3.304 \(\int \frac{\left (a+b x^3\right )^8}{x^{37}} \, dx\)

Optimal. Leaf size=84 \[ \frac{b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac{b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac{\left (a+b x^3\right )^9}{36 a x^{36}} \]

[Out]

-(a + b*x^3)^9/(36*a*x^36) + (b*(a + b*x^3)^9)/(132*a^2*x^33) - (b^2*(a + b*x^3)
^9)/(660*a^3*x^30) + (b^3*(a + b*x^3)^9)/(5940*a^4*x^27)

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Rubi [A]  time = 0.10607, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b^3 \left (a+b x^3\right )^9}{5940 a^4 x^{27}}-\frac{b^2 \left (a+b x^3\right )^9}{660 a^3 x^{30}}+\frac{b \left (a+b x^3\right )^9}{132 a^2 x^{33}}-\frac{\left (a+b x^3\right )^9}{36 a x^{36}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^37,x]

[Out]

-(a + b*x^3)^9/(36*a*x^36) + (b*(a + b*x^3)^9)/(132*a^2*x^33) - (b^2*(a + b*x^3)
^9)/(660*a^3*x^30) + (b^3*(a + b*x^3)^9)/(5940*a^4*x^27)

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Rubi in Sympy [A]  time = 12.6375, size = 73, normalized size = 0.87 \[ - \frac{\left (a + b x^{3}\right )^{9}}{36 a x^{36}} + \frac{b \left (a + b x^{3}\right )^{9}}{132 a^{2} x^{33}} - \frac{b^{2} \left (a + b x^{3}\right )^{9}}{660 a^{3} x^{30}} + \frac{b^{3} \left (a + b x^{3}\right )^{9}}{5940 a^{4} x^{27}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**37,x)

[Out]

-(a + b*x**3)**9/(36*a*x**36) + b*(a + b*x**3)**9/(132*a**2*x**33) - b**2*(a + b
*x**3)**9/(660*a**3*x**30) + b**3*(a + b*x**3)**9/(5940*a**4*x**27)

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Mathematica [A]  time = 0.00787638, size = 108, normalized size = 1.29 \[ -\frac{a^8}{36 x^{36}}-\frac{8 a^7 b}{33 x^{33}}-\frac{14 a^6 b^2}{15 x^{30}}-\frac{56 a^5 b^3}{27 x^{27}}-\frac{35 a^4 b^4}{12 x^{24}}-\frac{8 a^3 b^5}{3 x^{21}}-\frac{14 a^2 b^6}{9 x^{18}}-\frac{8 a b^7}{15 x^{15}}-\frac{b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^37,x]

[Out]

-a^8/(36*x^36) - (8*a^7*b)/(33*x^33) - (14*a^6*b^2)/(15*x^30) - (56*a^5*b^3)/(27
*x^27) - (35*a^4*b^4)/(12*x^24) - (8*a^3*b^5)/(3*x^21) - (14*a^2*b^6)/(9*x^18) -
(8*a*b^7)/(15*x^15) - b^8/(12*x^12)

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Maple [A]  time = 0.01, size = 91, normalized size = 1.1 \[ -{\frac{{b}^{8}}{12\,{x}^{12}}}-{\frac{8\,a{b}^{7}}{15\,{x}^{15}}}-{\frac{14\,{a}^{2}{b}^{6}}{9\,{x}^{18}}}-{\frac{35\,{a}^{4}{b}^{4}}{12\,{x}^{24}}}-{\frac{56\,{a}^{5}{b}^{3}}{27\,{x}^{27}}}-{\frac{14\,{a}^{6}{b}^{2}}{15\,{x}^{30}}}-{\frac{8\,{a}^{3}{b}^{5}}{3\,{x}^{21}}}-{\frac{8\,{a}^{7}b}{33\,{x}^{33}}}-{\frac{{a}^{8}}{36\,{x}^{36}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^37,x)

[Out]

-1/12*b^8/x^12-8/15*a*b^7/x^15-14/9*a^2*b^6/x^18-35/12*a^4*b^4/x^24-56/27*a^5*b^
3/x^27-14/15*a^6*b^2/x^30-8/3*a^3*b^5/x^21-8/33*a^7*b/x^33-1/36*a^8/x^36

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Maxima [A]  time = 1.42196, size = 124, normalized size = 1.48 \[ -\frac{495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^37,x, algorithm="maxima")

[Out]

-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^5*x^15
+ 17325*a^4*b^4*x^12 + 12320*a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440*a^7*b*x^3 +
165*a^8)/x^36

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Fricas [A]  time = 0.20289, size = 124, normalized size = 1.48 \[ -\frac{495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^37,x, algorithm="fricas")

[Out]

-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^5*x^15
+ 17325*a^4*b^4*x^12 + 12320*a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440*a^7*b*x^3 +
165*a^8)/x^36

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Sympy [A]  time = 6.09571, size = 99, normalized size = 1.18 \[ - \frac{165 a^{8} + 1440 a^{7} b x^{3} + 5544 a^{6} b^{2} x^{6} + 12320 a^{5} b^{3} x^{9} + 17325 a^{4} b^{4} x^{12} + 15840 a^{3} b^{5} x^{15} + 9240 a^{2} b^{6} x^{18} + 3168 a b^{7} x^{21} + 495 b^{8} x^{24}}{5940 x^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**37,x)

[Out]

-(165*a**8 + 1440*a**7*b*x**3 + 5544*a**6*b**2*x**6 + 12320*a**5*b**3*x**9 + 173
25*a**4*b**4*x**12 + 15840*a**3*b**5*x**15 + 9240*a**2*b**6*x**18 + 3168*a*b**7*
x**21 + 495*b**8*x**24)/(5940*x**36)

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GIAC/XCAS [A]  time = 0.222435, size = 124, normalized size = 1.48 \[ -\frac{495 \, b^{8} x^{24} + 3168 \, a b^{7} x^{21} + 9240 \, a^{2} b^{6} x^{18} + 15840 \, a^{3} b^{5} x^{15} + 17325 \, a^{4} b^{4} x^{12} + 12320 \, a^{5} b^{3} x^{9} + 5544 \, a^{6} b^{2} x^{6} + 1440 \, a^{7} b x^{3} + 165 \, a^{8}}{5940 \, x^{36}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^37,x, algorithm="giac")

[Out]

-1/5940*(495*b^8*x^24 + 3168*a*b^7*x^21 + 9240*a^2*b^6*x^18 + 15840*a^3*b^5*x^15
+ 17325*a^4*b^4*x^12 + 12320*a^5*b^3*x^9 + 5544*a^6*b^2*x^6 + 1440*a^7*b*x^3 +
165*a^8)/x^36