3.61 \(\int \frac{x^4 (d+e x)}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=55 \[ -\frac{c d-2 b e}{c^3 (b+c x)}+\frac{b (c d-b e)}{2 c^3 (b+c x)^2}+\frac{e \log (b+c x)}{c^3} \]

[Out]

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

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Rubi [A]  time = 0.0466188, antiderivative size = 55, 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} \[ -\frac{c d-2 b e}{c^3 (b+c x)}+\frac{b (c d-b e)}{2 c^3 (b+c x)^2}+\frac{e \log (b+c x)}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0163263, size = 54, normalized size = 0.98 \[ \frac{3 b^2 e-b c (d-4 e x)+2 e (b+c x)^2 \log (b+c x)-2 c^2 d x}{2 c^3 (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.006, size = 70, normalized size = 1.3 \begin{align*} 2\,{\frac{be}{{c}^{3} \left ( cx+b \right ) }}-{\frac{d}{{c}^{2} \left ( cx+b \right ) }}+{\frac{e\ln \left ( cx+b \right ) }{{c}^{3}}}-{\frac{{b}^{2}e}{2\,{c}^{3} \left ( cx+b \right ) ^{2}}}+{\frac{bd}{2\,{c}^{2} \left ( cx+b \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)/(c*x^2+b*x)^3,x)

[Out]

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

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Maxima [A]  time = 1.01698, size = 85, normalized size = 1.55 \begin{align*} -\frac{b c d - 3 \, b^{2} e + 2 \,{\left (c^{2} d - 2 \, b c e\right )} x}{2 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\right )}} + \frac{e \log \left (c x + b\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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.73018, size = 174, normalized size = 3.16 \begin{align*} -\frac{b c d - 3 \, b^{2} e + 2 \,{\left (c^{2} d - 2 \, b c e\right )} x - 2 \,{\left (c^{2} e x^{2} + 2 \, b c e x + b^{2} e\right )} \log \left (c x + b\right )}{2 \,{\left (c^{5} x^{2} + 2 \, b c^{4} x + b^{2} c^{3}\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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.89074, size = 63, normalized size = 1.15 \begin{align*} \frac{3 b^{2} e - b c d + x \left (4 b c e - 2 c^{2} d\right )}{2 b^{2} c^{3} + 4 b c^{4} x + 2 c^{5} x^{2}} + \frac{e \log{\left (b + c x \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)**3,x)

[Out]

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

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Giac [A]  time = 1.13753, size = 74, normalized size = 1.35 \begin{align*} \frac{e \log \left ({\left | c x + b \right |}\right )}{c^{3}} - \frac{2 \,{\left (c d - 2 \, b e\right )} x + \frac{b c d - 3 \, b^{2} e}{c}}{2 \,{\left (c x + b\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

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