3.293 \(\int \frac{\left (a+b x^3\right )^8}{x^4} \, dx\)

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

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.155611, 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 \[ -\frac{a^8}{3 x^3}+8 a^7 b \log (x)+\frac{28}{3} a^6 b^2 x^3+\frac{28}{3} a^5 b^3 x^6+\frac{70}{9} a^4 b^4 x^9+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{15} a^2 b^6 x^{15}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^8/x^4,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{3 x^{3}} + \frac{8 a^{7} b \log{\left (x^{3} \right )}}{3} + \frac{28 a^{6} b^{2} x^{3}}{3} + \frac{56 a^{5} b^{3} \int ^{x^{3}} x\, dx}{3} + \frac{70 a^{4} b^{4} x^{9}}{9} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{15}}{15} + \frac{4 a b^{7} x^{18}}{9} + \frac{b^{8} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**8/x**4,x)

[Out]

-a**8/(3*x**3) + 8*a**7*b*log(x**3)/3 + 28*a**6*b**2*x**3/3 + 56*a**5*b**3*Integ
ral(x, (x, x**3))/3 + 70*a**4*b**4*x**9/9 + 14*a**3*b**5*x**12/3 + 28*a**2*b**6*
x**15/15 + 4*a*b**7*x**18/9 + b**8*x**21/21

_______________________________________________________________________________________

Mathematica [A]  time = 0.0166772, size = 105, normalized size = 1. \[ -\frac{a^8}{3 x^3}+8 a^7 b \log (x)+\frac{28}{3} a^6 b^2 x^3+\frac{28}{3} a^5 b^3 x^6+\frac{70}{9} a^4 b^4 x^9+\frac{14}{3} a^3 b^5 x^{12}+\frac{28}{15} a^2 b^6 x^{15}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{21}}{21} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^8/x^4,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 90, normalized size = 0.9 \[ -{\frac{{a}^{8}}{3\,{x}^{3}}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{3}}{3}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{6}}{3}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+{\frac{14\,{a}^{3}{b}^{5}{x}^{12}}{3}}+{\frac{28\,{a}^{2}{b}^{6}{x}^{15}}{15}}+{\frac{4\,a{b}^{7}{x}^{18}}{9}}+{\frac{{b}^{8}{x}^{21}}{21}}+8\,{a}^{7}b\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^8/x^4,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.4256, size = 123, normalized size = 1.17 \[ \frac{1}{21} \, b^{8} x^{21} + \frac{4}{9} \, a b^{7} x^{18} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{70}{9} \, a^{4} b^{4} x^{9} + \frac{28}{3} \, a^{5} b^{3} x^{6} + \frac{28}{3} \, a^{6} b^{2} x^{3} + \frac{8}{3} \, a^{7} b \log \left (x^{3}\right ) - \frac{a^{8}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^4,x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.210162, size = 127, normalized size = 1.21 \[ \frac{15 \, b^{8} x^{24} + 140 \, a b^{7} x^{21} + 588 \, a^{2} b^{6} x^{18} + 1470 \, a^{3} b^{5} x^{15} + 2450 \, a^{4} b^{4} x^{12} + 2940 \, a^{5} b^{3} x^{9} + 2940 \, a^{6} b^{2} x^{6} + 2520 \, a^{7} b x^{3} \log \left (x\right ) - 105 \, a^{8}}{315 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^4,x, algorithm="fricas")

[Out]

1/315*(15*b^8*x^24 + 140*a*b^7*x^21 + 588*a^2*b^6*x^18 + 1470*a^3*b^5*x^15 + 245
0*a^4*b^4*x^12 + 2940*a^5*b^3*x^9 + 2940*a^6*b^2*x^6 + 2520*a^7*b*x^3*log(x) - 1
05*a^8)/x^3

_______________________________________________________________________________________

Sympy [A]  time = 1.52058, size = 105, normalized size = 1. \[ - \frac{a^{8}}{3 x^{3}} + 8 a^{7} b \log{\left (x \right )} + \frac{28 a^{6} b^{2} x^{3}}{3} + \frac{28 a^{5} b^{3} x^{6}}{3} + \frac{70 a^{4} b^{4} x^{9}}{9} + \frac{14 a^{3} b^{5} x^{12}}{3} + \frac{28 a^{2} b^{6} x^{15}}{15} + \frac{4 a b^{7} x^{18}}{9} + \frac{b^{8} x^{21}}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**8/x**4,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.224818, size = 135, normalized size = 1.29 \[ \frac{1}{21} \, b^{8} x^{21} + \frac{4}{9} \, a b^{7} x^{18} + \frac{28}{15} \, a^{2} b^{6} x^{15} + \frac{14}{3} \, a^{3} b^{5} x^{12} + \frac{70}{9} \, a^{4} b^{4} x^{9} + \frac{28}{3} \, a^{5} b^{3} x^{6} + \frac{28}{3} \, a^{6} b^{2} x^{3} + 8 \, a^{7} b{\rm ln}\left ({\left | x \right |}\right ) - \frac{8 \, a^{7} b x^{3} + a^{8}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^8/x^4,x, algorithm="giac")

[Out]

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