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

Optimal. Leaf size=57 \[ \frac{d}{b^2 (b+c x)}-\frac{d \log (b+c x)}{b^3}+\frac{d \log (x)}{b^3}+\frac{c d-b e}{2 b c (b+c x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0437102, antiderivative size = 57, 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} \[ \frac{d}{b^2 (b+c x)}-\frac{d \log (b+c x)}{b^3}+\frac{d \log (x)}{b^3}+\frac{c d-b e}{2 b c (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0416078, size = 53, normalized size = 0.93 \[ \frac{\frac{b \left (b^2 (-e)+3 b c d+2 c^2 d x\right )}{c (b+c x)^2}-2 d \log (b+c x)+2 d \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.008, size = 59, normalized size = 1. \begin{align*}{\frac{d\ln \left ( x \right ) }{{b}^{3}}}-{\frac{e}{2\,c \left ( cx+b \right ) ^{2}}}+{\frac{d}{2\,b \left ( cx+b \right ) ^{2}}}-{\frac{d\ln \left ( cx+b \right ) }{{b}^{3}}}+{\frac{d}{{b}^{2} \left ( cx+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.10209, size = 92, normalized size = 1.61 \begin{align*} \frac{2 \, c^{2} d x + 3 \, b c d - b^{2} e}{2 \,{\left (b^{2} c^{3} x^{2} + 2 \, b^{3} c^{2} x + b^{4} c\right )}} - \frac{d \log \left (c x + b\right )}{b^{3}} + \frac{d \log \left (x\right )}{b^{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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.76518, size = 236, normalized size = 4.14 \begin{align*} \frac{2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e - 2 \,{\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \left (c x + b\right ) + 2 \,{\left (c^{3} d x^{2} + 2 \, b c^{2} d x + b^{2} c d\right )} \log \left (x\right )}{2 \,{\left (b^{3} c^{3} x^{2} + 2 \, b^{4} c^{2} x + b^{5} c\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.89887, size = 63, normalized size = 1.11 \begin{align*} \frac{- b^{2} e + 3 b c d + 2 c^{2} d x}{2 b^{4} c + 4 b^{3} c^{2} x + 2 b^{2} c^{3} x^{2}} + \frac{d \left (\log{\left (x \right )} - \log{\left (\frac{b}{c} + x \right )}\right )}{b^{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)**3,x)

[Out]

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

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Giac [A]  time = 1.12754, size = 81, normalized size = 1.42 \begin{align*} -\frac{d \log \left ({\left | c x + b \right |}\right )}{b^{3}} + \frac{d \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{2 \, b c^{2} d x + 3 \, b^{2} c d - b^{3} e}{2 \,{\left (c x + b\right )}^{2} b^{3} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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