3.1608 \(\int \frac{x^3}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=57 \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 x}{a^4}+\frac{b^4 \log (a x+b)}{a^5}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]

[Out]

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

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Rubi [A]  time = 0.0267549, antiderivative size = 57, 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} \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 x}{a^4}+\frac{b^4 \log (a x+b)}{a^5}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b/x),x]

[Out]

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

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

Mathematica [A]  time = 0.0040021, size = 57, normalized size = 1. \[ \frac{b^2 x^2}{2 a^3}-\frac{b^3 x}{a^4}+\frac{b^4 \log (a x+b)}{a^5}-\frac{b x^3}{3 a^2}+\frac{x^4}{4 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.004, size = 52, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}x}{{a}^{4}}}+{\frac{{b}^{2}{x}^{2}}{2\,{a}^{3}}}-{\frac{b{x}^{3}}{3\,{a}^{2}}}+{\frac{{x}^{4}}{4\,a}}+{\frac{{b}^{4}\ln \left ( ax+b \right ) }{{a}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.0212, size = 70, normalized size = 1.23 \begin{align*} \frac{b^{4} \log \left (a x + b\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{12 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.42849, size = 117, normalized size = 2.05 \begin{align*} \frac{3 \, a^{4} x^{4} - 4 \, a^{3} b x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a b^{3} x + 12 \, b^{4} \log \left (a x + b\right )}{12 \, a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.272151, size = 49, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4 a} - \frac{b x^{3}}{3 a^{2}} + \frac{b^{2} x^{2}}{2 a^{3}} - \frac{b^{3} x}{a^{4}} + \frac{b^{4} \log{\left (a x + b \right )}}{a^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10112, size = 72, normalized size = 1.26 \begin{align*} \frac{b^{4} \log \left ({\left | a x + b \right |}\right )}{a^{5}} + \frac{3 \, a^{3} x^{4} - 4 \, a^{2} b x^{3} + 6 \, a b^{2} x^{2} - 12 \, b^{3} x}{12 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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