3.1607 \(\int \frac{x^4}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=70 \[ -\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}+\frac{b^4 x}{a^5}-\frac{b^5 \log (a x+b)}{a^6}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]

[Out]

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

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Rubi [A]  time = 0.0372933, antiderivative size = 70, 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} \[ -\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}+\frac{b^4 x}{a^5}-\frac{b^5 \log (a x+b)}{a^6}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]

Antiderivative was successfully verified.

[In]

Int[x^4/(a + b/x),x]

[Out]

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

Mathematica [A]  time = 0.004117, size = 70, normalized size = 1. \[ -\frac{b^3 x^2}{2 a^4}+\frac{b^2 x^3}{3 a^3}+\frac{b^4 x}{a^5}-\frac{b^5 \log (a x+b)}{a^6}-\frac{b x^4}{4 a^2}+\frac{x^5}{5 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4/(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.003, size = 63, normalized size = 0.9 \begin{align*}{\frac{{b}^{4}x}{{a}^{5}}}-{\frac{{b}^{3}{x}^{2}}{2\,{a}^{4}}}+{\frac{{b}^{2}{x}^{3}}{3\,{a}^{3}}}-{\frac{b{x}^{4}}{4\,{a}^{2}}}+{\frac{{x}^{5}}{5\,a}}-{\frac{{b}^{5}\ln \left ( ax+b \right ) }{{a}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01661, size = 86, normalized size = 1.23 \begin{align*} -\frac{b^{5} \log \left (a x + b\right )}{a^{6}} + \frac{12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^5*log(a*x + b)/a^6 + 1/60*(12*a^4*x^5 - 15*a^3*b*x^4 + 20*a^2*b^2*x^3 - 30*a*b^3*x^2 + 60*b^4*x)/a^5

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Fricas [A]  time = 1.3889, size = 144, normalized size = 2.06 \begin{align*} \frac{12 \, a^{5} x^{5} - 15 \, a^{4} b x^{4} + 20 \, a^{3} b^{2} x^{3} - 30 \, a^{2} b^{3} x^{2} + 60 \, a b^{4} x - 60 \, b^{5} \log \left (a x + b\right )}{60 \, a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*a^5*x^5 - 15*a^4*b*x^4 + 20*a^3*b^2*x^3 - 30*a^2*b^3*x^2 + 60*a*b^4*x - 60*b^5*log(a*x + b))/a^6

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Sympy [A]  time = 0.283082, size = 61, normalized size = 0.87 \begin{align*} \frac{x^{5}}{5 a} - \frac{b x^{4}}{4 a^{2}} + \frac{b^{2} x^{3}}{3 a^{3}} - \frac{b^{3} x^{2}}{2 a^{4}} + \frac{b^{4} x}{a^{5}} - \frac{b^{5} \log{\left (a x + b \right )}}{a^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10506, size = 88, normalized size = 1.26 \begin{align*} -\frac{b^{5} \log \left ({\left | a x + b \right |}\right )}{a^{6}} + \frac{12 \, a^{4} x^{5} - 15 \, a^{3} b x^{4} + 20 \, a^{2} b^{2} x^{3} - 30 \, a b^{3} x^{2} + 60 \, b^{4} x}{60 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^5*log(abs(a*x + b))/a^6 + 1/60*(12*a^4*x^5 - 15*a^3*b*x^4 + 20*a^2*b^2*x^3 - 30*a*b^3*x^2 + 60*b^4*x)/a^5