3.1619 \(\int \frac{x^5}{(a+\frac{b}{x})^2} \, dx\)

Optimal. Leaf size=98 \[ \frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}+\frac{b^7}{a^8 (a x+b)}-\frac{6 b^5 x}{a^7}+\frac{7 b^6 \log (a x+b)}{a^8}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0655405, antiderivative size = 98, 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, Rules used = {263, 43} \[ \frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}+\frac{b^7}{a^8 (a x+b)}-\frac{6 b^5 x}{a^7}+\frac{7 b^6 \log (a x+b)}{a^8}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^5/(a + b/x)^2,x]

[Out]

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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}{\left (a+\frac{b}{x}\right )^2} \, dx &=\int \frac{x^7}{(b+a x)^2} \, dx\\ &=\int \left (-\frac{6 b^5}{a^7}+\frac{5 b^4 x}{a^6}-\frac{4 b^3 x^2}{a^5}+\frac{3 b^2 x^3}{a^4}-\frac{2 b x^4}{a^3}+\frac{x^5}{a^2}-\frac{b^7}{a^7 (b+a x)^2}+\frac{7 b^6}{a^7 (b+a x)}\right ) \, dx\\ &=-\frac{6 b^5 x}{a^7}+\frac{5 b^4 x^2}{2 a^6}-\frac{4 b^3 x^3}{3 a^5}+\frac{3 b^2 x^4}{4 a^4}-\frac{2 b x^5}{5 a^3}+\frac{x^6}{6 a^2}+\frac{b^7}{a^8 (b+a x)}+\frac{7 b^6 \log (b+a x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.0206441, size = 88, normalized size = 0.9 \[ \frac{150 a^2 b^4 x^2-80 a^3 b^3 x^3+45 a^4 b^2 x^4-24 a^5 b x^5+10 a^6 x^6+\frac{60 b^7}{a x+b}-360 a b^5 x+420 b^6 \log (a x+b)}{60 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/(a + b/x)^2,x]

[Out]

(-360*a*b^5*x + 150*a^2*b^4*x^2 - 80*a^3*b^3*x^3 + 45*a^4*b^2*x^4 - 24*a^5*b*x^5 + 10*a^6*x^6 + (60*b^7)/(b +
a*x) + 420*b^6*Log[b + a*x])/(60*a^8)

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Maple [A]  time = 0.005, size = 89, normalized size = 0.9 \begin{align*} -6\,{\frac{{b}^{5}x}{{a}^{7}}}+{\frac{5\,{b}^{4}{x}^{2}}{2\,{a}^{6}}}-{\frac{4\,{b}^{3}{x}^{3}}{3\,{a}^{5}}}+{\frac{3\,{b}^{2}{x}^{4}}{4\,{a}^{4}}}-{\frac{2\,b{x}^{5}}{5\,{a}^{3}}}+{\frac{{x}^{6}}{6\,{a}^{2}}}+{\frac{{b}^{7}}{{a}^{8} \left ( ax+b \right ) }}+7\,{\frac{{b}^{6}\ln \left ( ax+b \right ) }{{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-6*b^5*x/a^7+5/2*b^4*x^2/a^6-4/3*b^3*x^3/a^5+3/4*b^2*x^4/a^4-2/5*b*x^5/a^3+1/6*x^6/a^2+b^7/a^8/(a*x+b)+7*b^6*l
n(a*x+b)/a^8

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Maxima [A]  time = 1.02057, size = 124, normalized size = 1.27 \begin{align*} \frac{b^{7}}{a^{9} x + a^{8} b} + \frac{7 \, b^{6} \log \left (a x + b\right )}{a^{8}} + \frac{10 \, a^{5} x^{6} - 24 \, a^{4} b x^{5} + 45 \, a^{3} b^{2} x^{4} - 80 \, a^{2} b^{3} x^{3} + 150 \, a b^{4} x^{2} - 360 \, b^{5} x}{60 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^7/(a^9*x + a^8*b) + 7*b^6*log(a*x + b)/a^8 + 1/60*(10*a^5*x^6 - 24*a^4*b*x^5 + 45*a^3*b^2*x^4 - 80*a^2*b^3*x
^3 + 150*a*b^4*x^2 - 360*b^5*x)/a^7

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Fricas [A]  time = 1.44011, size = 239, normalized size = 2.44 \begin{align*} \frac{10 \, a^{7} x^{7} - 14 \, a^{6} b x^{6} + 21 \, a^{5} b^{2} x^{5} - 35 \, a^{4} b^{3} x^{4} + 70 \, a^{3} b^{4} x^{3} - 210 \, a^{2} b^{5} x^{2} - 360 \, a b^{6} x + 60 \, b^{7} + 420 \,{\left (a b^{6} x + b^{7}\right )} \log \left (a x + b\right )}{60 \,{\left (a^{9} x + a^{8} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(10*a^7*x^7 - 14*a^6*b*x^6 + 21*a^5*b^2*x^5 - 35*a^4*b^3*x^4 + 70*a^3*b^4*x^3 - 210*a^2*b^5*x^2 - 360*a*b
^6*x + 60*b^7 + 420*(a*b^6*x + b^7)*log(a*x + b))/(a^9*x + a^8*b)

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Sympy [A]  time = 0.384867, size = 99, normalized size = 1.01 \begin{align*} \frac{b^{7}}{a^{9} x + a^{8} b} + \frac{x^{6}}{6 a^{2}} - \frac{2 b x^{5}}{5 a^{3}} + \frac{3 b^{2} x^{4}}{4 a^{4}} - \frac{4 b^{3} x^{3}}{3 a^{5}} + \frac{5 b^{4} x^{2}}{2 a^{6}} - \frac{6 b^{5} x}{a^{7}} + \frac{7 b^{6} \log{\left (a x + b \right )}}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.08777, size = 128, normalized size = 1.31 \begin{align*} \frac{7 \, b^{6} \log \left ({\left | a x + b \right |}\right )}{a^{8}} + \frac{b^{7}}{{\left (a x + b\right )} a^{8}} + \frac{10 \, a^{10} x^{6} - 24 \, a^{9} b x^{5} + 45 \, a^{8} b^{2} x^{4} - 80 \, a^{7} b^{3} x^{3} + 150 \, a^{6} b^{4} x^{2} - 360 \, a^{5} b^{5} x}{60 \, a^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7*b^6*log(abs(a*x + b))/a^8 + b^7/((a*x + b)*a^8) + 1/60*(10*a^10*x^6 - 24*a^9*b*x^5 + 45*a^8*b^2*x^4 - 80*a^7
*b^3*x^3 + 150*a^6*b^4*x^2 - 360*a^5*b^5*x)/a^12