\(\int \frac {x^5}{(a+\frac {b}{x})^2} \, dx\) [1619]

Optimal result

Integrand size = 13, antiderivative size = 98 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=-\frac {6 b^5 x}{a^7}+\frac {5 b^4 x^2}{2 a^6}-\frac {4 b^3 x^3}{3 a^5}+\frac {3 b^2 x^4}{4 a^4}-\frac {2 b x^5}{5 a^3}+\frac {x^6}{6 a^2}+\frac {b^7}{a^8 (b+a x)}+\frac {7 b^6 \log (b+a x)}{a^8} \]

[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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {b^7}{a^8 (a x+b)}+\frac {7 b^6 \log (a x+b)}{a^8}-\frac {6 b^5 x}{a^7}+\frac {5 b^4 x^2}{2 a^6}-\frac {4 b^3 x^3}{3 a^5}+\frac {3 b^2 x^4}{4 a^4}-\frac {2 b x^5}{5 a^3}+\frac {x^6}{6 a^2} \]

[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 45

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 269

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*} \text {integral}& = \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] (verified)

Time = 0.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {-360 a b^5 x+150 a^2 b^4 x^2-80 a^3 b^3 x^3+45 a^4 b^2 x^4-24 a^5 b x^5+10 a^6 x^6+\frac {60 b^7}{b+a x}+420 b^6 \log (b+a x)}{60 a^8} \]

[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)

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {\frac {1}{6} a^{5} x^{6}-\frac {2}{5} a^{4} b \,x^{5}+\frac {3}{4} a^{3} b^{2} x^{4}-\frac {4}{3} a^{2} b^{3} x^{3}+\frac {5}{2} b^{4} x^{2} a -6 b^{5} x}{a^{7}}+\frac {b^{7}}{a^{8} \left (a x +b \right )}+\frac {7 b^{6} \ln \left (a x +b \right )}{a^{8}}\)	\(89\)
risch	\(-\frac {6 b^{5} x}{a^{7}}+\frac {5 b^{4} x^{2}}{2 a^{6}}-\frac {4 b^{3} x^{3}}{3 a^{5}}+\frac {3 b^{2} x^{4}}{4 a^{4}}-\frac {2 b \,x^{5}}{5 a^{3}}+\frac {x^{6}}{6 a^{2}}+\frac {b^{7}}{a^{8} \left (a x +b \right )}+\frac {7 b^{6} \ln \left (a x +b \right )}{a^{8}}\)	\(89\)
norman	\(\frac {\frac {7 b^{7}}{a^{8}}+\frac {x^{7}}{6 a}-\frac {7 b \,x^{6}}{30 a^{2}}+\frac {7 b^{2} x^{5}}{20 a^{3}}-\frac {7 b^{3} x^{4}}{12 a^{4}}+\frac {7 b^{4} x^{3}}{6 a^{5}}-\frac {7 b^{5} x^{2}}{2 a^{6}}}{a x +b}+\frac {7 b^{6} \ln \left (a x +b \right )}{a^{8}}\)	\(94\)
parallelrisch	\(\frac {10 a^{7} x^{7}-14 a^{6} b \,x^{6}+21 a^{5} b^{2} x^{5}-35 a^{4} b^{3} x^{4}+70 a^{3} b^{4} x^{3}+420 \ln \left (a x +b \right ) x a \,b^{6}-210 b^{5} x^{2} a^{2}+420 \ln \left (a x +b \right ) b^{7}+420 b^{7}}{60 a^{8} \left (a x +b \right )}\)	\(104\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {10 \, a^{7} x^{7} - 14 \, a^{6} b x^{6} + 21 \, a^{5} b^{2} x^{5} - 35 \, a^{4} b^{3} x^{4} + 70 \, a^{3} b^{4} x^{3} - 210 \, a^{2} b^{5} x^{2} - 360 \, a b^{6} x + 60 \, b^{7} + 420 \, {\left (a b^{6} x + b^{7}\right )} \log \left (a x + b\right )}{60 \, {\left (a^{9} x + a^{8} b\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {b^{7}}{a^{9} x + a^{8} b} + \frac {x^{6}}{6 a^{2}} - \frac {2 b x^{5}}{5 a^{3}} + \frac {3 b^{2} x^{4}}{4 a^{4}} - \frac {4 b^{3} x^{3}}{3 a^{5}} + \frac {5 b^{4} x^{2}}{2 a^{6}} - \frac {6 b^{5} x}{a^{7}} + \frac {7 b^{6} \log {\left (a x + b \right )}}{a^{8}} \]

[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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.94 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {b^{7}}{a^{9} x + a^{8} b} + \frac {7 \, b^{6} \log \left (a x + b\right )}{a^{8}} + \frac {10 \, a^{5} x^{6} - 24 \, a^{4} b x^{5} + 45 \, a^{3} b^{2} x^{4} - 80 \, a^{2} b^{3} x^{3} + 150 \, a b^{4} x^{2} - 360 \, b^{5} x}{60 \, a^{7}} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {7 \, b^{6} \log \left ({\left | a x + b \right |}\right )}{a^{8}} + \frac {b^{7}}{{\left (a x + b\right )} a^{8}} + \frac {10 \, a^{10} x^{6} - 24 \, a^{9} b x^{5} + 45 \, a^{8} b^{2} x^{4} - 80 \, a^{7} b^{3} x^{3} + 150 \, a^{6} b^{4} x^{2} - 360 \, a^{5} b^{5} x}{60 \, a^{12}} \]

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.96 \[ \int \frac {x^5}{\left (a+\frac {b}{x}\right )^2} \, dx=\frac {x^6}{6\,a^2}+\frac {7\,b^6\,\ln \left (b+a\,x\right )}{a^8}-\frac {2\,b\,x^5}{5\,a^3}-\frac {6\,b^5\,x}{a^7}+\frac {3\,b^2\,x^4}{4\,a^4}-\frac {4\,b^3\,x^3}{3\,a^5}+\frac {5\,b^4\,x^2}{2\,a^6}+\frac {b^7}{a\,\left (x\,a^8+b\,a^7\right )} \]

[In]

int(x^5/(a + b/x)^2,x)

[Out]

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