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

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

[Out]

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

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Rubi [A]  time = 0.0430754, 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} \[ \frac{b (c d-b e)}{c^3 (b+c x)}+\frac{(c d-2 b e) \log (b+c x)}{c^3}+\frac{e x}{c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0259708, size = 41, normalized size = 0.91 \[ \frac{\frac{b (c d-b e)}{b+c x}+(c d-2 b e) \log (b+c x)+c e x}{c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 61, normalized size = 1.4 \begin{align*}{\frac{ex}{{c}^{2}}}-{\frac{{b}^{2}e}{{c}^{3} \left ( cx+b \right ) }}+{\frac{bd}{{c}^{2} \left ( cx+b \right ) }}-2\,{\frac{\ln \left ( cx+b \right ) be}{{c}^{3}}}+{\frac{\ln \left ( cx+b \right ) d}{{c}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.11642, size = 68, normalized size = 1.51 \begin{align*} \frac{b c d - b^{2} e}{c^{4} x + b c^{3}} + \frac{e x}{c^{2}} + \frac{{\left (c d - 2 \, b 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^3*(e*x+d)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64778, size = 149, normalized size = 3.31 \begin{align*} \frac{c^{2} e x^{2} + b c e x + b c d - b^{2} e +{\left (b c d - 2 \, b^{2} e +{\left (c^{2} d - 2 \, b c e\right )} x\right )} \log \left (c x + b\right )}{c^{4} x + b c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.611978, size = 44, normalized size = 0.98 \begin{align*} - \frac{b^{2} e - b c d}{b c^{3} + c^{4} x} + \frac{e x}{c^{2}} - \frac{\left (2 b e - c d\right ) \log{\left (b + c x \right )}}{c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13689, size = 69, normalized size = 1.53 \begin{align*} \frac{x e}{c^{2}} + \frac{{\left (c d - 2 \, b e\right )} \log \left ({\left | c x + b \right |}\right )}{c^{3}} + \frac{b c d - b^{2} e}{{\left (c x + b\right )} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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