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

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

[Out]

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

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Rubi [A]  time = 0.0404068, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {765} \[ \frac{x (c d-b e)}{c^2}-\frac{b (c d-b e) \log (b+c x)}{c^3}+\frac{e x^2}{2 c} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*(d + e*x))/(b*x + c*x^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0131524, size = 41, normalized size = 0.91 \[ \frac{c x (-2 b e+2 c d+c e x)+2 b (b e-c d) \log (b+c x)}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.08209, size = 62, normalized size = 1.38 \begin{align*} \frac{c e x^{2} + 2 \,{\left (c d - b e\right )} x}{2 \, c^{2}} - \frac{{\left (b c d - b^{2} e\right )} \log \left (c x + b\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.83609, size = 103, normalized size = 2.29 \begin{align*} \frac{c^{2} e x^{2} + 2 \,{\left (c^{2} d - b c e\right )} x - 2 \,{\left (b c d - b^{2} e\right )} \log \left (c x + b\right )}{2 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.650462, size = 37, normalized size = 0.82 \begin{align*} \frac{b \left (b e - c d\right ) \log{\left (b + c x \right )}}{c^{3}} + \frac{e x^{2}}{2 c} - \frac{x \left (b e - c d\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12361, size = 66, normalized size = 1.47 \begin{align*} \frac{c x^{2} e + 2 \, c d x - 2 \, b x e}{2 \, c^{2}} - \frac{{\left (b c d - b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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