3.305 \(\int \frac{\left (a+b x^3\right )^8}{x^{40}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}+\frac{b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{\left (a+b x^3\right )^9}{39 a x^{39}} \]

[Out]

-(a + b*x^3)^9/(39*a*x^39) + (b*(a + b*x^3)^9)/(117*a^2*x^36) - (b^2*(a + b*x^3)
^9)/(429*a^3*x^33) + (b^3*(a + b*x^3)^9)/(2145*a^4*x^30) - (b^4*(a + b*x^3)^9)/(
19305*a^5*x^27)

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Rubi [A]  time = 0.137152, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{b^4 \left (a+b x^3\right )^9}{19305 a^5 x^{27}}+\frac{b^3 \left (a+b x^3\right )^9}{2145 a^4 x^{30}}-\frac{b^2 \left (a+b x^3\right )^9}{429 a^3 x^{33}}+\frac{b \left (a+b x^3\right )^9}{117 a^2 x^{36}}-\frac{\left (a+b x^3\right )^9}{39 a x^{39}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^40,x]

[Out]

-(a + b*x^3)^9/(39*a*x^39) + (b*(a + b*x^3)^9)/(117*a^2*x^36) - (b^2*(a + b*x^3)
^9)/(429*a^3*x^33) + (b^3*(a + b*x^3)^9)/(2145*a^4*x^30) - (b^4*(a + b*x^3)^9)/(
19305*a^5*x^27)

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Rubi in Sympy [A]  time = 23.7686, size = 109, normalized size = 1.03 \[ - \frac{a^{8}}{39 x^{39}} - \frac{2 a^{7} b}{9 x^{36}} - \frac{28 a^{6} b^{2}}{33 x^{33}} - \frac{28 a^{5} b^{3}}{15 x^{30}} - \frac{70 a^{4} b^{4}}{27 x^{27}} - \frac{7 a^{3} b^{5}}{3 x^{24}} - \frac{4 a^{2} b^{6}}{3 x^{21}} - \frac{4 a b^{7}}{9 x^{18}} - \frac{b^{8}}{15 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**40,x)

[Out]

-a**8/(39*x**39) - 2*a**7*b/(9*x**36) - 28*a**6*b**2/(33*x**33) - 28*a**5*b**3/(
15*x**30) - 70*a**4*b**4/(27*x**27) - 7*a**3*b**5/(3*x**24) - 4*a**2*b**6/(3*x**
21) - 4*a*b**7/(9*x**18) - b**8/(15*x**15)

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Mathematica [A]  time = 0.0121738, size = 108, normalized size = 1.02 \[ -\frac{a^8}{39 x^{39}}-\frac{2 a^7 b}{9 x^{36}}-\frac{28 a^6 b^2}{33 x^{33}}-\frac{28 a^5 b^3}{15 x^{30}}-\frac{70 a^4 b^4}{27 x^{27}}-\frac{7 a^3 b^5}{3 x^{24}}-\frac{4 a^2 b^6}{3 x^{21}}-\frac{4 a b^7}{9 x^{18}}-\frac{b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^40,x]

[Out]

-a^8/(39*x^39) - (2*a^7*b)/(9*x^36) - (28*a^6*b^2)/(33*x^33) - (28*a^5*b^3)/(15*
x^30) - (70*a^4*b^4)/(27*x^27) - (7*a^3*b^5)/(3*x^24) - (4*a^2*b^6)/(3*x^21) - (
4*a*b^7)/(9*x^18) - b^8/(15*x^15)

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Maple [A]  time = 0.01, size = 91, normalized size = 0.9 \[ -{\frac{{b}^{8}}{15\,{x}^{15}}}-{\frac{28\,{a}^{5}{b}^{3}}{15\,{x}^{30}}}-{\frac{2\,{a}^{7}b}{9\,{x}^{36}}}-{\frac{{a}^{8}}{39\,{x}^{39}}}-{\frac{28\,{a}^{6}{b}^{2}}{33\,{x}^{33}}}-{\frac{4\,a{b}^{7}}{9\,{x}^{18}}}-{\frac{7\,{a}^{3}{b}^{5}}{3\,{x}^{24}}}-{\frac{70\,{a}^{4}{b}^{4}}{27\,{x}^{27}}}-{\frac{4\,{a}^{2}{b}^{6}}{3\,{x}^{21}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^40,x)

[Out]

-1/15*b^8/x^15-28/15*a^5*b^3/x^30-2/9*a^7*b/x^36-1/39*a^8/x^39-28/33*a^6*b^2/x^3
3-4/9*a*b^7/x^18-7/3*a^3*b^5/x^24-70/27*a^4*b^4/x^27-4/3*a^2*b^6/x^21

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Maxima [A]  time = 1.45825, size = 124, normalized size = 1.17 \[ -\frac{1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^40,x, algorithm="maxima")

[Out]

-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3*b^5*x
^15 + 50050*a^4*b^4*x^12 + 36036*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4290*a^7*b*x^
3 + 495*a^8)/x^39

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Fricas [A]  time = 0.204735, size = 124, normalized size = 1.17 \[ -\frac{1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^40,x, algorithm="fricas")

[Out]

-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3*b^5*x
^15 + 50050*a^4*b^4*x^12 + 36036*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4290*a^7*b*x^
3 + 495*a^8)/x^39

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Sympy [A]  time = 6.49634, size = 99, normalized size = 0.93 \[ - \frac{495 a^{8} + 4290 a^{7} b x^{3} + 16380 a^{6} b^{2} x^{6} + 36036 a^{5} b^{3} x^{9} + 50050 a^{4} b^{4} x^{12} + 45045 a^{3} b^{5} x^{15} + 25740 a^{2} b^{6} x^{18} + 8580 a b^{7} x^{21} + 1287 b^{8} x^{24}}{19305 x^{39}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**40,x)

[Out]

-(495*a**8 + 4290*a**7*b*x**3 + 16380*a**6*b**2*x**6 + 36036*a**5*b**3*x**9 + 50
050*a**4*b**4*x**12 + 45045*a**3*b**5*x**15 + 25740*a**2*b**6*x**18 + 8580*a*b**
7*x**21 + 1287*b**8*x**24)/(19305*x**39)

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GIAC/XCAS [A]  time = 0.217514, size = 124, normalized size = 1.17 \[ -\frac{1287 \, b^{8} x^{24} + 8580 \, a b^{7} x^{21} + 25740 \, a^{2} b^{6} x^{18} + 45045 \, a^{3} b^{5} x^{15} + 50050 \, a^{4} b^{4} x^{12} + 36036 \, a^{5} b^{3} x^{9} + 16380 \, a^{6} b^{2} x^{6} + 4290 \, a^{7} b x^{3} + 495 \, a^{8}}{19305 \, x^{39}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^40,x, algorithm="giac")

[Out]

-1/19305*(1287*b^8*x^24 + 8580*a*b^7*x^21 + 25740*a^2*b^6*x^18 + 45045*a^3*b^5*x
^15 + 50050*a^4*b^4*x^12 + 36036*a^5*b^3*x^9 + 16380*a^6*b^2*x^6 + 4290*a^7*b*x^
3 + 495*a^8)/x^39