3.1512 \(\int \frac{(b+2 c x) (a+b x+c x^2)^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=217 \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)^2}+\frac{4 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}-\frac{c^2 x (8 c d-5 b e)}{e^5}+\frac{c^3 x^2}{e^4} \]

[Out]

-((c^2*(8*c*d - 5*b*e)*x)/e^5) + (c^3*x^2)/e^4 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(3*e^6*(d + e*x)^3)
- ((c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^6*(d + e*x)^2) + ((2*c*d - b*e)*(10*
c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(e^6*(d + e*x)) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*L
og[d + e*x])/e^6

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Rubi [A]  time = 0.225815, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {771} \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)^2}+\frac{4 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}-\frac{c^2 x (8 c d-5 b e)}{e^5}+\frac{c^3 x^2}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((c^2*(8*c*d - 5*b*e)*x)/e^5) + (c^3*x^2)/e^4 + ((2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)^2)/(3*e^6*(d + e*x)^3)
- ((c*d^2 - b*d*e + a*e^2)*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))/(e^6*(d + e*x)^2) + ((2*c*d - b*e)*(10*
c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)))/(e^6*(d + e*x)) + (4*c*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e))*L
og[d + e*x])/e^6

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 )^2}{(d+e x)^4} \, dx &=\int \left (-\frac{c^2 (8 c d-5 b e)}{e^5}+\frac{2 c^3 x}{e^4}+\frac{(-2 c d+b e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^4}+\frac{2 \left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2-5 b c d e+b^2 e^2+a c e^2\right )}{e^5 (d+e x)^3}+\frac{(2 c d-b e) \left (-10 c^2 d^2-b^2 e^2+2 c e (5 b d-3 a e)\right )}{e^5 (d+e x)^2}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{c^2 (8 c d-5 b e) x}{e^5}+\frac{c^3 x^2}{e^4}+\frac{(2 c d-b e) \left (c d^2-b d e+a e^2\right )^2}{3 e^6 (d+e x)^3}-\frac{\left (c d^2-b d e+a e^2\right ) \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^6 (d+e x)^2}+\frac{(2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )}{e^6 (d+e x)}+\frac{4 c \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right ) \log (d+e x)}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.139732, size = 300, normalized size = 1.38 \[ \frac{-c e^2 \left (a^2 e^2 (d+3 e x)+6 a b e \left (d^2+3 d e x+3 e^2 x^2\right )-2 b^2 d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )-b e^3 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+12 c (d+e x)^3 \log (d+e x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+c^2 e \left (2 a d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-5 b \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )\right )+c^3 \left (-9 d^3 e^2 x^2-63 d^2 e^3 x^3+81 d^4 e x+47 d^5-15 d e^4 x^4+3 e^5 x^5\right )}{3 e^6 (d+e x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c^3*(47*d^5 + 81*d^4*e*x - 9*d^3*e^2*x^2 - 63*d^2*e^3*x^3 - 15*d*e^4*x^4 + 3*e^5*x^5) - b*e^3*(a^2*e^2 + a*b*
e*(d + 3*e*x) + b^2*(d^2 + 3*d*e*x + 3*e^2*x^2)) - c*e^2*(a^2*e^2*(d + 3*e*x) + 6*a*b*e*(d^2 + 3*d*e*x + 3*e^2
*x^2) - 2*b^2*d*(11*d^2 + 27*d*e*x + 18*e^2*x^2)) + c^2*e*(2*a*d*e*(11*d^2 + 27*d*e*x + 18*e^2*x^2) - 5*b*(13*
d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3*e^4*x^4)) + 12*c*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))
*(d + e*x)^3*Log[d + e*x])/(3*e^6*(d + e*x)^3)

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Maple [B]  time = 0.01, size = 495, normalized size = 2.3 \begin{align*} -6\,{\frac{{b}^{2}{d}^{2}c}{{e}^{4} \left ( ex+d \right ) ^{2}}}-6\,{\frac{a{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}-20\,{\frac{{c}^{2}\ln \left ( ex+d \right ) bd}{{e}^{5}}}-6\,{\frac{abc}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{a{c}^{2}d}{{e}^{4} \left ( ex+d \right ) }}+12\,{\frac{{b}^{2}cd}{{e}^{4} \left ( ex+d \right ) }}+10\,{\frac{b{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{2\,{b}^{2}da}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{b}^{2}{d}^{3}c}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-30\,{\frac{b{d}^{2}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{2\,d{a}^{2}c}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{5\,b{d}^{4}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{d}^{3}a{c}^{2}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{{c}^{3}{x}^{2}}{{e}^{4}}}+6\,{\frac{abcd}{{e}^{3} \left ( ex+d \right ) ^{2}}}-2\,{\frac{{d}^{2}abc}{{e}^{3} \left ( ex+d \right ) ^{3}}}+20\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{6}}}-{\frac{c{a}^{2}}{{e}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{{b}^{2}a}{{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{{b}^{3}d}{{e}^{3} \left ( ex+d \right ) ^{2}}}-5\,{\frac{{c}^{3}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+5\,{\frac{b{c}^{2}x}{{e}^{4}}}-8\,{\frac{{c}^{3}dx}{{e}^{5}}}+20\,{\frac{{c}^{3}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}-{\frac{{a}^{2}b}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{{d}^{2}{b}^{3}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{c}^{3}{d}^{5}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+4\,{\frac{{c}^{2}\ln \left ( ex+d \right ) a}{{e}^{4}}}+4\,{\frac{c\ln \left ( ex+d \right ){b}^{2}}{{e}^{4}}}-{\frac{{b}^{3}}{{e}^{3} \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)^2/(e*x+d)^4,x)

[Out]

-6/e^4/(e*x+d)^2*b^2*c*d^2-6/e^4/(e*x+d)^2*a*c^2*d^2-20*c^2/e^5*ln(e*x+d)*b*d-6/e^3/(e*x+d)*c*a*b+12/e^4/(e*x+
d)*a*c^2*d+12/e^4/(e*x+d)*b^2*c*d+10/e^5/(e*x+d)^2*b*c^2*d^3+2/3/e^2/(e*x+d)^3*d*a*b^2+4/3/e^4/(e*x+d)^3*d^3*b
^2*c-30/e^5/(e*x+d)*b*c^2*d^2+2/3/e^2/(e*x+d)^3*d*a^2*c-5/3/e^5/(e*x+d)^3*b*c^2*d^4+4/3/e^4/(e*x+d)^3*d^3*a*c^
2+c^3*x^2/e^4+6/e^3/(e*x+d)^2*a*b*c*d-2/e^3/(e*x+d)^3*d^2*a*b*c+20*c^3/e^6*ln(e*x+d)*d^2-1/e^2/(e*x+d)^2*c*a^2
-1/e^2/(e*x+d)^2*a*b^2+1/e^3/(e*x+d)^2*b^3*d-5/e^6/(e*x+d)^2*c^3*d^4+5*c^2/e^4*b*x-8*c^3/e^5*d*x+20/e^6/(e*x+d
)*c^3*d^3-1/3/e/(e*x+d)^3*a^2*b-1/3/e^3/(e*x+d)^3*d^2*b^3+2/3/e^6/(e*x+d)^3*c^3*d^5+4*c^2/e^4*ln(e*x+d)*a+4*c/
e^4*ln(e*x+d)*b^2-1/e^3/(e*x+d)*b^3

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Maxima [A]  time = 1.07844, size = 441, normalized size = 2.03 \begin{align*} \frac{47 \, c^{3} d^{5} - 65 \, b c^{2} d^{4} e - a^{2} b e^{5} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} -{\left (a b^{2} + a^{2} c\right )} d e^{4} + 3 \,{\left (20 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 3 \,{\left (35 \, c^{3} d^{4} e - 50 \, b c^{2} d^{3} e^{2} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} -{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x}{3 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{c^{3} e x^{2} -{\left (8 \, c^{3} d - 5 \, b c^{2} e\right )} x}{e^{5}} + \frac{4 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(47*c^3*d^5 - 65*b*c^2*d^4*e - a^2*b*e^5 + 22*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 - (a*b^2 +
a^2*c)*d*e^4 + 3*(20*c^3*d^3*e^2 - 30*b*c^2*d^2*e^3 + 12*(b^2*c + a*c^2)*d*e^4 - (b^3 + 6*a*b*c)*e^5)*x^2 + 3
*(35*c^3*d^4*e - 50*b*c^2*d^3*e^2 + 18*(b^2*c + a*c^2)*d^2*e^3 - (b^3 + 6*a*b*c)*d*e^4 - (a*b^2 + a^2*c)*e^5)*
x)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) + (c^3*e*x^2 - (8*c^3*d - 5*b*c^2*e)*x)/e^5 + 4*(5*c^3*d^2
- 5*b*c^2*d*e + (b^2*c + a*c^2)*e^2)*log(e*x + d)/e^6

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Fricas [B]  time = 1.43047, size = 988, normalized size = 4.55 \begin{align*} \frac{3 \, c^{3} e^{5} x^{5} + 47 \, c^{3} d^{5} - 65 \, b c^{2} d^{4} e - a^{2} b e^{5} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} -{\left (a b^{2} + a^{2} c\right )} d e^{4} - 15 \,{\left (c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{4} - 9 \,{\left (7 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, c^{3} d^{3} e^{2} + 15 \, b c^{2} d^{2} e^{3} - 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} +{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 3 \,{\left (27 \, c^{3} d^{4} e - 45 \, b c^{2} d^{3} e^{2} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} -{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e +{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} +{\left (5 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} +{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (5 \, c^{3} d^{3} e^{2} - 5 \, b c^{2} d^{2} e^{3} +{\left (b^{2} c + a c^{2}\right )} d e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{4} e - 5 \, b c^{2} d^{3} e^{2} +{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{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)^2/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^3*e^5*x^5 + 47*c^3*d^5 - 65*b*c^2*d^4*e - a^2*b*e^5 + 22*(b^2*c + a*c^2)*d^3*e^2 - (b^3 + 6*a*b*c)*d^
2*e^3 - (a*b^2 + a^2*c)*d*e^4 - 15*(c^3*d*e^4 - b*c^2*e^5)*x^4 - 9*(7*c^3*d^2*e^3 - 5*b*c^2*d*e^4)*x^3 - 3*(3*
c^3*d^3*e^2 + 15*b*c^2*d^2*e^3 - 12*(b^2*c + a*c^2)*d*e^4 + (b^3 + 6*a*b*c)*e^5)*x^2 + 3*(27*c^3*d^4*e - 45*b*
c^2*d^3*e^2 + 18*(b^2*c + a*c^2)*d^2*e^3 - (b^3 + 6*a*b*c)*d*e^4 - (a*b^2 + a^2*c)*e^5)*x + 12*(5*c^3*d^5 - 5*
b*c^2*d^4*e + (b^2*c + a*c^2)*d^3*e^2 + (5*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + (b^2*c + a*c^2)*e^5)*x^3 + 3*(5*c^3*d
^3*e^2 - 5*b*c^2*d^2*e^3 + (b^2*c + a*c^2)*d*e^4)*x^2 + 3*(5*c^3*d^4*e - 5*b*c^2*d^3*e^2 + (b^2*c + a*c^2)*d^2
*e^3)*x)*log(e*x + d))/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)

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Sympy [A]  time = 23.787, size = 377, normalized size = 1.74 \begin{align*} \frac{c^{3} x^{2}}{e^{4}} + \frac{4 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{a^{2} b e^{5} + a^{2} c d e^{4} + a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 22 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 22 b^{2} c d^{3} e^{2} + 65 b c^{2} d^{4} e - 47 c^{3} d^{5} + x^{2} \left (18 a b c e^{5} - 36 a c^{2} d e^{4} + 3 b^{3} e^{5} - 36 b^{2} c d e^{4} + 90 b c^{2} d^{2} e^{3} - 60 c^{3} d^{3} e^{2}\right ) + x \left (3 a^{2} c e^{5} + 3 a b^{2} e^{5} + 18 a b c d e^{4} - 54 a c^{2} d^{2} e^{3} + 3 b^{3} d e^{4} - 54 b^{2} c d^{2} e^{3} + 150 b c^{2} d^{3} e^{2} - 105 c^{3} d^{4} e\right )}{3 d^{3} e^{6} + 9 d^{2} e^{7} x + 9 d e^{8} x^{2} + 3 e^{9} x^{3}} + \frac{x \left (5 b c^{2} e - 8 c^{3} d\right )}{e^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x+a)**2/(e*x+d)**4,x)

[Out]

c**3*x**2/e**4 + 4*c*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)*log(d + e*x)/e**6 - (a**2*b*e**5 + a**2*
c*d*e**4 + a*b**2*d*e**4 + 6*a*b*c*d**2*e**3 - 22*a*c**2*d**3*e**2 + b**3*d**2*e**3 - 22*b**2*c*d**3*e**2 + 65
*b*c**2*d**4*e - 47*c**3*d**5 + x**2*(18*a*b*c*e**5 - 36*a*c**2*d*e**4 + 3*b**3*e**5 - 36*b**2*c*d*e**4 + 90*b
*c**2*d**2*e**3 - 60*c**3*d**3*e**2) + x*(3*a**2*c*e**5 + 3*a*b**2*e**5 + 18*a*b*c*d*e**4 - 54*a*c**2*d**2*e**
3 + 3*b**3*d*e**4 - 54*b**2*c*d**2*e**3 + 150*b*c**2*d**3*e**2 - 105*c**3*d**4*e))/(3*d**3*e**6 + 9*d**2*e**7*
x + 9*d*e**8*x**2 + 3*e**9*x**3) + x*(5*b*c**2*e - 8*c**3*d)/e**5

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Giac [A]  time = 1.1441, size = 424, normalized size = 1.95 \begin{align*} 4 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} e^{\left (-6\right )} \log \left ({\left | x e + d \right |}\right ) +{\left (c^{3} x^{2} e^{4} - 8 \, c^{3} d x e^{3} + 5 \, b c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, c^{3} d^{5} - 65 \, b c^{2} d^{4} e + 22 \, b^{2} c d^{3} e^{2} + 22 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - a b^{2} d e^{4} - a^{2} c d e^{4} - a^{2} b e^{5} + 3 \,{\left (20 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} + 12 \, b^{2} c d e^{4} + 12 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{2} + 3 \,{\left (35 \, c^{3} d^{4} e - 50 \, b c^{2} d^{3} e^{2} + 18 \, b^{2} c d^{2} e^{3} + 18 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - a b^{2} e^{5} - a^{2} c e^{5}\right )} x\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*(5*c^3*d^2 - 5*b*c^2*d*e + b^2*c*e^2 + a*c^2*e^2)*e^(-6)*log(abs(x*e + d)) + (c^3*x^2*e^4 - 8*c^3*d*x*e^3 +
5*b*c^2*x*e^4)*e^(-8) + 1/3*(47*c^3*d^5 - 65*b*c^2*d^4*e + 22*b^2*c*d^3*e^2 + 22*a*c^2*d^3*e^2 - b^3*d^2*e^3 -
6*a*b*c*d^2*e^3 - a*b^2*d*e^4 - a^2*c*d*e^4 - a^2*b*e^5 + 3*(20*c^3*d^3*e^2 - 30*b*c^2*d^2*e^3 + 12*b^2*c*d*e
^4 + 12*a*c^2*d*e^4 - b^3*e^5 - 6*a*b*c*e^5)*x^2 + 3*(35*c^3*d^4*e - 50*b*c^2*d^3*e^2 + 18*b^2*c*d^2*e^3 + 18*
a*c^2*d^2*e^3 - b^3*d*e^4 - 6*a*b*c*d*e^4 - a*b^2*e^5 - a^2*c*e^5)*x)*e^(-6)/(x*e + d)^3