3.1609 \(\int \frac{x^2}{a+\frac{b}{x}} \, dx\)

Optimal. Leaf size=44 \[ \frac{b^2 x}{a^3}-\frac{b^3 \log (a x+b)}{a^4}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]

[Out]

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

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Rubi [A]  time = 0.0222153, antiderivative size = 44, 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} \[ \frac{b^2 x}{a^3}-\frac{b^3 \log (a x+b)}{a^4}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b/x),x]

[Out]

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

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

Mathematica [A]  time = 0.0034744, size = 44, normalized size = 1. \[ \frac{b^2 x}{a^3}-\frac{b^3 \log (a x+b)}{a^4}-\frac{b x^2}{2 a^2}+\frac{x^3}{3 a} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b/x),x]

[Out]

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

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Maple [A]  time = 0.002, size = 41, normalized size = 0.9 \begin{align*}{\frac{{b}^{2}x}{{a}^{3}}}-{\frac{b{x}^{2}}{2\,{a}^{2}}}+{\frac{{x}^{3}}{3\,a}}-{\frac{{b}^{3}\ln \left ( ax+b \right ) }{{a}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a+b/x),x)

[Out]

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

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Maxima [A]  time = 1.01937, size = 57, normalized size = 1.3 \begin{align*} -\frac{b^{3} \log \left (a x + b\right )}{a^{4}} + \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.41328, size = 92, normalized size = 2.09 \begin{align*} \frac{2 \, a^{3} x^{3} - 3 \, a^{2} b x^{2} + 6 \, a b^{2} x - 6 \, b^{3} \log \left (a x + b\right )}{6 \, a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.268416, size = 37, normalized size = 0.84 \begin{align*} \frac{x^{3}}{3 a} - \frac{b x^{2}}{2 a^{2}} + \frac{b^{2} x}{a^{3}} - \frac{b^{3} \log{\left (a x + b \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(a+b/x),x)

[Out]

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

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Giac [A]  time = 1.09174, size = 58, normalized size = 1.32 \begin{align*} -\frac{b^{3} \log \left ({\left | a x + b \right |}\right )}{a^{4}} + \frac{2 \, a^{2} x^{3} - 3 \, a b x^{2} + 6 \, b^{2} x}{6 \, a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(a+b/x),x, algorithm="giac")

[Out]

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