3.207 \(\int \frac{x (c+d x)}{a+b x} \, dx\)

Optimal. Leaf size=45 \[ \frac{x (b c-a d)}{b^2}-\frac{a (b c-a d) \log (a+b x)}{b^3}+\frac{d x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0323585, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {77} \[ \frac{x (b c-a d)}{b^2}-\frac{a (b c-a d) \log (a+b x)}{b^3}+\frac{d x^2}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(x*(c + d*x))/(a + b*x),x]

[Out]

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

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

Mathematica [A]  time = 0.0127889, size = 41, normalized size = 0.91 \[ \frac{b x (-2 a d+2 b c+b d x)+2 a (a d-b c) \log (a+b x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(c + d*x))/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.001, size = 52, normalized size = 1.2 \begin{align*}{\frac{d{x}^{2}}{2\,b}}-{\frac{adx}{{b}^{2}}}+{\frac{cx}{b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) d}{{b}^{3}}}-{\frac{a\ln \left ( bx+a \right ) c}{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(d*x+c)/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.08073, size = 62, normalized size = 1.38 \begin{align*} \frac{b d x^{2} + 2 \,{\left (b c - a d\right )} x}{2 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)/(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.89706, size = 103, normalized size = 2.29 \begin{align*} \frac{b^{2} d x^{2} + 2 \,{\left (b^{2} c - a b d\right )} x - 2 \,{\left (a b c - a^{2} d\right )} \log \left (b x + a\right )}{2 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)/(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.388438, size = 37, normalized size = 0.82 \begin{align*} \frac{a \left (a d - b c\right ) \log{\left (a + b x \right )}}{b^{3}} + \frac{d x^{2}}{2 b} - \frac{x \left (a d - b c\right )}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)/(b*x+a),x)

[Out]

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

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Giac [A]  time = 1.23569, size = 62, normalized size = 1.38 \begin{align*} \frac{b d x^{2} + 2 \, b c x - 2 \, a d x}{2 \, b^{2}} - \frac{{\left (a b c - a^{2} d\right )} \log \left ({\left | b x + a \right |}\right )}{b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(d*x+c)/(b*x+a),x, algorithm="giac")

[Out]

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