3.1633 \(\int \frac{x^4}{(a+\frac{b}{x})^3} \, dx\)

Optimal. Leaf size=99 \[ -\frac{5 b^3 x^2}{a^6}+\frac{2 b^2 x^3}{a^5}+\frac{b^7}{2 a^8 (a x+b)^2}-\frac{7 b^6}{a^8 (a x+b)}+\frac{15 b^4 x}{a^7}-\frac{21 b^5 \log (a x+b)}{a^8}-\frac{3 b x^4}{4 a^4}+\frac{x^5}{5 a^3} \]

[Out]

(15*b^4*x)/a^7 - (5*b^3*x^2)/a^6 + (2*b^2*x^3)/a^5 - (3*b*x^4)/(4*a^4) + x^5/(5*a^3) + b^7/(2*a^8*(b + a*x)^2)
- (7*b^6)/(a^8*(b + a*x)) - (21*b^5*Log[b + a*x])/a^8

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Rubi [A]  time = 0.0703092, antiderivative size = 99, 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} \[ -\frac{5 b^3 x^2}{a^6}+\frac{2 b^2 x^3}{a^5}+\frac{b^7}{2 a^8 (a x+b)^2}-\frac{7 b^6}{a^8 (a x+b)}+\frac{15 b^4 x}{a^7}-\frac{21 b^5 \log (a x+b)}{a^8}-\frac{3 b x^4}{4 a^4}+\frac{x^5}{5 a^3} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x)^3,x]

[Out]

(15*b^4*x)/a^7 - (5*b^3*x^2)/a^6 + (2*b^2*x^3)/a^5 - (3*b*x^4)/(4*a^4) + x^5/(5*a^3) + b^7/(2*a^8*(b + a*x)^2)
- (7*b^6)/(a^8*(b + a*x)) - (21*b^5*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^4}{\left (a+\frac{b}{x}\right )^3} \, dx &=\int \frac{x^7}{(b+a x)^3} \, dx\\ &=\int \left (\frac{15 b^4}{a^7}-\frac{10 b^3 x}{a^6}+\frac{6 b^2 x^2}{a^5}-\frac{3 b x^3}{a^4}+\frac{x^4}{a^3}-\frac{b^7}{a^7 (b+a x)^3}+\frac{7 b^6}{a^7 (b+a x)^2}-\frac{21 b^5}{a^7 (b+a x)}\right ) \, dx\\ &=\frac{15 b^4 x}{a^7}-\frac{5 b^3 x^2}{a^6}+\frac{2 b^2 x^3}{a^5}-\frac{3 b x^4}{4 a^4}+\frac{x^5}{5 a^3}+\frac{b^7}{2 a^8 (b+a x)^2}-\frac{7 b^6}{a^8 (b+a x)}-\frac{21 b^5 \log (b+a x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.0487932, size = 85, normalized size = 0.86 \[ \frac{-100 a^2 b^3 x^2+40 a^3 b^2 x^3-15 a^4 b x^4+4 a^5 x^5-\frac{10 b^6 (14 a x+13 b)}{(a x+b)^2}+300 a b^4 x-420 b^5 \log (a x+b)}{20 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x)^3,x]

[Out]

(300*a*b^4*x - 100*a^2*b^3*x^2 + 40*a^3*b^2*x^3 - 15*a^4*b*x^4 + 4*a^5*x^5 - (10*b^6*(13*b + 14*a*x))/(b + a*x
)^2 - 420*b^5*Log[b + a*x])/(20*a^8)

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Maple [A]  time = 0.007, size = 94, normalized size = 1. \begin{align*} 15\,{\frac{{b}^{4}x}{{a}^{7}}}-5\,{\frac{{b}^{3}{x}^{2}}{{a}^{6}}}+2\,{\frac{{b}^{2}{x}^{3}}{{a}^{5}}}-{\frac{3\,b{x}^{4}}{4\,{a}^{4}}}+{\frac{{x}^{5}}{5\,{a}^{3}}}+{\frac{{b}^{7}}{2\,{a}^{8} \left ( ax+b \right ) ^{2}}}-7\,{\frac{{b}^{6}}{{a}^{8} \left ( ax+b \right ) }}-21\,{\frac{{b}^{5}\ln \left ( ax+b \right ) }{{a}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4/(a+b/x)^3,x)

[Out]

15*b^4*x/a^7-5*b^3*x^2/a^6+2*b^2*x^3/a^5-3/4*b*x^4/a^4+1/5*x^5/a^3+1/2*b^7/a^8/(a*x+b)^2-7*b^6/a^8/(a*x+b)-21*
b^5*ln(a*x+b)/a^8

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Maxima [A]  time = 1.01566, size = 139, normalized size = 1.4 \begin{align*} -\frac{14 \, a b^{6} x + 13 \, b^{7}}{2 \,{\left (a^{10} x^{2} + 2 \, a^{9} b x + a^{8} b^{2}\right )}} - \frac{21 \, b^{5} \log \left (a x + b\right )}{a^{8}} + \frac{4 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 40 \, a^{2} b^{2} x^{3} - 100 \, a b^{3} x^{2} + 300 \, b^{4} x}{20 \, a^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x)^3,x, algorithm="maxima")

[Out]

-1/2*(14*a*b^6*x + 13*b^7)/(a^10*x^2 + 2*a^9*b*x + a^8*b^2) - 21*b^5*log(a*x + b)/a^8 + 1/20*(4*a^4*x^5 - 15*a
^3*b*x^4 + 40*a^2*b^2*x^3 - 100*a*b^3*x^2 + 300*b^4*x)/a^7

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Fricas [A]  time = 1.43907, size = 284, normalized size = 2.87 \begin{align*} \frac{4 \, a^{7} x^{7} - 7 \, a^{6} b x^{6} + 14 \, a^{5} b^{2} x^{5} - 35 \, a^{4} b^{3} x^{4} + 140 \, a^{3} b^{4} x^{3} + 500 \, a^{2} b^{5} x^{2} + 160 \, a b^{6} x - 130 \, b^{7} - 420 \,{\left (a^{2} b^{5} x^{2} + 2 \, a b^{6} x + b^{7}\right )} \log \left (a x + b\right )}{20 \,{\left (a^{10} x^{2} + 2 \, a^{9} b x + a^{8} b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x)^3,x, algorithm="fricas")

[Out]

1/20*(4*a^7*x^7 - 7*a^6*b*x^6 + 14*a^5*b^2*x^5 - 35*a^4*b^3*x^4 + 140*a^3*b^4*x^3 + 500*a^2*b^5*x^2 + 160*a*b^
6*x - 130*b^7 - 420*(a^2*b^5*x^2 + 2*a*b^6*x + b^7)*log(a*x + b))/(a^10*x^2 + 2*a^9*b*x + a^8*b^2)

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Sympy [A]  time = 0.483557, size = 107, normalized size = 1.08 \begin{align*} - \frac{14 a b^{6} x + 13 b^{7}}{2 a^{10} x^{2} + 4 a^{9} b x + 2 a^{8} b^{2}} + \frac{x^{5}}{5 a^{3}} - \frac{3 b x^{4}}{4 a^{4}} + \frac{2 b^{2} x^{3}}{a^{5}} - \frac{5 b^{3} x^{2}}{a^{6}} + \frac{15 b^{4} x}{a^{7}} - \frac{21 b^{5} \log{\left (a x + b \right )}}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4/(a+b/x)**3,x)

[Out]

-(14*a*b**6*x + 13*b**7)/(2*a**10*x**2 + 4*a**9*b*x + 2*a**8*b**2) + x**5/(5*a**3) - 3*b*x**4/(4*a**4) + 2*b**
2*x**3/a**5 - 5*b**3*x**2/a**6 + 15*b**4*x/a**7 - 21*b**5*log(a*x + b)/a**8

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Giac [A]  time = 1.1001, size = 128, normalized size = 1.29 \begin{align*} -\frac{21 \, b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{8}} - \frac{14 \, a b^{6} x + 13 \, b^{7}}{2 \,{\left (a x + b\right )}^{2} a^{8}} + \frac{4 \, a^{12} x^{5} - 15 \, a^{11} b x^{4} + 40 \, a^{10} b^{2} x^{3} - 100 \, a^{9} b^{3} x^{2} + 300 \, a^{8} b^{4} x}{20 \, a^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4/(a+b/x)^3,x, algorithm="giac")

[Out]

-21*b^5*log(abs(a*x + b))/a^8 - 1/2*(14*a*b^6*x + 13*b^7)/((a*x + b)^2*a^8) + 1/20*(4*a^12*x^5 - 15*a^11*b*x^4
+ 40*a^10*b^2*x^3 - 100*a^9*b^3*x^2 + 300*a^8*b^4*x)/a^15