3.299 \(\int \frac{\left (a+b x^3\right )^8}{x^{22}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^8}{21 x^{21}}-\frac{4 a^7 b}{9 x^{18}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}+8 a b^7 \log (x)+\frac{b^8 x^3}{3} \]

[Out]

-a^8/(21*x^21) - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3)/(3*x
^12) - (70*a^4*b^4)/(9*x^9) - (28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3*x^3) + (b^8
*x^3)/3 + 8*a*b^7*Log[x]

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Rubi [A]  time = 0.121104, 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 \[ -\frac{a^8}{21 x^{21}}-\frac{4 a^7 b}{9 x^{18}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}+8 a b^7 \log (x)+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^22,x]

[Out]

-a^8/(21*x^21) - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3)/(3*x
^12) - (70*a^4*b^4)/(9*x^9) - (28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3*x^3) + (b^8
*x^3)/3 + 8*a*b^7*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{21 x^{21}} - \frac{4 a^{7} b}{9 x^{18}} - \frac{28 a^{6} b^{2}}{15 x^{15}} - \frac{14 a^{5} b^{3}}{3 x^{12}} - \frac{70 a^{4} b^{4}}{9 x^{9}} - \frac{28 a^{3} b^{5}}{3 x^{6}} - \frac{28 a^{2} b^{6}}{3 x^{3}} + \frac{8 a b^{7} \log{\left (x^{3} \right )}}{3} + \frac{\int ^{x^{3}} b^{8}\, dx}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**22,x)

[Out]

-a**8/(21*x**21) - 4*a**7*b/(9*x**18) - 28*a**6*b**2/(15*x**15) - 14*a**5*b**3/(
3*x**12) - 70*a**4*b**4/(9*x**9) - 28*a**3*b**5/(3*x**6) - 28*a**2*b**6/(3*x**3)
+ 8*a*b**7*log(x**3)/3 + Integral(b**8, (x, x**3))/3

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Mathematica [A]  time = 0.0181568, size = 105, normalized size = 1. \[ -\frac{a^8}{21 x^{21}}-\frac{4 a^7 b}{9 x^{18}}-\frac{28 a^6 b^2}{15 x^{15}}-\frac{14 a^5 b^3}{3 x^{12}}-\frac{70 a^4 b^4}{9 x^9}-\frac{28 a^3 b^5}{3 x^6}-\frac{28 a^2 b^6}{3 x^3}+8 a b^7 \log (x)+\frac{b^8 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^22,x]

[Out]

-a^8/(21*x^21) - (4*a^7*b)/(9*x^18) - (28*a^6*b^2)/(15*x^15) - (14*a^5*b^3)/(3*x
^12) - (70*a^4*b^4)/(9*x^9) - (28*a^3*b^5)/(3*x^6) - (28*a^2*b^6)/(3*x^3) + (b^8
*x^3)/3 + 8*a*b^7*Log[x]

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Maple [A]  time = 0.013, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{21\,{x}^{21}}}-{\frac{4\,{a}^{7}b}{9\,{x}^{18}}}-{\frac{28\,{a}^{6}{b}^{2}}{15\,{x}^{15}}}-{\frac{14\,{a}^{5}{b}^{3}}{3\,{x}^{12}}}-{\frac{70\,{a}^{4}{b}^{4}}{9\,{x}^{9}}}-{\frac{28\,{a}^{3}{b}^{5}}{3\,{x}^{6}}}-{\frac{28\,{a}^{2}{b}^{6}}{3\,{x}^{3}}}+{\frac{{b}^{8}{x}^{3}}{3}}+8\,a{b}^{7}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^22,x)

[Out]

-1/21*a^8/x^21-4/9*a^7*b/x^18-28/15*a^6*b^2/x^15-14/3*a^5*b^3/x^12-70/9*a^4*b^4/
x^9-28/3*a^3*b^5/x^6-28/3*a^2*b^6/x^3+1/3*b^8*x^3+8*a*b^7*ln(x)

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Maxima [A]  time = 1.42445, size = 127, normalized size = 1.21 \[ \frac{1}{3} \, b^{8} x^{3} + \frac{8}{3} \, a b^{7} \log \left (x^{3}\right ) - \frac{2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^22,x, algorithm="maxima")

[Out]

1/3*b^8*x^3 + 8/3*a*b^7*log(x^3) - 1/315*(2940*a^2*b^6*x^18 + 2940*a^3*b^5*x^15
+ 2450*a^4*b^4*x^12 + 1470*a^5*b^3*x^9 + 588*a^6*b^2*x^6 + 140*a^7*b*x^3 + 15*a^
8)/x^21

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Fricas [A]  time = 0.210282, size = 127, normalized size = 1.21 \[ \frac{105 \, b^{8} x^{24} + 2520 \, a b^{7} x^{21} \log \left (x\right ) - 2940 \, a^{2} b^{6} x^{18} - 2940 \, a^{3} b^{5} x^{15} - 2450 \, a^{4} b^{4} x^{12} - 1470 \, a^{5} b^{3} x^{9} - 588 \, a^{6} b^{2} x^{6} - 140 \, a^{7} b x^{3} - 15 \, a^{8}}{315 \, x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^22,x, algorithm="fricas")

[Out]

1/315*(105*b^8*x^24 + 2520*a*b^7*x^21*log(x) - 2940*a^2*b^6*x^18 - 2940*a^3*b^5*
x^15 - 2450*a^4*b^4*x^12 - 1470*a^5*b^3*x^9 - 588*a^6*b^2*x^6 - 140*a^7*b*x^3 -
15*a^8)/x^21

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Sympy [A]  time = 3.95809, size = 97, normalized size = 0.92 \[ 8 a b^{7} \log{\left (x \right )} + \frac{b^{8} x^{3}}{3} - \frac{15 a^{8} + 140 a^{7} b x^{3} + 588 a^{6} b^{2} x^{6} + 1470 a^{5} b^{3} x^{9} + 2450 a^{4} b^{4} x^{12} + 2940 a^{3} b^{5} x^{15} + 2940 a^{2} b^{6} x^{18}}{315 x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**22,x)

[Out]

8*a*b**7*log(x) + b**8*x**3/3 - (15*a**8 + 140*a**7*b*x**3 + 588*a**6*b**2*x**6
+ 1470*a**5*b**3*x**9 + 2450*a**4*b**4*x**12 + 2940*a**3*b**5*x**15 + 2940*a**2*
b**6*x**18)/(315*x**21)

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GIAC/XCAS [A]  time = 0.216382, size = 138, normalized size = 1.31 \[ \frac{1}{3} \, b^{8} x^{3} + 8 \, a b^{7}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2178 \, a b^{7} x^{21} + 2940 \, a^{2} b^{6} x^{18} + 2940 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 1470 \, a^{5} b^{3} x^{9} + 588 \, a^{6} b^{2} x^{6} + 140 \, a^{7} b x^{3} + 15 \, a^{8}}{315 \, x^{21}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^22,x, algorithm="giac")

[Out]

1/3*b^8*x^3 + 8*a*b^7*ln(abs(x)) - 1/315*(2178*a*b^7*x^21 + 2940*a^2*b^6*x^18 +
2940*a^3*b^5*x^15 + 2450*a^4*b^4*x^12 + 1470*a^5*b^3*x^9 + 588*a^6*b^2*x^6 + 140
*a^7*b*x^3 + 15*a^8)/x^21