3.1620 \(\int \frac{x^4}{(a+\frac{b}{x})^2} \, dx\)

Optimal. Leaf size=81 \[ -\frac{2 b^3 x^2}{a^5}+\frac{b^2 x^3}{a^4}-\frac{b^6}{a^7 (a x+b)}+\frac{5 b^4 x}{a^6}-\frac{6 b^5 \log (a x+b)}{a^7}-\frac{b x^4}{2 a^3}+\frac{x^5}{5 a^2} \]

[Out]

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

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Rubi [A]  time = 0.0527342, antiderivative size = 81, 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} \[ -\frac{2 b^3 x^2}{a^5}+\frac{b^2 x^3}{a^4}-\frac{b^6}{a^7 (a x+b)}+\frac{5 b^4 x}{a^6}-\frac{6 b^5 \log (a x+b)}{a^7}-\frac{b x^4}{2 a^3}+\frac{x^5}{5 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0229799, size = 77, normalized size = 0.95 \[ \frac{-20 a^2 b^3 x^2+10 a^3 b^2 x^3-5 a^4 b x^4+2 a^5 x^5-\frac{10 b^6}{a x+b}+50 a b^4 x-60 b^5 \log (a x+b)}{10 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x)^2,x]

[Out]

(50*a*b^4*x - 20*a^2*b^3*x^2 + 10*a^3*b^2*x^3 - 5*a^4*b*x^4 + 2*a^5*x^5 - (10*b^6)/(b + a*x) - 60*b^5*Log[b +
a*x])/(10*a^7)

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Maple [A]  time = 0.006, size = 78, normalized size = 1. \begin{align*} 5\,{\frac{{b}^{4}x}{{a}^{6}}}-2\,{\frac{{b}^{3}{x}^{2}}{{a}^{5}}}+{\frac{{b}^{2}{x}^{3}}{{a}^{4}}}-{\frac{b{x}^{4}}{2\,{a}^{3}}}+{\frac{{x}^{5}}{5\,{a}^{2}}}-{\frac{{b}^{6}}{{a}^{7} \left ( ax+b \right ) }}-6\,{\frac{{b}^{5}\ln \left ( ax+b \right ) }{{a}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+b/x)^2,x)

[Out]

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

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Maxima [A]  time = 1.11633, size = 111, normalized size = 1.37 \begin{align*} -\frac{b^{6}}{a^{8} x + a^{7} b} - \frac{6 \, b^{5} \log \left (a x + b\right )}{a^{7}} + \frac{2 \, a^{4} x^{5} - 5 \, a^{3} b x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a b^{3} x^{2} + 50 \, b^{4} x}{10 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x)^2,x, algorithm="maxima")

[Out]

-b^6/(a^8*x + a^7*b) - 6*b^5*log(a*x + b)/a^7 + 1/10*(2*a^4*x^5 - 5*a^3*b*x^4 + 10*a^2*b^2*x^3 - 20*a*b^3*x^2
+ 50*b^4*x)/a^6

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Fricas [A]  time = 1.39507, size = 208, normalized size = 2.57 \begin{align*} \frac{2 \, a^{6} x^{6} - 3 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{2} b^{4} x^{2} + 50 \, a b^{5} x - 10 \, b^{6} - 60 \,{\left (a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{10 \,{\left (a^{8} x + a^{7} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x)^2,x, algorithm="fricas")

[Out]

1/10*(2*a^6*x^6 - 3*a^5*b*x^5 + 5*a^4*b^2*x^4 - 10*a^3*b^3*x^3 + 30*a^2*b^4*x^2 + 50*a*b^5*x - 10*b^6 - 60*(a*
b^5*x + b^6)*log(a*x + b))/(a^8*x + a^7*b)

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Sympy [A]  time = 0.369558, size = 78, normalized size = 0.96 \begin{align*} - \frac{b^{6}}{a^{8} x + a^{7} b} + \frac{x^{5}}{5 a^{2}} - \frac{b x^{4}}{2 a^{3}} + \frac{b^{2} x^{3}}{a^{4}} - \frac{2 b^{3} x^{2}}{a^{5}} + \frac{5 b^{4} x}{a^{6}} - \frac{6 b^{5} \log{\left (a x + b \right )}}{a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b/x)**2,x)

[Out]

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

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Giac [A]  time = 1.11966, size = 115, normalized size = 1.42 \begin{align*} -\frac{6 \, b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{7}} - \frac{b^{6}}{{\left (a x + b\right )} a^{7}} + \frac{2 \, a^{8} x^{5} - 5 \, a^{7} b x^{4} + 10 \, a^{6} b^{2} x^{3} - 20 \, a^{5} b^{3} x^{2} + 50 \, a^{4} b^{4} x}{10 \, a^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x)^2,x, algorithm="giac")

[Out]

-6*b^5*log(abs(a*x + b))/a^7 - b^6/((a*x + b)*a^7) + 1/10*(2*a^8*x^5 - 5*a^7*b*x^4 + 10*a^6*b^2*x^3 - 20*a^5*b
^3*x^2 + 50*a^4*b^4*x)/a^10