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

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

[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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Rubi [A]  time = 0.05, antiderivative size = 81, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.03, size = 77, normalized size = 0.95 \[ \frac {2 a^5 x^5-5 a^4 b x^4+10 a^3 b^2 x^3-20 a^2 b^3 x^2-\frac {10 b^6}{a x+b}-60 b^5 \log (a x+b)+50 a b^4 x}{10 a^7} \]

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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fricas [A]  time = 1.14, size = 96, normalized size = 1.19 \[ \frac {2 \, a^{6} x^{6} - 3 \, a^{5} b x^{5} + 5 \, a^{4} b^{2} x^{4} - 10 \, a^{3} b^{3} x^{3} + 30 \, a^{2} b^{4} x^{2} + 50 \, a b^{5} x - 10 \, b^{6} - 60 \, {\left (a b^{5} x + b^{6}\right )} \log \left (a x + b\right )}{10 \, {\left (a^{8} x + a^{7} b\right )}} \]

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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giac [A]  time = 0.15, size = 85, normalized size = 1.05 \[ -\frac {6 \, b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{7}} - \frac {b^{6}}{{\left (a x + b\right )} a^{7}} + \frac {2 \, a^{8} x^{5} - 5 \, a^{7} b x^{4} + 10 \, a^{6} b^{2} x^{3} - 20 \, a^{5} b^{3} x^{2} + 50 \, a^{4} b^{4} x}{10 \, a^{10}} \]

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

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maple [A]  time = 0.01, size = 78, normalized size = 0.96 \[ \frac {x^{5}}{5 a^{2}}-\frac {b \,x^{4}}{2 a^{3}}+\frac {b^{2} x^{3}}{a^{4}}-\frac {2 b^{3} x^{2}}{a^{5}}+\frac {5 b^{4} x}{a^{6}}-\frac {b^{6}}{\left (a x +b \right ) a^{7}}-\frac {6 b^{5} \ln \left (a x +b \right )}{a^{7}} \]

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 = 1.18, size = 82, normalized size = 1.01 \[ -\frac {b^{6}}{a^{8} x + a^{7} b} - \frac {6 \, b^{5} \log \left (a x + b\right )}{a^{7}} + \frac {2 \, a^{4} x^{5} - 5 \, a^{3} b x^{4} + 10 \, a^{2} b^{2} x^{3} - 20 \, a b^{3} x^{2} + 50 \, b^{4} x}{10 \, a^{6}} \]

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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mupad [B]  time = 0.05, size = 83, normalized size = 1.02 \[ \frac {x^5}{5\,a^2}-\frac {6\,b^5\,\ln \left (b+a\,x\right )}{a^7}-\frac {b\,x^4}{2\,a^3}+\frac {5\,b^4\,x}{a^6}+\frac {b^2\,x^3}{a^4}-\frac {2\,b^3\,x^2}{a^5}-\frac {b^6}{a\,\left (x\,a^7+b\,a^6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a + b/x)^2,x)

[Out]

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

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sympy [A]  time = 0.26, size = 78, normalized size = 0.96 \[ - \frac {b^{6}}{a^{8} x + a^{7} b} + \frac {x^{5}}{5 a^{2}} - \frac {b x^{4}}{2 a^{3}} + \frac {b^{2} x^{3}}{a^{4}} - \frac {2 b^{3} x^{2}}{a^{5}} + \frac {5 b^{4} x}{a^{6}} - \frac {6 b^{5} \log {\left (a x + b \right )}}{a^{7}} \]

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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