3.295 \(\int \frac{\left (a+b x^3\right )^8}{x^{10}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.130433, 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 \[ -\frac{a^8}{9 x^9}-\frac{4 a^7 b}{3 x^6}-\frac{28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)+\frac{70}{3} a^4 b^4 x^3+\frac{28}{3} a^3 b^5 x^6+\frac{28}{9} a^2 b^6 x^9+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^10,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**10,x)

[Out]

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

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Mathematica [A]  time = 0.0182771, size = 105, normalized size = 1. \[ -\frac{a^8}{9 x^9}-\frac{4 a^7 b}{3 x^6}-\frac{28 a^6 b^2}{3 x^3}+56 a^5 b^3 \log (x)+\frac{70}{3} a^4 b^4 x^3+\frac{28}{3} a^3 b^5 x^6+\frac{28}{9} a^2 b^6 x^9+\frac{2}{3} a b^7 x^{12}+\frac{b^8 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^10,x]

[Out]

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

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Maple [A]  time = 0.01, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{9\,{x}^{9}}}-{\frac{4\,{a}^{7}b}{3\,{x}^{6}}}-{\frac{28\,{a}^{6}{b}^{2}}{3\,{x}^{3}}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{3}}{3}}+{\frac{28\,{a}^{3}{b}^{5}{x}^{6}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{9}}{9}}+{\frac{2\,a{b}^{7}{x}^{12}}{3}}+{\frac{{b}^{8}{x}^{15}}{15}}+56\,{a}^{5}{b}^{3}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^10,x)

[Out]

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

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Maxima [A]  time = 1.42643, size = 124, normalized size = 1.18 \[ \frac{1}{15} \, b^{8} x^{15} + \frac{2}{3} \, a b^{7} x^{12} + \frac{28}{9} \, a^{2} b^{6} x^{9} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{70}{3} \, a^{4} b^{4} x^{3} + \frac{56}{3} \, a^{5} b^{3} \log \left (x^{3}\right ) - \frac{84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^10,x, algorithm="maxima")

[Out]

1/15*b^8*x^15 + 2/3*a*b^7*x^12 + 28/9*a^2*b^6*x^9 + 28/3*a^3*b^5*x^6 + 70/3*a^4*
b^4*x^3 + 56/3*a^5*b^3*log(x^3) - 1/9*(84*a^6*b^2*x^6 + 12*a^7*b*x^3 + a^8)/x^9

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Fricas [A]  time = 0.215436, size = 127, normalized size = 1.21 \[ \frac{3 \, b^{8} x^{24} + 30 \, a b^{7} x^{21} + 140 \, a^{2} b^{6} x^{18} + 420 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 2520 \, a^{5} b^{3} x^{9} \log \left (x\right ) - 420 \, a^{6} b^{2} x^{6} - 60 \, a^{7} b x^{3} - 5 \, a^{8}}{45 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.04845, size = 102, normalized size = 0.97 \[ 56 a^{5} b^{3} \log{\left (x \right )} + \frac{70 a^{4} b^{4} x^{3}}{3} + \frac{28 a^{3} b^{5} x^{6}}{3} + \frac{28 a^{2} b^{6} x^{9}}{9} + \frac{2 a b^{7} x^{12}}{3} + \frac{b^{8} x^{15}}{15} - \frac{a^{8} + 12 a^{7} b x^{3} + 84 a^{6} b^{2} x^{6}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**10,x)

[Out]

56*a**5*b**3*log(x) + 70*a**4*b**4*x**3/3 + 28*a**3*b**5*x**6/3 + 28*a**2*b**6*x
**9/9 + 2*a*b**7*x**12/3 + b**8*x**15/15 - (a**8 + 12*a**7*b*x**3 + 84*a**6*b**2
*x**6)/(9*x**9)

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GIAC/XCAS [A]  time = 0.223587, size = 138, normalized size = 1.31 \[ \frac{1}{15} \, b^{8} x^{15} + \frac{2}{3} \, a b^{7} x^{12} + \frac{28}{9} \, a^{2} b^{6} x^{9} + \frac{28}{3} \, a^{3} b^{5} x^{6} + \frac{70}{3} \, a^{4} b^{4} x^{3} + 56 \, a^{5} b^{3}{\rm ln}\left ({\left | x \right |}\right ) - \frac{308 \, a^{5} b^{3} x^{9} + 84 \, a^{6} b^{2} x^{6} + 12 \, a^{7} b x^{3} + a^{8}}{9 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^10,x, algorithm="giac")

[Out]

1/15*b^8*x^15 + 2/3*a*b^7*x^12 + 28/9*a^2*b^6*x^9 + 28/3*a^3*b^5*x^6 + 70/3*a^4*
b^4*x^3 + 56*a^5*b^3*ln(abs(x)) - 1/9*(308*a^5*b^3*x^9 + 84*a^6*b^2*x^6 + 12*a^7
*b*x^3 + a^8)/x^9