3.296 \(\int \frac{\left (a+b x^3\right )^8}{x^{13}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^8}{12 x^{12}}-\frac{8 a^7 b}{9 x^9}-\frac{14 a^6 b^2}{3 x^6}-\frac{56 a^5 b^3}{3 x^3}+70 a^4 b^4 \log (x)+\frac{56}{3} a^3 b^5 x^3+\frac{14}{3} a^2 b^6 x^6+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]

[Out]

-a^8/(12*x^12) - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/(3*x^3)
+ (56*a^3*b^5*x^3)/3 + (14*a^2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8*x^12)/12 + 7
0*a^4*b^4*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.127354, 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 \[ -\frac{a^8}{12 x^{12}}-\frac{8 a^7 b}{9 x^9}-\frac{14 a^6 b^2}{3 x^6}-\frac{56 a^5 b^3}{3 x^3}+70 a^4 b^4 \log (x)+\frac{56}{3} a^3 b^5 x^3+\frac{14}{3} a^2 b^6 x^6+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^13,x]

[Out]

-a^8/(12*x^12) - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/(3*x^3)
+ (56*a^3*b^5*x^3)/3 + (14*a^2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8*x^12)/12 + 7
0*a^4*b^4*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{12 x^{12}} - \frac{8 a^{7} b}{9 x^{9}} - \frac{14 a^{6} b^{2}}{3 x^{6}} - \frac{56 a^{5} b^{3}}{3 x^{3}} + \frac{70 a^{4} b^{4} \log{\left (x^{3} \right )}}{3} + \frac{56 a^{3} b^{5} x^{3}}{3} + \frac{28 a^{2} b^{6} \int ^{x^{3}} x\, dx}{3} + \frac{8 a b^{7} x^{9}}{9} + \frac{b^{8} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**13,x)

[Out]

-a**8/(12*x**12) - 8*a**7*b/(9*x**9) - 14*a**6*b**2/(3*x**6) - 56*a**5*b**3/(3*x
**3) + 70*a**4*b**4*log(x**3)/3 + 56*a**3*b**5*x**3/3 + 28*a**2*b**6*Integral(x,
(x, x**3))/3 + 8*a*b**7*x**9/9 + b**8*x**12/12

_______________________________________________________________________________________

Mathematica [A]  time = 0.00874738, size = 105, normalized size = 1. \[ -\frac{a^8}{12 x^{12}}-\frac{8 a^7 b}{9 x^9}-\frac{14 a^6 b^2}{3 x^6}-\frac{56 a^5 b^3}{3 x^3}+70 a^4 b^4 \log (x)+\frac{56}{3} a^3 b^5 x^3+\frac{14}{3} a^2 b^6 x^6+\frac{8}{9} a b^7 x^9+\frac{b^8 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^13,x]

[Out]

-a^8/(12*x^12) - (8*a^7*b)/(9*x^9) - (14*a^6*b^2)/(3*x^6) - (56*a^5*b^3)/(3*x^3)
+ (56*a^3*b^5*x^3)/3 + (14*a^2*b^6*x^6)/3 + (8*a*b^7*x^9)/9 + (b^8*x^12)/12 + 7
0*a^4*b^4*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{12\,{x}^{12}}}-{\frac{8\,{a}^{7}b}{9\,{x}^{9}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,{x}^{6}}}-{\frac{56\,{a}^{5}{b}^{3}}{3\,{x}^{3}}}+{\frac{56\,{a}^{3}{b}^{5}{x}^{3}}{3}}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+{\frac{8\,a{b}^{7}{x}^{9}}{9}}+{\frac{{b}^{8}{x}^{12}}{12}}+70\,{a}^{4}{b}^{4}\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^13,x)

[Out]

-1/12*a^8/x^12-8/9*a^7*b/x^9-14/3*a^6*b^2/x^6-56/3*a^5*b^3/x^3+56/3*a^3*b^5*x^3+
14/3*a^2*b^6*x^6+8/9*a*b^7*x^9+1/12*b^8*x^12+70*a^4*b^4*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.45092, size = 127, normalized size = 1.21 \[ \frac{1}{12} \, b^{8} x^{12} + \frac{8}{9} \, a b^{7} x^{9} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{56}{3} \, a^{3} b^{5} x^{3} + \frac{70}{3} \, a^{4} b^{4} \log \left (x^{3}\right ) - \frac{672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^13,x, algorithm="maxima")

[Out]

1/12*b^8*x^12 + 8/9*a*b^7*x^9 + 14/3*a^2*b^6*x^6 + 56/3*a^3*b^5*x^3 + 70/3*a^4*b
^4*log(x^3) - 1/36*(672*a^5*b^3*x^9 + 168*a^6*b^2*x^6 + 32*a^7*b*x^3 + 3*a^8)/x^
12

_______________________________________________________________________________________

Fricas [A]  time = 0.212814, size = 127, normalized size = 1.21 \[ \frac{3 \, b^{8} x^{24} + 32 \, a b^{7} x^{21} + 168 \, a^{2} b^{6} x^{18} + 672 \, a^{3} b^{5} x^{15} + 2520 \, a^{4} b^{4} x^{12} \log \left (x\right ) - 672 \, a^{5} b^{3} x^{9} - 168 \, a^{6} b^{2} x^{6} - 32 \, a^{7} b x^{3} - 3 \, a^{8}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^13,x, algorithm="fricas")

[Out]

1/36*(3*b^8*x^24 + 32*a*b^7*x^21 + 168*a^2*b^6*x^18 + 672*a^3*b^5*x^15 + 2520*a^
4*b^4*x^12*log(x) - 672*a^5*b^3*x^9 - 168*a^6*b^2*x^6 - 32*a^7*b*x^3 - 3*a^8)/x^
12

_______________________________________________________________________________________

Sympy [A]  time = 2.50088, size = 102, normalized size = 0.97 \[ 70 a^{4} b^{4} \log{\left (x \right )} + \frac{56 a^{3} b^{5} x^{3}}{3} + \frac{14 a^{2} b^{6} x^{6}}{3} + \frac{8 a b^{7} x^{9}}{9} + \frac{b^{8} x^{12}}{12} - \frac{3 a^{8} + 32 a^{7} b x^{3} + 168 a^{6} b^{2} x^{6} + 672 a^{5} b^{3} x^{9}}{36 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**13,x)

[Out]

70*a**4*b**4*log(x) + 56*a**3*b**5*x**3/3 + 14*a**2*b**6*x**6/3 + 8*a*b**7*x**9/
9 + b**8*x**12/12 - (3*a**8 + 32*a**7*b*x**3 + 168*a**6*b**2*x**6 + 672*a**5*b**
3*x**9)/(36*x**12)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.219508, size = 140, normalized size = 1.33 \[ \frac{1}{12} \, b^{8} x^{12} + \frac{8}{9} \, a b^{7} x^{9} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{56}{3} \, a^{3} b^{5} x^{3} + 70 \, a^{4} b^{4}{\rm ln}\left ({\left | x \right |}\right ) - \frac{1750 \, a^{4} b^{4} x^{12} + 672 \, a^{5} b^{3} x^{9} + 168 \, a^{6} b^{2} x^{6} + 32 \, a^{7} b x^{3} + 3 \, a^{8}}{36 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^13,x, algorithm="giac")

[Out]

1/12*b^8*x^12 + 8/9*a*b^7*x^9 + 14/3*a^2*b^6*x^6 + 56/3*a^3*b^5*x^3 + 70*a^4*b^4
*ln(abs(x)) - 1/36*(1750*a^4*b^4*x^12 + 672*a^5*b^3*x^9 + 168*a^6*b^2*x^6 + 32*a
^7*b*x^3 + 3*a^8)/x^12