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

Optimal. Leaf size=87 \[ -\frac{(c d-b e)^3}{b^2 c^2 (b+c x)}+\frac{(c d-b e)^2 (b e+2 c d) \log (b+c x)}{b^3 c^2}-\frac{d^2 \log (x) (2 c d-3 b e)}{b^3}-\frac{d^3}{b^2 x} \]

[Out]

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

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Rubi [A]  time = 0.081705, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {698} \[ -\frac{(c d-b e)^3}{b^2 c^2 (b+c x)}+\frac{(c d-b e)^2 (b e+2 c d) \log (b+c x)}{b^3 c^2}-\frac{d^2 \log (x) (2 c d-3 b e)}{b^3}-\frac{d^3}{b^2 x} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3/(b*x + c*x^2)^2,x]

[Out]

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

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.0754549, size = 79, normalized size = 0.91 \[ \frac{\frac{b (b e-c d)^3}{c^2 (b+c x)}+\frac{(c d-b e)^2 (b e+2 c d) \log (b+c x)}{c^2}+d^2 \log (x) (3 b e-2 c d)-\frac{b d^3}{x}}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.057, size = 141, normalized size = 1.6 \begin{align*} -{\frac{{d}^{3}}{{b}^{2}x}}+3\,{\frac{{d}^{2}\ln \left ( x \right ) e}{{b}^{2}}}-2\,{\frac{{d}^{3}\ln \left ( x \right ) c}{{b}^{3}}}+{\frac{\ln \left ( cx+b \right ){e}^{3}}{{c}^{2}}}-3\,{\frac{\ln \left ( cx+b \right ){d}^{2}e}{{b}^{2}}}+2\,{\frac{c\ln \left ( cx+b \right ){d}^{3}}{{b}^{3}}}+{\frac{b{e}^{3}}{{c}^{2} \left ( cx+b \right ) }}-3\,{\frac{d{e}^{2}}{c \left ( cx+b \right ) }}+3\,{\frac{{d}^{2}e}{b \left ( cx+b \right ) }}-{\frac{{d}^{3}c}{{b}^{2} \left ( cx+b \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.14688, size = 178, normalized size = 2.05 \begin{align*} -\frac{b c^{2} d^{3} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{b^{2} c^{3} x^{2} + b^{3} c^{2} x} - \frac{{\left (2 \, c d^{3} - 3 \, b d^{2} e\right )} \log \left (x\right )}{b^{3}} + \frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} \log \left (c x + b\right )}{b^{3} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.72213, size = 392, normalized size = 4.51 \begin{align*} -\frac{b^{2} c^{2} d^{3} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + 3 \, b^{3} c d e^{2} - b^{4} e^{3}\right )} x -{\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e + b^{3} c e^{3}\right )} x^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e + b^{4} e^{3}\right )} x\right )} \log \left (c x + b\right ) +{\left ({\left (2 \, c^{4} d^{3} - 3 \, b c^{3} d^{2} e\right )} x^{2} +{\left (2 \, b c^{3} d^{3} - 3 \, b^{2} c^{2} d^{2} e\right )} x\right )} \log \left (x\right )}{b^{3} c^{3} x^{2} + b^{4} c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.59428, size = 250, normalized size = 2.87 \begin{align*} \frac{- b c^{2} d^{3} + x \left (b^{3} e^{3} - 3 b^{2} c d e^{2} + 3 b c^{2} d^{2} e - 2 c^{3} d^{3}\right )}{b^{3} c^{2} x + b^{2} c^{3} x^{2}} + \frac{d^{2} \left (3 b e - 2 c d\right ) \log{\left (x + \frac{- 3 b^{2} c d^{2} e + 2 b c^{2} d^{3} + b c d^{2} \left (3 b e - 2 c d\right )}{b^{3} e^{3} - 6 b c^{2} d^{2} e + 4 c^{3} d^{3}} \right )}}{b^{3}} + \frac{\left (b e - c d\right )^{2} \left (b e + 2 c d\right ) \log{\left (x + \frac{- 3 b^{2} c d^{2} e + 2 b c^{2} d^{3} + \frac{b \left (b e - c d\right )^{2} \left (b e + 2 c d\right )}{c}}{b^{3} e^{3} - 6 b c^{2} d^{2} e + 4 c^{3} d^{3}} \right )}}{b^{3} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-b*c**2*d**3 + x*(b**3*e**3 - 3*b**2*c*d*e**2 + 3*b*c**2*d**2*e - 2*c**3*d**3))/(b**3*c**2*x + b**2*c**3*x**2
) + d**2*(3*b*e - 2*c*d)*log(x + (-3*b**2*c*d**2*e + 2*b*c**2*d**3 + b*c*d**2*(3*b*e - 2*c*d))/(b**3*e**3 - 6*
b*c**2*d**2*e + 4*c**3*d**3))/b**3 + (b*e - c*d)**2*(b*e + 2*c*d)*log(x + (-3*b**2*c*d**2*e + 2*b*c**2*d**3 +
b*(b*e - c*d)**2*(b*e + 2*c*d)/c)/(b**3*e**3 - 6*b*c**2*d**2*e + 4*c**3*d**3))/(b**3*c**2)

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Giac [A]  time = 1.37846, size = 174, normalized size = 2. \begin{align*} -\frac{{\left (2 \, c d^{3} - 3 \, b d^{2} e\right )} \log \left ({\left | x \right |}\right )}{b^{3}} + \frac{{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + b^{3} e^{3}\right )} \log \left ({\left | c x + b \right |}\right )}{b^{3} c^{2}} - \frac{b c^{2} d^{3} +{\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} x}{{\left (c x + b\right )} b^{2} c^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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