3.294 \(\int \frac{\left (a+b x^3\right )^8}{x^7} \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^8}{6 x^6}-\frac{8 a^7 b}{3 x^3}+28 a^6 b^2 \log (x)+\frac{56}{3} a^5 b^3 x^3+\frac{35}{3} a^4 b^4 x^6+\frac{56}{9} a^3 b^5 x^9+\frac{7}{3} a^2 b^6 x^{12}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]

[Out]

-a^8/(6*x^6) - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56
*a^3*b^5*x^9)/9 + (7*a^2*b^6*x^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^
6*b^2*Log[x]

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Rubi [A]  time = 0.133875, 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 \[ -\frac{a^8}{6 x^6}-\frac{8 a^7 b}{3 x^3}+28 a^6 b^2 \log (x)+\frac{56}{3} a^5 b^3 x^3+\frac{35}{3} a^4 b^4 x^6+\frac{56}{9} a^3 b^5 x^9+\frac{7}{3} a^2 b^6 x^{12}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^7,x]

[Out]

-a^8/(6*x^6) - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56
*a^3*b^5*x^9)/9 + (7*a^2*b^6*x^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^
6*b^2*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{6 x^{6}} - \frac{8 a^{7} b}{3 x^{3}} + \frac{28 a^{6} b^{2} \log{\left (x^{3} \right )}}{3} + \frac{56 a^{5} b^{3} x^{3}}{3} + \frac{70 a^{4} b^{4} \int ^{x^{3}} x\, dx}{3} + \frac{56 a^{3} b^{5} x^{9}}{9} + \frac{7 a^{2} b^{6} x^{12}}{3} + \frac{8 a b^{7} x^{15}}{15} + \frac{b^{8} x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**7,x)

[Out]

-a**8/(6*x**6) - 8*a**7*b/(3*x**3) + 28*a**6*b**2*log(x**3)/3 + 56*a**5*b**3*x**
3/3 + 70*a**4*b**4*Integral(x, (x, x**3))/3 + 56*a**3*b**5*x**9/9 + 7*a**2*b**6*
x**12/3 + 8*a*b**7*x**15/15 + b**8*x**18/18

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Mathematica [A]  time = 0.00911952, size = 105, normalized size = 1. \[ -\frac{a^8}{6 x^6}-\frac{8 a^7 b}{3 x^3}+28 a^6 b^2 \log (x)+\frac{56}{3} a^5 b^3 x^3+\frac{35}{3} a^4 b^4 x^6+\frac{56}{9} a^3 b^5 x^9+\frac{7}{3} a^2 b^6 x^{12}+\frac{8}{15} a b^7 x^{15}+\frac{b^8 x^{18}}{18} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^7,x]

[Out]

-a^8/(6*x^6) - (8*a^7*b)/(3*x^3) + (56*a^5*b^3*x^3)/3 + (35*a^4*b^4*x^6)/3 + (56
*a^3*b^5*x^9)/9 + (7*a^2*b^6*x^12)/3 + (8*a*b^7*x^15)/15 + (b^8*x^18)/18 + 28*a^
6*b^2*Log[x]

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Maple [A]  time = 0.009, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{6\,{x}^{6}}}-{\frac{8\,{a}^{7}b}{3\,{x}^{3}}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{3}}{3}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{6}}{3}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{9}}{9}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{12}}{3}}+{\frac{8\,a{b}^{7}{x}^{15}}{15}}+{\frac{{b}^{8}{x}^{18}}{18}}+28\,{a}^{6}{b}^{2}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^7,x)

[Out]

-1/6*a^8/x^6-8/3*a^7*b/x^3+56/3*a^5*b^3*x^3+35/3*a^4*b^4*x^6+56/9*a^3*b^5*x^9+7/
3*a^2*b^6*x^12+8/15*a*b^7*x^15+1/18*b^8*x^18+28*a^6*b^2*ln(x)

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Maxima [A]  time = 1.43207, size = 124, normalized size = 1.18 \[ \frac{1}{18} \, b^{8} x^{18} + \frac{8}{15} \, a b^{7} x^{15} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{3} \, a^{5} b^{3} x^{3} + \frac{28}{3} \, a^{6} b^{2} \log \left (x^{3}\right ) - \frac{16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^7,x, algorithm="maxima")

[Out]

1/18*b^8*x^18 + 8/15*a*b^7*x^15 + 7/3*a^2*b^6*x^12 + 56/9*a^3*b^5*x^9 + 35/3*a^4
*b^4*x^6 + 56/3*a^5*b^3*x^3 + 28/3*a^6*b^2*log(x^3) - 1/6*(16*a^7*b*x^3 + a^8)/x
^6

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Fricas [A]  time = 0.215054, size = 127, normalized size = 1.21 \[ \frac{5 \, b^{8} x^{24} + 48 \, a b^{7} x^{21} + 210 \, a^{2} b^{6} x^{18} + 560 \, a^{3} b^{5} x^{15} + 1050 \, a^{4} b^{4} x^{12} + 1680 \, a^{5} b^{3} x^{9} + 2520 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 240 \, a^{7} b x^{3} - 15 \, a^{8}}{90 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^7,x, algorithm="fricas")

[Out]

1/90*(5*b^8*x^24 + 48*a*b^7*x^21 + 210*a^2*b^6*x^18 + 560*a^3*b^5*x^15 + 1050*a^
4*b^4*x^12 + 1680*a^5*b^3*x^9 + 2520*a^6*b^2*x^6*log(x) - 240*a^7*b*x^3 - 15*a^8
)/x^6

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Sympy [A]  time = 1.75869, size = 104, normalized size = 0.99 \[ 28 a^{6} b^{2} \log{\left (x \right )} + \frac{56 a^{5} b^{3} x^{3}}{3} + \frac{35 a^{4} b^{4} x^{6}}{3} + \frac{56 a^{3} b^{5} x^{9}}{9} + \frac{7 a^{2} b^{6} x^{12}}{3} + \frac{8 a b^{7} x^{15}}{15} + \frac{b^{8} x^{18}}{18} - \frac{a^{8} + 16 a^{7} b x^{3}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**7,x)

[Out]

28*a**6*b**2*log(x) + 56*a**5*b**3*x**3/3 + 35*a**4*b**4*x**6/3 + 56*a**3*b**5*x
**9/9 + 7*a**2*b**6*x**12/3 + 8*a*b**7*x**15/15 + b**8*x**18/18 - (a**8 + 16*a**
7*b*x**3)/(6*x**6)

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GIAC/XCAS [A]  time = 0.216023, size = 138, normalized size = 1.31 \[ \frac{1}{18} \, b^{8} x^{18} + \frac{8}{15} \, a b^{7} x^{15} + \frac{7}{3} \, a^{2} b^{6} x^{12} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{35}{3} \, a^{4} b^{4} x^{6} + \frac{56}{3} \, a^{5} b^{3} x^{3} + 28 \, a^{6} b^{2}{\rm ln}\left ({\left | x \right |}\right ) - \frac{84 \, a^{6} b^{2} x^{6} + 16 \, a^{7} b x^{3} + a^{8}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^7,x, algorithm="giac")

[Out]

1/18*b^8*x^18 + 8/15*a*b^7*x^15 + 7/3*a^2*b^6*x^12 + 56/9*a^3*b^5*x^9 + 35/3*a^4
*b^4*x^6 + 56/3*a^5*b^3*x^3 + 28*a^6*b^2*ln(abs(x)) - 1/6*(84*a^6*b^2*x^6 + 16*a
^7*b*x^3 + a^8)/x^6