3.52 \(\int \frac{x^4 (d+e x)}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{b^2 (c d-b e)}{c^4 (b+c x)}+\frac{x (c d-2 b e)}{c^3}-\frac{b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac{e x^2}{2 c^2} \]

[Out]

((c*d - 2*b*e)*x)/c^3 + (e*x^2)/(2*c^2) - (b^2*(c*d - b*e))/(c^4*(b + c*x)) - (b*(2*c*d - 3*b*e)*Log[b + c*x])
/c^4

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Rubi [A]  time = 0.0686247, antiderivative size = 69, 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} \[ -\frac{b^2 (c d-b e)}{c^4 (b+c x)}+\frac{x (c d-2 b e)}{c^3}-\frac{b (2 c d-3 b e) \log (b+c x)}{c^4}+\frac{e x^2}{2 c^2} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

((c*d - 2*b*e)*x)/c^3 + (e*x^2)/(2*c^2) - (b^2*(c*d - b*e))/(c^4*(b + c*x)) - (b*(2*c*d - 3*b*e)*Log[b + c*x])
/c^4

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

Mathematica [A]  time = 0.0500609, size = 66, normalized size = 0.96 \[ \frac{\frac{2 b^2 (b e-c d)}{b+c x}+2 c x (c d-2 b e)+2 b (3 b e-2 c d) \log (b+c x)+c^2 e x^2}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(d + e*x))/(b*x + c*x^2)^2,x]

[Out]

(2*c*(c*d - 2*b*e)*x + c^2*e*x^2 + (2*b^2*(-(c*d) + b*e))/(b + c*x) + 2*b*(-2*c*d + 3*b*e)*Log[b + c*x])/(2*c^
4)

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Maple [A]  time = 0.008, size = 84, normalized size = 1.2 \begin{align*}{\frac{e{x}^{2}}{2\,{c}^{2}}}-2\,{\frac{bex}{{c}^{3}}}+{\frac{dx}{{c}^{2}}}+{\frac{{b}^{3}e}{{c}^{4} \left ( cx+b \right ) }}-{\frac{{b}^{2}d}{{c}^{3} \left ( cx+b \right ) }}+3\,{\frac{{b}^{2}\ln \left ( cx+b \right ) e}{{c}^{4}}}-2\,{\frac{b\ln \left ( cx+b \right ) d}{{c}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)/(c*x^2+b*x)^2,x)

[Out]

1/2*e*x^2/c^2-2/c^3*b*e*x+d*x/c^2+b^3/c^4/(c*x+b)*e-b^2/c^3/(c*x+b)*d+3*b^2/c^4*ln(c*x+b)*e-2*b/c^3*ln(c*x+b)*
d

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Maxima [A]  time = 1.0835, size = 101, normalized size = 1.46 \begin{align*} -\frac{b^{2} c d - b^{3} e}{c^{5} x + b c^{4}} + \frac{c e x^{2} + 2 \,{\left (c d - 2 \, b e\right )} x}{2 \, c^{3}} - \frac{{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left (c x + b\right )}{c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

-(b^2*c*d - b^3*e)/(c^5*x + b*c^4) + 1/2*(c*e*x^2 + 2*(c*d - 2*b*e)*x)/c^3 - (2*b*c*d - 3*b^2*e)*log(c*x + b)/
c^4

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Fricas [A]  time = 1.85371, size = 240, normalized size = 3.48 \begin{align*} \frac{c^{3} e x^{3} - 2 \, b^{2} c d + 2 \, b^{3} e +{\left (2 \, c^{3} d - 3 \, b c^{2} e\right )} x^{2} + 2 \,{\left (b c^{2} d - 2 \, b^{2} c e\right )} x - 2 \,{\left (2 \, b^{2} c d - 3 \, b^{3} e +{\left (2 \, b c^{2} d - 3 \, b^{2} c e\right )} x\right )} \log \left (c x + b\right )}{2 \,{\left (c^{5} x + b c^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

1/2*(c^3*e*x^3 - 2*b^2*c*d + 2*b^3*e + (2*c^3*d - 3*b*c^2*e)*x^2 + 2*(b*c^2*d - 2*b^2*c*e)*x - 2*(2*b^2*c*d -
3*b^3*e + (2*b*c^2*d - 3*b^2*c*e)*x)*log(c*x + b))/(c^5*x + b*c^4)

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Sympy [A]  time = 0.612942, size = 66, normalized size = 0.96 \begin{align*} \frac{b \left (3 b e - 2 c d\right ) \log{\left (b + c x \right )}}{c^{4}} + \frac{b^{3} e - b^{2} c d}{b c^{4} + c^{5} x} + \frac{e x^{2}}{2 c^{2}} - \frac{x \left (2 b e - c d\right )}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)/(c*x**2+b*x)**2,x)

[Out]

b*(3*b*e - 2*c*d)*log(b + c*x)/c**4 + (b**3*e - b**2*c*d)/(b*c**4 + c**5*x) + e*x**2/(2*c**2) - x*(2*b*e - c*d
)/c**3

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Giac [A]  time = 1.20667, size = 109, normalized size = 1.58 \begin{align*} -\frac{{\left (2 \, b c d - 3 \, b^{2} e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{4}} + \frac{c^{2} x^{2} e + 2 \, c^{2} d x - 4 \, b c x e}{2 \, c^{4}} - \frac{b^{2} c d - b^{3} e}{{\left (c x + b\right )} c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

-(2*b*c*d - 3*b^2*e)*log(abs(c*x + b))/c^4 + 1/2*(c^2*x^2*e + 2*c^2*d*x - 4*b*c*x*e)/c^4 - (b^2*c*d - b^3*e)/(
(c*x + b)*c^4)