3.1500 \(\int \frac{(b+2 c x) (a+b x+c x^2)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=102 \[ \frac{\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{c x (4 c d-3 b e)}{e^3}+\frac{c^2 x^2}{e^2} \]

[Out]

-((c*(4*c*d - 3*b*e)*x)/e^3) + (c^2*x^2)/e^2 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(e^4*(d + e*x)) + ((6*c
^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*Log[d + e*x])/e^4

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Rubi [A]  time = 0.102549, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{\log (d+e x) \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{e^4}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )}{e^4 (d+e x)}-\frac{c x (4 c d-3 b e)}{e^3}+\frac{c^2 x^2}{e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c*(4*c*d - 3*b*e)*x)/e^3) + (c^2*x^2)/e^2 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2))/(e^4*(d + e*x)) + ((6*c
^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e))*Log[d + e*x])/e^4

Rule 771

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

Rubi steps

\begin{align*} \int \frac{(b+2 c x) \left (a+b x+c x^2\right )}{(d+e x)^2} \, dx &=\int \left (-\frac{c (4 c d-3 b e)}{e^3}+\frac{2 c^2 x}{e^2}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )}{e^3 (d+e x)^2}+\frac{6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)}{e^3 (d+e x)}\right ) \, dx\\ &=-\frac{c (4 c d-3 b e) x}{e^3}+\frac{c^2 x^2}{e^2}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )}{e^4 (d+e x)}+\frac{\left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right ) \log (d+e x)}{e^4}\\ \end{align*}

Mathematica [A]  time = 0.0698493, size = 97, normalized size = 0.95 \[ \frac{\log (d+e x) \left (2 c e (a e-3 b d)+b^2 e^2+6 c^2 d^2\right )+\frac{(2 c d-b e) \left (e (a e-b d)+c d^2\right )}{d+e x}-c e x (4 c d-3 b e)+c^2 e^2 x^2}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(c*e*(4*c*d - 3*b*e)*x) + c^2*e^2*x^2 + ((2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e)))/(d + e*x) + (6*c^2*d^2 +
b^2*e^2 + 2*c*e*(-3*b*d + a*e))*Log[d + e*x])/e^4

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Maple [A]  time = 0.007, size = 166, normalized size = 1.6 \begin{align*}{\frac{{c}^{2}{x}^{2}}{{e}^{2}}}+3\,{\frac{bcx}{{e}^{2}}}-4\,{\frac{{c}^{2}dx}{{e}^{3}}}+2\,{\frac{\ln \left ( ex+d \right ) ac}{{e}^{2}}}+{\frac{\ln \left ( ex+d \right ){b}^{2}}{{e}^{2}}}-6\,{\frac{\ln \left ( ex+d \right ) bcd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{2}}{{e}^{4}}}-{\frac{ab}{e \left ( ex+d \right ) }}+2\,{\frac{acd}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{b}^{2}d}{{e}^{2} \left ( ex+d \right ) }}-3\,{\frac{b{d}^{2}c}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{{c}^{2}{d}^{3}}{{e}^{4} \left ( ex+d \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^2*x^2/e^2+3*c/e^2*b*x-4*c^2/e^3*d*x+2/e^2*ln(e*x+d)*a*c+1/e^2*ln(e*x+d)*b^2-6/e^3*ln(e*x+d)*b*c*d+6/e^4*ln(e
*x+d)*c^2*d^2-1/e/(e*x+d)*a*b+2/e^2/(e*x+d)*d*a*c+1/e^2/(e*x+d)*d*b^2-3/e^3/(e*x+d)*b*c*d^2+2/e^4/(e*x+d)*c^2*
d^3

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Maxima [A]  time = 1.04406, size = 158, normalized size = 1.55 \begin{align*} \frac{2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}}{e^{5} x + d e^{4}} + \frac{c^{2} e x^{2} -{\left (4 \, c^{2} d - 3 \, b c e\right )} x}{e^{3}} + \frac{{\left (6 \, c^{2} d^{2} - 6 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)/(e^5*x + d*e^4) + (c^2*e*x^2 - (4*c^2*d - 3*b*c*e)*x
)/e^3 + (6*c^2*d^2 - 6*b*c*d*e + (b^2 + 2*a*c)*e^2)*log(e*x + d)/e^4

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Fricas [A]  time = 1.69476, size = 359, normalized size = 3.52 \begin{align*} \frac{c^{2} e^{3} x^{3} + 2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2} - 3 \,{\left (c^{2} d e^{2} - b c e^{3}\right )} x^{2} -{\left (4 \, c^{2} d^{2} e - 3 \, b c d e^{2}\right )} x +{\left (6 \, c^{2} d^{3} - 6 \, b c d^{2} e +{\left (b^{2} + 2 \, a c\right )} d e^{2} +{\left (6 \, c^{2} d^{2} e - 6 \, b c d e^{2} +{\left (b^{2} + 2 \, a c\right )} e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{5} x + d e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*e^3*x^3 + 2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2 - 3*(c^2*d*e^2 - b*c*e^3)*x^2 - (4*c^2*
d^2*e - 3*b*c*d*e^2)*x + (6*c^2*d^3 - 6*b*c*d^2*e + (b^2 + 2*a*c)*d*e^2 + (6*c^2*d^2*e - 6*b*c*d*e^2 + (b^2 +
2*a*c)*e^3)*x)*log(e*x + d))/(e^5*x + d*e^4)

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Sympy [A]  time = 0.912448, size = 124, normalized size = 1.22 \begin{align*} \frac{c^{2} x^{2}}{e^{2}} - \frac{a b e^{3} - 2 a c d e^{2} - b^{2} d e^{2} + 3 b c d^{2} e - 2 c^{2} d^{3}}{d e^{4} + e^{5} x} + \frac{x \left (3 b c e - 4 c^{2} d\right )}{e^{3}} + \frac{\left (2 a c e^{2} + b^{2} e^{2} - 6 b c d e + 6 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*x**2/e**2 - (a*b*e**3 - 2*a*c*d*e**2 - b**2*d*e**2 + 3*b*c*d**2*e - 2*c**2*d**3)/(d*e**4 + e**5*x) + x*(3
*b*c*e - 4*c**2*d)/e**3 + (2*a*c*e**2 + b**2*e**2 - 6*b*c*d*e + 6*c**2*d**2)*log(d + e*x)/e**4

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Giac [A]  time = 1.15624, size = 239, normalized size = 2.34 \begin{align*}{\left (c^{2} - \frac{3 \,{\left (2 \, c^{2} d e - b c e^{2}\right )} e^{\left (-1\right )}}{x e + d}\right )}{\left (x e + d\right )}^{2} e^{\left (-4\right )} -{\left (6 \, c^{2} d^{2} - 6 \, b c d e + b^{2} e^{2} + 2 \, a c e^{2}\right )} e^{\left (-4\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{2 \, c^{2} d^{3} e^{2}}{x e + d} - \frac{3 \, b c d^{2} e^{3}}{x e + d} + \frac{b^{2} d e^{4}}{x e + d} + \frac{2 \, a c d e^{4}}{x e + d} - \frac{a b e^{5}}{x e + d}\right )} e^{\left (-6\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2 - 3*(2*c^2*d*e - b*c*e^2)*e^(-1)/(x*e + d))*(x*e + d)^2*e^(-4) - (6*c^2*d^2 - 6*b*c*d*e + b^2*e^2 + 2*a*c
*e^2)*e^(-4)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (2*c^2*d^3*e^2/(x*e + d) - 3*b*c*d^2*e^3/(x*e + d) + b^2*d
*e^4/(x*e + d) + 2*a*c*d*e^4/(x*e + d) - a*b*e^5/(x*e + d))*e^(-6)