3.46 \(\int \frac{x (d+e x)}{b x+c x^2} \, dx\)

Optimal. Leaf size=25 \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]

[Out]

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

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Rubi [A]  time = 0.0230134, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056, Rules used = {765} \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]

Antiderivative was successfully verified.

[In]

Int[(x*(d + e*x))/(b*x + c*x^2),x]

[Out]

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

Rule 765

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand
Integrand[(e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, e, f, g, m}, x] && IntegerQ[p] && (
GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0071542, size = 25, normalized size = 1. \[ \frac{(c d-b e) \log (b+c x)}{c^2}+\frac{e x}{c} \]

Antiderivative was successfully verified.

[In]

Integrate[(x*(d + e*x))/(b*x + c*x^2),x]

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 1.3 \begin{align*}{\frac{ex}{c}}-{\frac{\ln \left ( cx+b \right ) be}{{c}^{2}}}+{\frac{\ln \left ( cx+b \right ) d}{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(e*x+d)/(c*x^2+b*x),x)

[Out]

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

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Maxima [A]  time = 1.16445, size = 34, normalized size = 1.36 \begin{align*} \frac{e x}{c} + \frac{{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(c*x^2+b*x),x, algorithm="maxima")

[Out]

e*x/c + (c*d - b*e)*log(c*x + b)/c^2

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Fricas [A]  time = 1.90932, size = 54, normalized size = 2.16 \begin{align*} \frac{c e x +{\left (c d - b e\right )} \log \left (c x + b\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(c*x^2+b*x),x, algorithm="fricas")

[Out]

(c*e*x + (c*d - b*e)*log(c*x + b))/c^2

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Sympy [A]  time = 0.492257, size = 20, normalized size = 0.8 \begin{align*} \frac{e x}{c} - \frac{\left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(c*x**2+b*x),x)

[Out]

e*x/c - (b*e - c*d)*log(b + c*x)/c**2

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Giac [A]  time = 1.16975, size = 38, normalized size = 1.52 \begin{align*} \frac{x e}{c} + \frac{{\left (c d - b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(e*x+d)/(c*x^2+b*x),x, algorithm="giac")

[Out]

x*e/c + (c*d - b*e)*log(abs(c*x + b))/c^2