3.169 \(\int \frac{x^5}{(a+b x)^2} \, dx\)

Optimal. Leaf size=72 \[ \frac{3 a^2 x^2}{2 b^4}+\frac{a^5}{b^6 (a+b x)}-\frac{4 a^3 x}{b^5}+\frac{5 a^4 \log (a+b x)}{b^6}-\frac{2 a x^3}{3 b^3}+\frac{x^4}{4 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0423157, antiderivative size = 72, 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, Rules used = {43} \[ \frac{3 a^2 x^2}{2 b^4}+\frac{a^5}{b^6 (a+b x)}-\frac{4 a^3 x}{b^5}+\frac{5 a^4 \log (a+b x)}{b^6}-\frac{2 a x^3}{3 b^3}+\frac{x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b*x)^2,x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{x^5}{(a+b x)^2} \, dx &=\int \left (-\frac{4 a^3}{b^5}+\frac{3 a^2 x}{b^4}-\frac{2 a x^2}{b^3}+\frac{x^3}{b^2}-\frac{a^5}{b^5 (a+b x)^2}+\frac{5 a^4}{b^5 (a+b x)}\right ) \, dx\\ &=-\frac{4 a^3 x}{b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{2 a x^3}{3 b^3}+\frac{x^4}{4 b^2}+\frac{a^5}{b^6 (a+b x)}+\frac{5 a^4 \log (a+b x)}{b^6}\\ \end{align*}

Mathematica [A]  time = 0.0295879, size = 66, normalized size = 0.92 \[ \frac{18 a^2 b^2 x^2+\frac{12 a^5}{a+b x}-48 a^3 b x+60 a^4 \log (a+b x)-8 a b^3 x^3+3 b^4 x^4}{12 b^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b*x)^2,x]

[Out]

(-48*a^3*b*x + 18*a^2*b^2*x^2 - 8*a*b^3*x^3 + 3*b^4*x^4 + (12*a^5)/(a + b*x) + 60*a^4*Log[a + b*x])/(12*b^6)

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Maple [A]  time = 0.006, size = 67, normalized size = 0.9 \begin{align*} -4\,{\frac{{a}^{3}x}{{b}^{5}}}+{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}}}-{\frac{2\,a{x}^{3}}{3\,{b}^{3}}}+{\frac{{x}^{4}}{4\,{b}^{2}}}+{\frac{{a}^{5}}{{b}^{6} \left ( bx+a \right ) }}+5\,{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5/(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.04987, size = 95, normalized size = 1.32 \begin{align*} \frac{a^{5}}{b^{7} x + a b^{6}} + \frac{5 \, a^{4} \log \left (b x + a\right )}{b^{6}} + \frac{3 \, b^{3} x^{4} - 8 \, a b^{2} x^{3} + 18 \, a^{2} b x^{2} - 48 \, a^{3} x}{12 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2,x, algorithm="maxima")

[Out]

a^5/(b^7*x + a*b^6) + 5*a^4*log(b*x + a)/b^6 + 1/12*(3*b^3*x^4 - 8*a*b^2*x^3 + 18*a^2*b*x^2 - 48*a^3*x)/b^5

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Fricas [A]  time = 1.58896, size = 186, normalized size = 2.58 \begin{align*} \frac{3 \, b^{5} x^{5} - 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} - 30 \, a^{3} b^{2} x^{2} - 48 \, a^{4} b x + 12 \, a^{5} + 60 \,{\left (a^{4} b x + a^{5}\right )} \log \left (b x + a\right )}{12 \,{\left (b^{7} x + a b^{6}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^5 - 5*a*b^4*x^4 + 10*a^2*b^3*x^3 - 30*a^3*b^2*x^2 - 48*a^4*b*x + 12*a^5 + 60*(a^4*b*x + a^5)*log
(b*x + a))/(b^7*x + a*b^6)

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Sympy [A]  time = 0.45716, size = 71, normalized size = 0.99 \begin{align*} \frac{a^{5}}{a b^{6} + b^{7} x} + \frac{5 a^{4} \log{\left (a + b x \right )}}{b^{6}} - \frac{4 a^{3} x}{b^{5}} + \frac{3 a^{2} x^{2}}{2 b^{4}} - \frac{2 a x^{3}}{3 b^{3}} + \frac{x^{4}}{4 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5/(b*x+a)**2,x)

[Out]

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

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Giac [A]  time = 1.17184, size = 122, normalized size = 1.69 \begin{align*} -\frac{{\left (b x + a\right )}^{4}{\left (\frac{20 \, a}{b x + a} - \frac{60 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac{120 \, a^{3}}{{\left (b x + a\right )}^{3}} - 3\right )}}{12 \, b^{6}} - \frac{5 \, a^{4} \log \left (\frac{{\left | b x + a \right |}}{{\left (b x + a\right )}^{2}{\left | b \right |}}\right )}{b^{6}} + \frac{a^{5}}{{\left (b x + a\right )} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5/(b*x+a)^2,x, algorithm="giac")

[Out]

-1/12*(b*x + a)^4*(20*a/(b*x + a) - 60*a^2/(b*x + a)^2 + 120*a^3/(b*x + a)^3 - 3)/b^6 - 5*a^4*log(abs(b*x + a)
/((b*x + a)^2*abs(b)))/b^6 + a^5/((b*x + a)*b^6)