3.193 \(\int \frac{x^8}{(a+b x)^4} \, dx\)

Optimal. Leaf size=114 \[ -\frac{a^8}{3 b^9 (a+b x)^3}+\frac{4 a^7}{b^9 (a+b x)^2}-\frac{28 a^6}{b^9 (a+b x)}-\frac{56 a^5 \log (a+b x)}{b^9}+\frac{35 a^4 x}{b^8}-\frac{10 a^3 x^2}{b^7}+\frac{10 a^2 x^3}{3 b^6}-\frac{a x^4}{b^5}+\frac{x^5}{5 b^4} \]

[Out]

(35*a^4*x)/b^8 - (10*a^3*x^2)/b^7 + (10*a^2*x^3)/(3*b^6) - (a*x^4)/b^5 + x^5/(5*
b^4) - a^8/(3*b^9*(a + b*x)^3) + (4*a^7)/(b^9*(a + b*x)^2) - (28*a^6)/(b^9*(a +
b*x)) - (56*a^5*Log[a + b*x])/b^9

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Rubi [A]  time = 0.182108, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^8}{3 b^9 (a+b x)^3}+\frac{4 a^7}{b^9 (a+b x)^2}-\frac{28 a^6}{b^9 (a+b x)}-\frac{56 a^5 \log (a+b x)}{b^9}+\frac{35 a^4 x}{b^8}-\frac{10 a^3 x^2}{b^7}+\frac{10 a^2 x^3}{3 b^6}-\frac{a x^4}{b^5}+\frac{x^5}{5 b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x)^4,x]

[Out]

(35*a^4*x)/b^8 - (10*a^3*x^2)/b^7 + (10*a^2*x^3)/(3*b^6) - (a*x^4)/b^5 + x^5/(5*
b^4) - a^8/(3*b^9*(a + b*x)^3) + (4*a^7)/(b^9*(a + b*x)^2) - (28*a^6)/(b^9*(a +
b*x)) - (56*a^5*Log[a + b*x])/b^9

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8}}{3 b^{9} \left (a + b x\right )^{3}} + \frac{4 a^{7}}{b^{9} \left (a + b x\right )^{2}} - \frac{28 a^{6}}{b^{9} \left (a + b x\right )} - \frac{56 a^{5} \log{\left (a + b x \right )}}{b^{9}} + \frac{35 a^{4} x}{b^{8}} - \frac{20 a^{3} \int x\, dx}{b^{7}} + \frac{10 a^{2} x^{3}}{3 b^{6}} - \frac{a x^{4}}{b^{5}} + \frac{x^{5}}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**8/(3*b**9*(a + b*x)**3) + 4*a**7/(b**9*(a + b*x)**2) - 28*a**6/(b**9*(a + b*
x)) - 56*a**5*log(a + b*x)/b**9 + 35*a**4*x/b**8 - 20*a**3*Integral(x, x)/b**7 +
10*a**2*x**3/(3*b**6) - a*x**4/b**5 + x**5/(5*b**4)

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Mathematica [A]  time = 0.0574478, size = 101, normalized size = 0.89 \[ \frac{-\frac{5 a^8}{(a+b x)^3}+\frac{60 a^7}{(a+b x)^2}-\frac{420 a^6}{a+b x}-840 a^5 \log (a+b x)+525 a^4 b x-150 a^3 b^2 x^2+50 a^2 b^3 x^3-15 a b^4 x^4+3 b^5 x^5}{15 b^9} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x)^4,x]

[Out]

(525*a^4*b*x - 150*a^3*b^2*x^2 + 50*a^2*b^3*x^3 - 15*a*b^4*x^4 + 3*b^5*x^5 - (5*
a^8)/(a + b*x)^3 + (60*a^7)/(a + b*x)^2 - (420*a^6)/(a + b*x) - 840*a^5*Log[a +
b*x])/(15*b^9)

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Maple [A]  time = 0.012, size = 109, normalized size = 1. \[ 35\,{\frac{{a}^{4}x}{{b}^{8}}}-10\,{\frac{{a}^{3}{x}^{2}}{{b}^{7}}}+{\frac{10\,{a}^{2}{x}^{3}}{3\,{b}^{6}}}-{\frac{a{x}^{4}}{{b}^{5}}}+{\frac{{x}^{5}}{5\,{b}^{4}}}-{\frac{{a}^{8}}{3\,{b}^{9} \left ( bx+a \right ) ^{3}}}+4\,{\frac{{a}^{7}}{{b}^{9} \left ( bx+a \right ) ^{2}}}-28\,{\frac{{a}^{6}}{{b}^{9} \left ( bx+a \right ) }}-56\,{\frac{{a}^{5}\ln \left ( bx+a \right ) }{{b}^{9}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x+a)^4,x)

[Out]

35*a^4*x/b^8-10*a^3*x^2/b^7+10/3*a^2*x^3/b^6-a*x^4/b^5+1/5*x^5/b^4-1/3*a^8/b^9/(
b*x+a)^3+4*a^7/b^9/(b*x+a)^2-28*a^6/b^9/(b*x+a)-56*a^5*ln(b*x+a)/b^9

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Maxima [A]  time = 1.34451, size = 169, normalized size = 1.48 \[ -\frac{84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \,{\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} - \frac{56 \, a^{5} \log \left (b x + a\right )}{b^{9}} + \frac{3 \, b^{4} x^{5} - 15 \, a b^{3} x^{4} + 50 \, a^{2} b^{2} x^{3} - 150 \, a^{3} b x^{2} + 525 \, a^{4} x}{15 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(84*a^6*b^2*x^2 + 156*a^7*b*x + 73*a^8)/(b^12*x^3 + 3*a*b^11*x^2 + 3*a^2*b^
10*x + a^3*b^9) - 56*a^5*log(b*x + a)/b^9 + 1/15*(3*b^4*x^5 - 15*a*b^3*x^4 + 50*
a^2*b^2*x^3 - 150*a^3*b*x^2 + 525*a^4*x)/b^8

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Fricas [A]  time = 0.210266, size = 219, normalized size = 1.92 \[ \frac{3 \, b^{8} x^{8} - 6 \, a b^{7} x^{7} + 14 \, a^{2} b^{6} x^{6} - 42 \, a^{3} b^{5} x^{5} + 210 \, a^{4} b^{4} x^{4} + 1175 \, a^{5} b^{3} x^{3} + 1005 \, a^{6} b^{2} x^{2} - 255 \, a^{7} b x - 365 \, a^{8} - 840 \,{\left (a^{5} b^{3} x^{3} + 3 \, a^{6} b^{2} x^{2} + 3 \, a^{7} b x + a^{8}\right )} \log \left (b x + a\right )}{15 \,{\left (b^{12} x^{3} + 3 \, a b^{11} x^{2} + 3 \, a^{2} b^{10} x + a^{3} b^{9}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*b^8*x^8 - 6*a*b^7*x^7 + 14*a^2*b^6*x^6 - 42*a^3*b^5*x^5 + 210*a^4*b^4*x^
4 + 1175*a^5*b^3*x^3 + 1005*a^6*b^2*x^2 - 255*a^7*b*x - 365*a^8 - 840*(a^5*b^3*x
^3 + 3*a^6*b^2*x^2 + 3*a^7*b*x + a^8)*log(b*x + a))/(b^12*x^3 + 3*a*b^11*x^2 + 3
*a^2*b^10*x + a^3*b^9)

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Sympy [A]  time = 2.34752, size = 129, normalized size = 1.13 \[ - \frac{56 a^{5} \log{\left (a + b x \right )}}{b^{9}} + \frac{35 a^{4} x}{b^{8}} - \frac{10 a^{3} x^{2}}{b^{7}} + \frac{10 a^{2} x^{3}}{3 b^{6}} - \frac{a x^{4}}{b^{5}} - \frac{73 a^{8} + 156 a^{7} b x + 84 a^{6} b^{2} x^{2}}{3 a^{3} b^{9} + 9 a^{2} b^{10} x + 9 a b^{11} x^{2} + 3 b^{12} x^{3}} + \frac{x^{5}}{5 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x+a)**4,x)

[Out]

-56*a**5*log(a + b*x)/b**9 + 35*a**4*x/b**8 - 10*a**3*x**2/b**7 + 10*a**2*x**3/(
3*b**6) - a*x**4/b**5 - (73*a**8 + 156*a**7*b*x + 84*a**6*b**2*x**2)/(3*a**3*b**
9 + 9*a**2*b**10*x + 9*a*b**11*x**2 + 3*b**12*x**3) + x**5/(5*b**4)

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GIAC/XCAS [A]  time = 0.202679, size = 143, normalized size = 1.25 \[ -\frac{56 \, a^{5}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{9}} - \frac{84 \, a^{6} b^{2} x^{2} + 156 \, a^{7} b x + 73 \, a^{8}}{3 \,{\left (b x + a\right )}^{3} b^{9}} + \frac{3 \, b^{16} x^{5} - 15 \, a b^{15} x^{4} + 50 \, a^{2} b^{14} x^{3} - 150 \, a^{3} b^{13} x^{2} + 525 \, a^{4} b^{12} x}{15 \, b^{20}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-56*a^5*ln(abs(b*x + a))/b^9 - 1/3*(84*a^6*b^2*x^2 + 156*a^7*b*x + 73*a^8)/((b*x
+ a)^3*b^9) + 1/15*(3*b^16*x^5 - 15*a*b^15*x^4 + 50*a^2*b^14*x^3 - 150*a^3*b^13
*x^2 + 525*a^4*b^12*x)/b^20