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

Optimal. Leaf size=104 \[ a^8 \log (x)+\frac{8}{3} a^7 b x^3+\frac{14}{3} a^6 b^2 x^6+\frac{56}{9} a^5 b^3 x^9+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{15} a^3 b^5 x^{15}+\frac{14}{9} a^2 b^6 x^{18}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24} \]

[Out]

(8*a^7*b*x^3)/3 + (14*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (35*a^4*b^4*x^12)/6
+ (56*a^3*b^5*x^15)/15 + (14*a^2*b^6*x^18)/9 + (8*a*b^7*x^21)/21 + (b^8*x^24)/24
+ a^8*Log[x]

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Rubi [A]  time = 0.11985, antiderivative size = 104, 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 \[ a^8 \log (x)+\frac{8}{3} a^7 b x^3+\frac{14}{3} a^6 b^2 x^6+\frac{56}{9} a^5 b^3 x^9+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{15} a^3 b^5 x^{15}+\frac{14}{9} a^2 b^6 x^{18}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x,x]

[Out]

(8*a^7*b*x^3)/3 + (14*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (35*a^4*b^4*x^12)/6
+ (56*a^3*b^5*x^15)/15 + (14*a^2*b^6*x^18)/9 + (8*a*b^7*x^21)/21 + (b^8*x^24)/24
+ a^8*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} \log{\left (x^{3} \right )}}{3} + \frac{8 a^{7} b x^{3}}{3} + \frac{28 a^{6} b^{2} \int ^{x^{3}} x\, dx}{3} + \frac{56 a^{5} b^{3} x^{9}}{9} + \frac{35 a^{4} b^{4} x^{12}}{6} + \frac{56 a^{3} b^{5} x^{15}}{15} + \frac{14 a^{2} b^{6} x^{18}}{9} + \frac{8 a b^{7} x^{21}}{21} + \frac{b^{8} x^{24}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**8*log(x**3)/3 + 8*a**7*b*x**3/3 + 28*a**6*b**2*Integral(x, (x, x**3))/3 + 56*
a**5*b**3*x**9/9 + 35*a**4*b**4*x**12/6 + 56*a**3*b**5*x**15/15 + 14*a**2*b**6*x
**18/9 + 8*a*b**7*x**21/21 + b**8*x**24/24

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Mathematica [A]  time = 0.00811701, size = 104, normalized size = 1. \[ a^8 \log (x)+\frac{8}{3} a^7 b x^3+\frac{14}{3} a^6 b^2 x^6+\frac{56}{9} a^5 b^3 x^9+\frac{35}{6} a^4 b^4 x^{12}+\frac{56}{15} a^3 b^5 x^{15}+\frac{14}{9} a^2 b^6 x^{18}+\frac{8}{21} a b^7 x^{21}+\frac{b^8 x^{24}}{24} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x,x]

[Out]

(8*a^7*b*x^3)/3 + (14*a^6*b^2*x^6)/3 + (56*a^5*b^3*x^9)/9 + (35*a^4*b^4*x^12)/6
+ (56*a^3*b^5*x^15)/15 + (14*a^2*b^6*x^18)/9 + (8*a*b^7*x^21)/21 + (b^8*x^24)/24
+ a^8*Log[x]

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Maple [A]  time = 0.005, size = 89, normalized size = 0.9 \[{\frac{8\,{a}^{7}b{x}^{3}}{3}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{6}}{3}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{9}}{9}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{12}}{6}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{15}}{15}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{18}}{9}}+{\frac{8\,a{b}^{7}{x}^{21}}{21}}+{\frac{{b}^{8}{x}^{24}}{24}}+{a}^{8}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x,x)

[Out]

8/3*a^7*b*x^3+14/3*a^6*b^2*x^6+56/9*a^5*b^3*x^9+35/6*a^4*b^4*x^12+56/15*a^3*b^5*
x^15+14/9*a^2*b^6*x^18+8/21*a*b^7*x^21+1/24*b^8*x^24+a^8*ln(x)

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Maxima [A]  time = 1.44793, size = 123, normalized size = 1.18 \[ \frac{1}{24} \, b^{8} x^{24} + \frac{8}{21} \, a b^{7} x^{21} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{56}{15} \, a^{3} b^{5} x^{15} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{8}{3} \, a^{7} b x^{3} + \frac{1}{3} \, a^{8} \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b^8*x^24 + 8/21*a*b^7*x^21 + 14/9*a^2*b^6*x^18 + 56/15*a^3*b^5*x^15 + 35/6*
a^4*b^4*x^12 + 56/9*a^5*b^3*x^9 + 14/3*a^6*b^2*x^6 + 8/3*a^7*b*x^3 + 1/3*a^8*log
(x^3)

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Fricas [A]  time = 0.213882, size = 119, normalized size = 1.14 \[ \frac{1}{24} \, b^{8} x^{24} + \frac{8}{21} \, a b^{7} x^{21} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{56}{15} \, a^{3} b^{5} x^{15} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{8}{3} \, a^{7} b x^{3} + a^{8} \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b^8*x^24 + 8/21*a*b^7*x^21 + 14/9*a^2*b^6*x^18 + 56/15*a^3*b^5*x^15 + 35/6*
a^4*b^4*x^12 + 56/9*a^5*b^3*x^9 + 14/3*a^6*b^2*x^6 + 8/3*a^7*b*x^3 + a^8*log(x)

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Sympy [A]  time = 1.32147, size = 105, normalized size = 1.01 \[ a^{8} \log{\left (x \right )} + \frac{8 a^{7} b x^{3}}{3} + \frac{14 a^{6} b^{2} x^{6}}{3} + \frac{56 a^{5} b^{3} x^{9}}{9} + \frac{35 a^{4} b^{4} x^{12}}{6} + \frac{56 a^{3} b^{5} x^{15}}{15} + \frac{14 a^{2} b^{6} x^{18}}{9} + \frac{8 a b^{7} x^{21}}{21} + \frac{b^{8} x^{24}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x,x)

[Out]

a**8*log(x) + 8*a**7*b*x**3/3 + 14*a**6*b**2*x**6/3 + 56*a**5*b**3*x**9/9 + 35*a
**4*b**4*x**12/6 + 56*a**3*b**5*x**15/15 + 14*a**2*b**6*x**18/9 + 8*a*b**7*x**21
/21 + b**8*x**24/24

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GIAC/XCAS [A]  time = 0.215843, size = 120, normalized size = 1.15 \[ \frac{1}{24} \, b^{8} x^{24} + \frac{8}{21} \, a b^{7} x^{21} + \frac{14}{9} \, a^{2} b^{6} x^{18} + \frac{56}{15} \, a^{3} b^{5} x^{15} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{14}{3} \, a^{6} b^{2} x^{6} + \frac{8}{3} \, a^{7} b x^{3} + a^{8}{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*b^8*x^24 + 8/21*a*b^7*x^21 + 14/9*a^2*b^6*x^18 + 56/15*a^3*b^5*x^15 + 35/6*
a^4*b^4*x^12 + 56/9*a^5*b^3*x^9 + 14/3*a^6*b^2*x^6 + 8/3*a^7*b*x^3 + a^8*ln(abs(
x))