3.1621 \(\int \frac{x^3}{(a+\frac{b}{x})^2} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0478724, antiderivative size = 72, 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} \[ \frac{3 b^2 x^2}{2 a^4}+\frac{b^5}{a^6 (a x+b)}-\frac{4 b^3 x}{a^5}+\frac{5 b^4 \log (a x+b)}{a^6}-\frac{2 b x^3}{3 a^3}+\frac{x^4}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x)^2,x]

[Out]

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

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

Mathematica [A]  time = 0.0163348, size = 66, normalized size = 0.92 \[ \frac{18 a^2 b^2 x^2-8 a^3 b x^3+3 a^4 x^4+\frac{12 b^5}{a x+b}-48 a b^3 x+60 b^4 \log (a x+b)}{12 a^6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x)^2,x]

[Out]

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

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Maple [A]  time = 0.007, size = 67, normalized size = 0.9 \begin{align*} -4\,{\frac{{b}^{3}x}{{a}^{5}}}+{\frac{3\,{b}^{2}{x}^{2}}{2\,{a}^{4}}}-{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+{\frac{{x}^{4}}{4\,{a}^{2}}}+{\frac{{b}^{5}}{{a}^{6} \left ( ax+b \right ) }}+5\,{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b/x)^2,x)

[Out]

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

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Maxima [A]  time = 1.03985, size = 95, normalized size = 1.32 \begin{align*} \frac{b^{5}}{a^{7} x + a^{6} b} + \frac{5 \, b^{4} \log \left (a x + b\right )}{a^{6}} + \frac{3 \, a^{3} x^{4} - 8 \, a^{2} b x^{3} + 18 \, a b^{2} x^{2} - 48 \, b^{3} x}{12 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b/x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.42428, size = 186, normalized size = 2.58 \begin{align*} \frac{3 \, a^{5} x^{5} - 5 \, a^{4} b x^{4} + 10 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} - 48 \, a b^{4} x + 12 \, b^{5} + 60 \,{\left (a b^{4} x + b^{5}\right )} \log \left (a x + b\right )}{12 \,{\left (a^{7} x + a^{6} b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b/x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.356564, size = 71, normalized size = 0.99 \begin{align*} \frac{b^{5}}{a^{7} x + a^{6} b} + \frac{x^{4}}{4 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{3 b^{2} x^{2}}{2 a^{4}} - \frac{4 b^{3} x}{a^{5}} + \frac{5 b^{4} \log{\left (a x + b \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b/x)**2,x)

[Out]

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

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Giac [A]  time = 1.14321, size = 99, normalized size = 1.38 \begin{align*} \frac{5 \, b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{6}} + \frac{b^{5}}{{\left (a x + b\right )} a^{6}} + \frac{3 \, a^{6} x^{4} - 8 \, a^{5} b x^{3} + 18 \, a^{4} b^{2} x^{2} - 48 \, a^{3} b^{3} x}{12 \, a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b/x)^2,x, algorithm="giac")

[Out]

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