3.1513 \(\int \frac{(b+2 c x) (a+b x+c x^2)^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=227 \[ -\frac{4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac{5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac{2 c^3 x}{e^5} \]

[Out]

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

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Rubi [A]  time = 0.208898, antiderivative size = 227, 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} \[ -\frac{4 c \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)}+\frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}-\frac{2 \left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{3 e^6 (d+e x)^3}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{4 e^6 (d+e x)^4}-\frac{5 c^2 (2 c d-b e) \log (d+e x)}{e^6}+\frac{2 c^3 x}{e^5} \]

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

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

Mathematica [A]  time = 0.177886, size = 292, normalized size = 1.29 \[ -\frac{2 c e^2 \left (a^2 e^2 (d+4 e x)+3 a b e \left (d^2+4 d e x+6 e^2 x^2\right )+6 b^2 \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )\right )+b e^3 \left (3 a^2 e^2+2 a b e (d+4 e x)+b^2 \left (d^2+4 d e x+6 e^2 x^2\right )\right )+c^2 e \left (12 a e \left (4 d^2 e x+d^3+6 d e^2 x^2+4 e^3 x^3\right )-5 b d \left (88 d^2 e x+25 d^3+108 d e^2 x^2+48 e^3 x^3\right )\right )+60 c^2 (d+e x)^4 (2 c d-b e) \log (d+e x)+2 c^3 \left (252 d^3 e^2 x^2+48 d^2 e^3 x^3+248 d^4 e x+77 d^5-48 d e^4 x^4-12 e^5 x^5\right )}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(a + b*x + c*x^2)^2)/(d + e*x)^5,x]

[Out]

-(2*c^3*(77*d^5 + 248*d^4*e*x + 252*d^3*e^2*x^2 + 48*d^2*e^3*x^3 - 48*d*e^4*x^4 - 12*e^5*x^5) + b*e^3*(3*a^2*e
^2 + 2*a*b*e*(d + 4*e*x) + b^2*(d^2 + 4*d*e*x + 6*e^2*x^2)) + 2*c*e^2*(a^2*e^2*(d + 4*e*x) + 3*a*b*e*(d^2 + 4*
d*e*x + 6*e^2*x^2) + 6*b^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3)) + c^2*e*(12*a*e*(d^3 + 4*d^2*e*x + 6*d
*e^2*x^2 + 4*e^3*x^3) - 5*b*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + 60*c^2*(2*c*d - b*e)*(d +
e*x)^4*Log[d + e*x])/(12*e^6*(d + e*x)^4)

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Maple [B]  time = 0.009, size = 507, normalized size = 2.2 \begin{align*} -15\,{\frac{b{d}^{2}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-{\frac{5\,b{d}^{4}{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{d{a}^{2}c}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+6\,{\frac{{b}^{2}cd}{{e}^{4} \left ( ex+d \right ) ^{2}}}+20\,{\frac{bd{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{{b}^{2}da}{2\,{e}^{2} \left ( ex+d \right ) ^{4}}}+{\frac{{d}^{3}a{c}^{2}}{{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{{b}^{2}{d}^{3}c}{{e}^{4} \left ( ex+d \right ) ^{4}}}-4\,{\frac{a{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{b}^{2}{d}^{2}c}{{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{20\,b{d}^{3}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+2\,{\frac{{c}^{3}x}{{e}^{5}}}-{\frac{3\,{d}^{2}abc}{2\,{e}^{3} \left ( ex+d \right ) ^{4}}}+4\,{\frac{abcd}{{e}^{3} \left ( ex+d \right ) ^{3}}}-20\,{\frac{{c}^{3}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}+5\,{\frac{{c}^{2}\ln \left ( ex+d \right ) b}{{e}^{5}}}-10\,{\frac{{c}^{3}\ln \left ( ex+d \right ) d}{{e}^{6}}}-{\frac{{a}^{2}b}{4\,e \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{2}{b}^{3}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{c}^{3}{d}^{5}}{2\,{e}^{6} \left ( ex+d \right ) ^{4}}}+10\,{\frac{{c}^{3}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{2\,{b}^{3}d}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{10\,{c}^{3}{d}^{4}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{2\,c{a}^{2}}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{2\,{b}^{2}a}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-4\,{\frac{a{c}^{2}}{{e}^{4} \left ( ex+d \right ) }}-4\,{\frac{{b}^{2}c}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{b}^{3}}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}+6\,{\frac{a{c}^{2}d}{{e}^{4} \left ( ex+d \right ) ^{2}}}-3\,{\frac{abc}{{e}^{3} \left ( ex+d \right ) ^{2}}} \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)^5,x)

[Out]

-15/e^5/(e*x+d)^2*b*c^2*d^2-5/4/e^5/(e*x+d)^4*d^4*b*c^2+1/2/e^2/(e*x+d)^4*d*a^2*c+6/e^4/(e*x+d)^2*b^2*c*d+20*c
^2/e^5/(e*x+d)*b*d+1/2/e^2/(e*x+d)^4*d*a*b^2+1/e^4/(e*x+d)^4*d^3*a*c^2+1/e^4/(e*x+d)^4*d^3*b^2*c-4/e^4/(e*x+d)
^3*a*c^2*d^2-4/e^4/(e*x+d)^3*b^2*c*d^2+20/3/e^5/(e*x+d)^3*b*c^2*d^3+2*c^3*x/e^5-3/2/e^3/(e*x+d)^4*d^2*a*b*c+4/
e^3/(e*x+d)^3*a*b*c*d-20*c^3/e^6/(e*x+d)*d^2+5*c^2/e^5*ln(e*x+d)*b-10*c^3/e^6*ln(e*x+d)*d-1/4/e/(e*x+d)^4*a^2*
b-1/4/e^3/(e*x+d)^4*d^2*b^3+1/2/e^6/(e*x+d)^4*c^3*d^5+10/e^6/(e*x+d)^2*c^3*d^3+2/3/e^3/(e*x+d)^3*b^3*d-10/3/e^
6/(e*x+d)^3*c^3*d^4-2/3/e^2/(e*x+d)^3*c*a^2-2/3/e^2/(e*x+d)^3*a*b^2-4*c^2/e^4/(e*x+d)*a-4*c/e^4/(e*x+d)*b^2-1/
2/e^3/(e*x+d)^2*b^3+6/e^4/(e*x+d)^2*a*c^2*d-3/e^3/(e*x+d)^2*c*a*b

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Maxima [A]  time = 1.05477, size = 456, normalized size = 2.01 \begin{align*} -\frac{154 \, c^{3} d^{5} - 125 \, b c^{2} d^{4} e + 3 \, a^{2} b e^{5} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} +{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} + 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} + 48 \,{\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} + 6 \,{\left (100 \, c^{3} d^{3} e^{2} - 90 \, 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} + 4 \,{\left (130 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac{2 \, c^{3} x}{e^{5}} - \frac{5 \,{\left (2 \, c^{3} d - b c^{2} e\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)^5,x, algorithm="maxima")

[Out]

-1/12*(154*c^3*d^5 - 125*b*c^2*d^4*e + 3*a^2*b*e^5 + 12*(b^2*c + a*c^2)*d^3*e^2 + (b^3 + 6*a*b*c)*d^2*e^3 + 2*
(a*b^2 + a^2*c)*d*e^4 + 48*(5*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + (b^2*c + a*c^2)*e^5)*x^3 + 6*(100*c^3*d^3*e^2 - 90
*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 + 4*(130*c^3*d^4*e - 110*b*c^2*d^3*e^2 +
12*(b^2*c + a*c^2)*d^2*e^3 + (b^3 + 6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*x)/(e^10*x^4 + 4*d*e^9*x^3 + 6*d^2
*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6) + 2*c^3*x/e^5 - 5*(2*c^3*d - b*c^2*e)*log(e*x + d)/e^6

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Fricas [B]  time = 1.43753, size = 944, normalized size = 4.16 \begin{align*} \frac{24 \, c^{3} e^{5} x^{5} + 96 \, c^{3} d e^{4} x^{4} - 154 \, c^{3} d^{5} + 125 \, b c^{2} d^{4} e - 3 \, a^{2} b e^{5} - 12 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} - 2 \,{\left (a b^{2} + a^{2} c\right )} d e^{4} - 48 \,{\left (2 \, 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} - 6 \,{\left (84 \, c^{3} d^{3} e^{2} - 90 \, 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} - 4 \,{\left (124 \, c^{3} d^{4} e - 110 \, b c^{2} d^{3} e^{2} + 12 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} +{\left (b^{3} + 6 \, a b c\right )} d e^{4} + 2 \,{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x - 60 \,{\left (2 \, c^{3} d^{5} - b c^{2} d^{4} e +{\left (2 \, c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{4} + 4 \,{\left (2 \, c^{3} d^{2} e^{3} - b c^{2} d e^{4}\right )} x^{3} + 6 \,{\left (2 \, c^{3} d^{3} e^{2} - b c^{2} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (2 \, c^{3} d^{4} e - b c^{2} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} 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)^5,x, algorithm="fricas")

[Out]

1/12*(24*c^3*e^5*x^5 + 96*c^3*d*e^4*x^4 - 154*c^3*d^5 + 125*b*c^2*d^4*e - 3*a^2*b*e^5 - 12*(b^2*c + a*c^2)*d^3
*e^2 - (b^3 + 6*a*b*c)*d^2*e^3 - 2*(a*b^2 + a^2*c)*d*e^4 - 48*(2*c^3*d^2*e^3 - 5*b*c^2*d*e^4 + (b^2*c + a*c^2)
*e^5)*x^3 - 6*(84*c^3*d^3*e^2 - 90*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 - 4*(12
4*c^3*d^4*e - 110*b*c^2*d^3*e^2 + 12*(b^2*c + a*c^2)*d^2*e^3 + (b^3 + 6*a*b*c)*d*e^4 + 2*(a*b^2 + a^2*c)*e^5)*
x - 60*(2*c^3*d^5 - b*c^2*d^4*e + (2*c^3*d*e^4 - b*c^2*e^5)*x^4 + 4*(2*c^3*d^2*e^3 - b*c^2*d*e^4)*x^3 + 6*(2*c
^3*d^3*e^2 - b*c^2*d^2*e^3)*x^2 + 4*(2*c^3*d^4*e - b*c^2*d^3*e^2)*x)*log(e*x + d))/(e^10*x^4 + 4*d*e^9*x^3 + 6
*d^2*e^8*x^2 + 4*d^3*e^7*x + d^4*e^6)

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Sympy [A]  time = 73.9333, size = 401, normalized size = 1.77 \begin{align*} \frac{2 c^{3} x}{e^{5}} + \frac{5 c^{2} \left (b e - 2 c d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{3 a^{2} b e^{5} + 2 a^{2} c d e^{4} + 2 a b^{2} d e^{4} + 6 a b c d^{2} e^{3} + 12 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} + 12 b^{2} c d^{3} e^{2} - 125 b c^{2} d^{4} e + 154 c^{3} d^{5} + x^{3} \left (48 a c^{2} e^{5} + 48 b^{2} c e^{5} - 240 b c^{2} d e^{4} + 240 c^{3} d^{2} e^{3}\right ) + x^{2} \left (36 a b c e^{5} + 72 a c^{2} d e^{4} + 6 b^{3} e^{5} + 72 b^{2} c d e^{4} - 540 b c^{2} d^{2} e^{3} + 600 c^{3} d^{3} e^{2}\right ) + x \left (8 a^{2} c e^{5} + 8 a b^{2} e^{5} + 24 a b c d e^{4} + 48 a c^{2} d^{2} e^{3} + 4 b^{3} d e^{4} + 48 b^{2} c d^{2} e^{3} - 440 b c^{2} d^{3} e^{2} + 520 c^{3} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{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)**2/(e*x+d)**5,x)

[Out]

2*c**3*x/e**5 + 5*c**2*(b*e - 2*c*d)*log(d + e*x)/e**6 - (3*a**2*b*e**5 + 2*a**2*c*d*e**4 + 2*a*b**2*d*e**4 +
6*a*b*c*d**2*e**3 + 12*a*c**2*d**3*e**2 + b**3*d**2*e**3 + 12*b**2*c*d**3*e**2 - 125*b*c**2*d**4*e + 154*c**3*
d**5 + x**3*(48*a*c**2*e**5 + 48*b**2*c*e**5 - 240*b*c**2*d*e**4 + 240*c**3*d**2*e**3) + x**2*(36*a*b*c*e**5 +
72*a*c**2*d*e**4 + 6*b**3*e**5 + 72*b**2*c*d*e**4 - 540*b*c**2*d**2*e**3 + 600*c**3*d**3*e**2) + x*(8*a**2*c*
e**5 + 8*a*b**2*e**5 + 24*a*b*c*d*e**4 + 48*a*c**2*d**2*e**3 + 4*b**3*d*e**4 + 48*b**2*c*d**2*e**3 - 440*b*c**
2*d**3*e**2 + 520*c**3*d**4*e))/(12*d**4*e**6 + 48*d**3*e**7*x + 72*d**2*e**8*x**2 + 48*d*e**9*x**3 + 12*e**10
*x**4)

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Giac [B]  time = 1.22479, size = 709, normalized size = 3.12 \begin{align*} 2 \,{\left (x e + d\right )} c^{3} e^{\left (-6\right )} + 5 \,{\left (2 \, c^{3} d - b c^{2} e\right )} e^{\left (-6\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{1}{12} \,{\left (\frac{240 \, c^{3} d^{2} e^{22}}{x e + d} - \frac{120 \, c^{3} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac{40 \, c^{3} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac{6 \, c^{3} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac{240 \, b c^{2} d e^{23}}{x e + d} + \frac{180 \, b c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac{80 \, b c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac{15 \, b c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{48 \, b^{2} c e^{24}}{x e + d} + \frac{48 \, a c^{2} e^{24}}{x e + d} - \frac{72 \, b^{2} c d e^{24}}{{\left (x e + d\right )}^{2}} - \frac{72 \, a c^{2} d e^{24}}{{\left (x e + d\right )}^{2}} + \frac{48 \, b^{2} c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac{48 \, a c^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac{12 \, b^{2} c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac{12 \, a c^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac{6 \, b^{3} e^{25}}{{\left (x e + d\right )}^{2}} + \frac{36 \, a b c e^{25}}{{\left (x e + d\right )}^{2}} - \frac{8 \, b^{3} d e^{25}}{{\left (x e + d\right )}^{3}} - \frac{48 \, a b c d e^{25}}{{\left (x e + d\right )}^{3}} + \frac{3 \, b^{3} d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac{18 \, a b c d^{2} e^{25}}{{\left (x e + d\right )}^{4}} + \frac{8 \, a b^{2} e^{26}}{{\left (x e + d\right )}^{3}} + \frac{8 \, a^{2} c e^{26}}{{\left (x e + d\right )}^{3}} - \frac{6 \, a b^{2} d e^{26}}{{\left (x e + d\right )}^{4}} - \frac{6 \, a^{2} c d e^{26}}{{\left (x e + d\right )}^{4}} + \frac{3 \, a^{2} b e^{27}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\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)^5,x, algorithm="giac")

[Out]

2*(x*e + d)*c^3*e^(-6) + 5*(2*c^3*d - b*c^2*e)*e^(-6)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(240*c^3*d^2
*e^22/(x*e + d) - 120*c^3*d^3*e^22/(x*e + d)^2 + 40*c^3*d^4*e^22/(x*e + d)^3 - 6*c^3*d^5*e^22/(x*e + d)^4 - 24
0*b*c^2*d*e^23/(x*e + d) + 180*b*c^2*d^2*e^23/(x*e + d)^2 - 80*b*c^2*d^3*e^23/(x*e + d)^3 + 15*b*c^2*d^4*e^23/
(x*e + d)^4 + 48*b^2*c*e^24/(x*e + d) + 48*a*c^2*e^24/(x*e + d) - 72*b^2*c*d*e^24/(x*e + d)^2 - 72*a*c^2*d*e^2
4/(x*e + d)^2 + 48*b^2*c*d^2*e^24/(x*e + d)^3 + 48*a*c^2*d^2*e^24/(x*e + d)^3 - 12*b^2*c*d^3*e^24/(x*e + d)^4
- 12*a*c^2*d^3*e^24/(x*e + d)^4 + 6*b^3*e^25/(x*e + d)^2 + 36*a*b*c*e^25/(x*e + d)^2 - 8*b^3*d*e^25/(x*e + d)^
3 - 48*a*b*c*d*e^25/(x*e + d)^3 + 3*b^3*d^2*e^25/(x*e + d)^4 + 18*a*b*c*d^2*e^25/(x*e + d)^4 + 8*a*b^2*e^26/(x
*e + d)^3 + 8*a^2*c*e^26/(x*e + d)^3 - 6*a*b^2*d*e^26/(x*e + d)^4 - 6*a^2*c*d*e^26/(x*e + d)^4 + 3*a^2*b*e^27/
(x*e + d)^4)*e^(-28)