35 a^4 b^4}{3 x^6}-\frac{56 a^3 b^5}{3 x^3}+28 a^2 b^6 \log (x)+\frac{8}{3} a b^7 x^3+\frac{b^8 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^19,x]

[Out]

-a^8/(18*x^18) - (8*a^7*b)/(15*x^15) - (7*a^6*b^2)/(3*x^12) - (56*a^5*b^3)/(9*x^
9) - (35*a^4*b^4)/(3*x^6) - (56*a^3*b^5)/(3*x^3) + (8*a*b^7*x^3)/3 + (b^8*x^6)/6
+ 28*a^2*b^6*Log[x]

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Maple [A]  time = 0.013, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{18\,{x}^{18}}}-{\frac{8\,{a}^{7}b}{15\,{x}^{15}}}-{\frac{7\,{a}^{6}{b}^{2}}{3\,{x}^{12}}}-{\frac{56\,{a}^{5}{b}^{3}}{9\,{x}^{9}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,{x}^{6}}}-{\frac{56\,{a}^{3}{b}^{5}}{3\,{x}^{3}}}+{\frac{8\,a{b}^{7}{x}^{3}}{3}}+{\frac{{b}^{8}{x}^{6}}{6}}+28\,{a}^{2}{b}^{6}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^19,x)

[Out]

-1/18*a^8/x^18-8/15*a^7*b/x^15-7/3*a^6*b^2/x^12-56/9*a^5*b^3/x^9-35/3*a^4*b^4/x^
6-56/3*a^3*b^5/x^3+8/3*a*b^7*x^3+1/6*b^8*x^6+28*a^2*b^6*ln(x)

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Maxima [A]  time = 1.45279, size = 127, normalized size = 1.21 \[ \frac{1}{6} \, b^{8} x^{6} + \frac{8}{3} \, a b^{7} x^{3} + \frac{28}{3} \, a^{2} b^{6} \log \left (x^{3}\right ) - \frac{1680 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 560 \, a^{5} b^{3} x^{9} + 210 \, a^{6} b^{2} x^{6} + 48 \, a^{7} b x^{3} + 5 \, a^{8}}{90 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^19,x, algorithm="maxima")

[Out]

1/6*b^8*x^6 + 8/3*a*b^7*x^3 + 28/3*a^2*b^6*log(x^3) - 1/90*(1680*a^3*b^5*x^15 +
1050*a^4*b^4*x^12 + 560*a^5*b^3*x^9 + 210*a^6*b^2*x^6 + 48*a^7*b*x^3 + 5*a^8)/x^
18

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Fricas [A]  time = 0.213185, size = 127, normalized size = 1.21 \[ \frac{15 \, b^{8} x^{24} + 240 \, a b^{7} x^{21} + 2520 \, a^{2} b^{6} x^{18} \log \left (x\right ) - 1680 \, a^{3} b^{5} x^{15} - 1050 \, a^{4} b^{4} x^{12} - 560 \, a^{5} b^{3} x^{9} - 210 \, a^{6} b^{2} x^{6} - 48 \, a^{7} b x^{3} - 5 \, a^{8}}{90 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^19,x, algorithm="fricas")

[Out]

1/90*(15*b^8*x^24 + 240*a*b^7*x^21 + 2520*a^2*b^6*x^18*log(x) - 1680*a^3*b^5*x^1
5 - 1050*a^4*b^4*x^12 - 560*a^5*b^3*x^9 - 210*a^6*b^2*x^6 - 48*a^7*b*x^3 - 5*a^8
)/x^18

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Sympy [A]  time = 3.41052, size = 99, normalized size = 0.94 \[ 28 a^{2} b^{6} \log{\left (x \right )} + \frac{8 a b^{7} x^{3}}{3} + \frac{b^{8} x^{6}}{6} - \frac{5 a^{8} + 48 a^{7} b x^{3} + 210 a^{6} b^{2} x^{6} + 560 a^{5} b^{3} x^{9} + 1050 a^{4} b^{4} x^{12} + 1680 a^{3} b^{5} x^{15}}{90 x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**19,x)

[Out]

28*a**2*b**6*log(x) + 8*a*b**7*x**3/3 + b**8*x**6/6 - (5*a**8 + 48*a**7*b*x**3 +
210*a**6*b**2*x**6 + 560*a**5*b**3*x**9 + 1050*a**4*b**4*x**12 + 1680*a**3*b**5
*x**15)/(90*x**18)

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GIAC/XCAS [A]  time = 0.218566, size = 140, normalized size = 1.33 \[ \frac{1}{6} \, b^{8} x^{6} + \frac{8}{3} \, a b^{7} x^{3} + 28 \, a^{2} b^{6}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2058 \, a^{2} b^{6} x^{18} + 1680 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 560 \, a^{5} b^{3} x^{9} + 210 \, a^{6} b^{2} x^{6} + 48 \, a^{7} b x^{3} + 5 \, a^{8}}{90 \, x^{18}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^19,x, algorithm="giac")

[Out]

1/6*b^8*x^6 + 8/3*a*b^7*x^3 + 28*a^2*b^6*ln(abs(x)) - 1/90*(2058*a^2*b^6*x^18 +
1680*a^3*b^5*x^15 + 1050*a^4*b^4*x^12 + 560*a^5*b^3*x^9 + 210*a^6*b^2*x^6 + 48*a
^7*b*x^3 + 5*a^8)/x^18