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

Optimal. Leaf size=31 \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.015678, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x),x]

[Out]

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

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

Mathematica [A]  time = 0.0032206, size = 31, normalized size = 1. \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.002, size = 30, normalized size = 1. \begin{align*} -{\frac{ax}{{b}^{2}}}+{\frac{{x}^{2}}{2\,b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.94316, size = 39, normalized size = 1.26 \begin{align*} \frac{a^{2} \log \left (b x + a\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.01677, size = 68, normalized size = 2.19 \begin{align*} \frac{b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.267409, size = 26, normalized size = 0.84 \begin{align*} \frac{a^{2} \log{\left (a + b x \right )}}{b^{3}} - \frac{a x}{b^{2}} + \frac{x^{2}}{2 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.07159, size = 41, normalized size = 1.32 \begin{align*} \frac{a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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