3.182 \(\int \frac{x^5}{(a+b x)^3} \, dx\)

Optimal. Leaf size=77 \[ \frac{a^5}{2 b^6 (a+b x)^2}-\frac{5 a^4}{b^6 (a+b x)}-\frac{10 a^3 \log (a+b x)}{b^6}+\frac{6 a^2 x}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{x^3}{3 b^3} \]

[Out]

(6*a^2*x)/b^5 - (3*a*x^2)/(2*b^4) + x^3/(3*b^3) + a^5/(2*b^6*(a + b*x)^2) - (5*a
^4)/(b^6*(a + b*x)) - (10*a^3*Log[a + b*x])/b^6

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Rubi [A]  time = 0.0987493, antiderivative size = 77, 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 \[ \frac{a^5}{2 b^6 (a+b x)^2}-\frac{5 a^4}{b^6 (a+b x)}-\frac{10 a^3 \log (a+b x)}{b^6}+\frac{6 a^2 x}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{x^3}{3 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x)^3,x]

[Out]

(6*a^2*x)/b^5 - (3*a*x^2)/(2*b^4) + x^3/(3*b^3) + a^5/(2*b^6*(a + b*x)^2) - (5*a
^4)/(b^6*(a + b*x)) - (10*a^3*Log[a + b*x])/b^6

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x+a)**3,x)

[Out]

a**5/(2*b**6*(a + b*x)**2) - 5*a**4/(b**6*(a + b*x)) - 10*a**3*log(a + b*x)/b**6
+ 6*a**2*x/b**5 - 3*a*Integral(x, x)/b**4 + x**3/(3*b**3)

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Mathematica [A]  time = 0.0475476, size = 67, normalized size = 0.87 \[ \frac{\frac{3 a^5}{(a+b x)^2}-\frac{30 a^4}{a+b x}-60 a^3 \log (a+b x)+36 a^2 b x-9 a b^2 x^2+2 b^3 x^3}{6 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x)^3,x]

[Out]

(36*a^2*b*x - 9*a*b^2*x^2 + 2*b^3*x^3 + (3*a^5)/(a + b*x)^2 - (30*a^4)/(a + b*x)
- 60*a^3*Log[a + b*x])/(6*b^6)

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Maple [A]  time = 0.01, size = 72, normalized size = 0.9 \[ 6\,{\frac{{a}^{2}x}{{b}^{5}}}-{\frac{3\,a{x}^{2}}{2\,{b}^{4}}}+{\frac{{x}^{3}}{3\,{b}^{3}}}+{\frac{{a}^{5}}{2\,{b}^{6} \left ( bx+a \right ) ^{2}}}-5\,{\frac{{a}^{4}}{{b}^{6} \left ( bx+a \right ) }}-10\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{6}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x+a)^3,x)

[Out]

6*a^2*x/b^5-3/2*a*x^2/b^4+1/3*x^3/b^3+1/2*a^5/b^6/(b*x+a)^2-5*a^4/b^6/(b*x+a)-10
*a^3*ln(b*x+a)/b^6

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Maxima [A]  time = 1.33174, size = 109, normalized size = 1.42 \[ -\frac{10 \, a^{4} b x + 9 \, a^{5}}{2 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} - \frac{10 \, a^{3} \log \left (b x + a\right )}{b^{6}} + \frac{2 \, b^{2} x^{3} - 9 \, a b x^{2} + 36 \, a^{2} x}{6 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x + a)^3,x, algorithm="maxima")

[Out]

-1/2*(10*a^4*b*x + 9*a^5)/(b^8*x^2 + 2*a*b^7*x + a^2*b^6) - 10*a^3*log(b*x + a)/
b^6 + 1/6*(2*b^2*x^3 - 9*a*b*x^2 + 36*a^2*x)/b^5

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Fricas [A]  time = 0.202211, size = 144, normalized size = 1.87 \[ \frac{2 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 20 \, a^{2} b^{3} x^{3} + 63 \, a^{3} b^{2} x^{2} + 6 \, a^{4} b x - 27 \, a^{5} - 60 \,{\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{6 \,{\left (b^{8} x^{2} + 2 \, a b^{7} x + a^{2} b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x + a)^3,x, algorithm="fricas")

[Out]

1/6*(2*b^5*x^5 - 5*a*b^4*x^4 + 20*a^2*b^3*x^3 + 63*a^3*b^2*x^2 + 6*a^4*b*x - 27*
a^5 - 60*(a^3*b^2*x^2 + 2*a^4*b*x + a^5)*log(b*x + a))/(b^8*x^2 + 2*a*b^7*x + a^
2*b^6)

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Sympy [A]  time = 1.76577, size = 83, normalized size = 1.08 \[ - \frac{10 a^{3} \log{\left (a + b x \right )}}{b^{6}} + \frac{6 a^{2} x}{b^{5}} - \frac{3 a x^{2}}{2 b^{4}} - \frac{9 a^{5} + 10 a^{4} b x}{2 a^{2} b^{6} + 4 a b^{7} x + 2 b^{8} x^{2}} + \frac{x^{3}}{3 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x+a)**3,x)

[Out]

-10*a**3*log(a + b*x)/b**6 + 6*a**2*x/b**5 - 3*a*x**2/(2*b**4) - (9*a**5 + 10*a*
*4*b*x)/(2*a**2*b**6 + 4*a*b**7*x + 2*b**8*x**2) + x**3/(3*b**3)

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GIAC/XCAS [A]  time = 0.205916, size = 99, normalized size = 1.29 \[ -\frac{10 \, a^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{6}} - \frac{10 \, a^{4} b x + 9 \, a^{5}}{2 \,{\left (b x + a\right )}^{2} b^{6}} + \frac{2 \, b^{6} x^{3} - 9 \, a b^{5} x^{2} + 36 \, a^{2} b^{4} x}{6 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x + a)^3,x, algorithm="giac")

[Out]

-10*a^3*ln(abs(b*x + a))/b^6 - 1/2*(10*a^4*b*x + 9*a^5)/((b*x + a)^2*b^6) + 1/6*
(2*b^6*x^3 - 9*a*b^5*x^2 + 36*a^2*b^4*x)/b^9