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

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))*Log[d + e*x])/e^6

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Rubi [A]  time = 0.580486, 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 \[ \frac{(2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6 (d+e x)^2}+\frac{4 c \log (d+e x) \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^6}+\frac{(2 c d-b e) \left (a e^2-b d e+c d^2\right )^2}{3 e^6 (d+e x)^3}-\frac{c^2 x (8 c d-5 b e)}{e^5}+\frac{c^3 x^2}{e^4} \]

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))*Log[d + e*x])/e^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c^{3} \int x\, dx}{e^{4}} + \frac{4 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} + \frac{\left (5 b e - 8 c d\right ) \int c^{2}\, dx}{e^{5}} - \frac{\left (b e - 2 c d\right ) \left (6 a c e^{2} + b^{2} e^{2} - 10 b c d e + 10 c^{2} d^{2}\right )}{e^{6} \left (d + e x\right )} - \frac{\left (a e^{2} - b d e + c d^{2}\right ) \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right )}{e^{6} \left (d + e x\right )^{2}} - \frac{\left (b e - 2 c d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}}{3 e^{6} \left (d + e x\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c**3*Integral(x, x)/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 + (5*b*e - 8*c*d)*Integral(c**2, x)/e**5 - (b*e - 2*c*d)*(6*
a*c*e**2 + b**2*e**2 - 10*b*c*d*e + 10*c**2*d**2)/(e**6*(d + e*x)) - (a*e**2 - b
*d*e + c*d**2)*(a*c*e**2 + b**2*e**2 - 5*b*c*d*e + 5*c**2*d**2)/(e**6*(d + e*x)*
*2) - (b*e - 2*c*d)*(a*e**2 - b*d*e + c*d**2)**2/(3*e**6*(d + e*x)**3)

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Mathematica [A]  time = 0.235926, size = 300, normalized size = 1.38 \[ \frac{-c e^2 \left (a^2 e^2 (d+3 e x)+6 a b e \left (d^2+3 d e x+3 e^2 x^2\right )-2 b^2 d \left (11 d^2+27 d e x+18 e^2 x^2\right )\right )-b e^3 \left (a^2 e^2+a b e (d+3 e x)+b^2 \left (d^2+3 d e x+3 e^2 x^2\right )\right )+12 c (d+e x)^3 \log (d+e x) \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+c^2 e \left (2 a d e \left (11 d^2+27 d e x+18 e^2 x^2\right )-5 b \left (13 d^4+27 d^3 e x+9 d^2 e^2 x^2-9 d e^3 x^3-3 e^4 x^4\right )\right )+c^3 \left (47 d^5+81 d^4 e x-9 d^3 e^2 x^2-63 d^2 e^3 x^3-15 d e^4 x^4+3 e^5 x^5\right )}{3 e^6 (d+e x)^3} \]

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 = 0.016, size = 495, normalized size = 2.3 \[{\frac{2\,a{b}^{2}d}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{c}^{3}{d}^{4}}{{e}^{6} \left ( ex+d \right ) ^{2}}}-8\,{\frac{{c}^{3}dx}{{e}^{5}}}-{\frac{{a}^{2}b}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3}{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{2\,{c}^{3}{d}^{5}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}c}{{e}^{2} \left ( ex+d \right ) ^{2}}}-30\,{\frac{b{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}-6\,{\frac{abc}{{e}^{3} \left ( ex+d \right ) }}+12\,{\frac{a{c}^{2}d}{{e}^{4} \left ( ex+d \right ) }}+12\,{\frac{{b}^{2}cd}{{e}^{4} \left ( ex+d \right ) }}-{\frac{a{b}^{2}}{{e}^{2} \left ( ex+d \right ) ^{2}}}+{\frac{d{b}^{3}}{{e}^{3} \left ( ex+d \right ) ^{2}}}+20\,{\frac{{c}^{3}{d}^{3}}{{e}^{6} \left ( ex+d \right ) }}-20\,{\frac{{c}^{2}\ln \left ( ex+d \right ) bd}{{e}^{5}}}+{\frac{{c}^{3}{x}^{2}}{{e}^{4}}}-2\,{\frac{abc{d}^{2}}{{e}^{3} \left ( ex+d \right ) ^{3}}}+6\,{\frac{cabd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+10\,{\frac{b{d}^{3}{c}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}-6\,{\frac{c{b}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{2\,{a}^{2}cd}{3\,{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{b}^{3}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{4\,a{c}^{2}{d}^{3}}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}+{\frac{4\,{b}^{2}{d}^{3}c}{3\,{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{5\,b{d}^{4}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-6\,{\frac{a{c}^{2}{d}^{2}}{{e}^{4} \left ( ex+d \right ) ^{2}}}+4\,{\frac{{c}^{2}\ln \left ( ex+d \right ) a}{{e}^{4}}}+4\,{\frac{c\ln \left ( ex+d \right ){b}^{2}}{{e}^{4}}}+20\,{\frac{{c}^{3}\ln \left ( ex+d \right ){d}^{2}}{{e}^{6}}}+5\,{\frac{b{c}^{2}x}{{e}^{4}}} \]

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]

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

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Maxima [A]  time = 0.734923, size = 441, normalized size = 2.03 \[ \frac{47 \, c^{3} d^{5} - 65 \, b c^{2} d^{4} e - a^{2} b e^{5} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} -{\left (a b^{2} + a^{2} c\right )} d e^{4} + 3 \,{\left (20 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} + 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 3 \,{\left (35 \, c^{3} d^{4} e - 50 \, b c^{2} d^{3} e^{2} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} -{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x}{3 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} + \frac{c^{3} e x^{2} -{\left (8 \, c^{3} d - 5 \, b c^{2} e\right )} x}{e^{5}} + \frac{4 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e +{\left (b^{2} c + a c^{2}\right )} e^{2}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(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 [A]  time = 0.261138, size = 657, normalized size = 3.03 \[ \frac{3 \, c^{3} e^{5} x^{5} + 47 \, c^{3} d^{5} - 65 \, b c^{2} d^{4} e - a^{2} b e^{5} + 22 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{3} -{\left (a b^{2} + a^{2} c\right )} d e^{4} - 15 \,{\left (c^{3} d e^{4} - b c^{2} e^{5}\right )} x^{4} - 9 \,{\left (7 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4}\right )} x^{3} - 3 \,{\left (3 \, c^{3} d^{3} e^{2} + 15 \, b c^{2} d^{2} e^{3} - 12 \,{\left (b^{2} c + a c^{2}\right )} d e^{4} +{\left (b^{3} + 6 \, a b c\right )} e^{5}\right )} x^{2} + 3 \,{\left (27 \, c^{3} d^{4} e - 45 \, b c^{2} d^{3} e^{2} + 18 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d e^{4} -{\left (a b^{2} + a^{2} c\right )} e^{5}\right )} x + 12 \,{\left (5 \, c^{3} d^{5} - 5 \, b c^{2} d^{4} e +{\left (b^{2} c + a c^{2}\right )} d^{3} e^{2} +{\left (5 \, c^{3} d^{2} e^{3} - 5 \, b c^{2} d e^{4} +{\left (b^{2} c + a c^{2}\right )} e^{5}\right )} x^{3} + 3 \,{\left (5 \, c^{3} d^{3} e^{2} - 5 \, b c^{2} d^{2} e^{3} +{\left (b^{2} c + a c^{2}\right )} d e^{4}\right )} x^{2} + 3 \,{\left (5 \, c^{3} d^{4} e - 5 \, b c^{2} d^{3} e^{2} +{\left (b^{2} c + a c^{2}\right )} d^{2} e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(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 = 69.3141, size = 377, normalized size = 1.74 \[ \frac{c^{3} x^{2}}{e^{4}} + \frac{4 c \left (a c e^{2} + b^{2} e^{2} - 5 b c d e + 5 c^{2} d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{a^{2} b e^{5} + a^{2} c d e^{4} + a b^{2} d e^{4} + 6 a b c d^{2} e^{3} - 22 a c^{2} d^{3} e^{2} + b^{3} d^{2} e^{3} - 22 b^{2} c d^{3} e^{2} + 65 b c^{2} d^{4} e - 47 c^{3} d^{5} + x^{2} \left (18 a b c e^{5} - 36 a c^{2} d e^{4} + 3 b^{3} e^{5} - 36 b^{2} c d e^{4} + 90 b c^{2} d^{2} e^{3} - 60 c^{3} d^{3} e^{2}\right ) + x \left (3 a^{2} c e^{5} + 3 a b^{2} e^{5} + 18 a b c d e^{4} - 54 a c^{2} d^{2} e^{3} + 3 b^{3} d e^{4} - 54 b^{2} c d^{2} e^{3} + 150 b c^{2} d^{3} e^{2} - 105 c^{3} d^{4} e\right )}{3 d^{3} e^{6} + 9 d^{2} e^{7} x + 9 d e^{8} x^{2} + 3 e^{9} x^{3}} + \frac{x \left (5 b c^{2} e - 8 c^{3} d\right )}{e^{5}} \]

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/XCAS [A]  time = 0.270582, size = 424, normalized size = 1.95 \[ 4 \,{\left (5 \, c^{3} d^{2} - 5 \, b c^{2} d e + b^{2} c e^{2} + a c^{2} e^{2}\right )} e^{\left (-6\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) +{\left (c^{3} x^{2} e^{4} - 8 \, c^{3} d x e^{3} + 5 \, b c^{2} x e^{4}\right )} e^{\left (-8\right )} + \frac{{\left (47 \, c^{3} d^{5} - 65 \, b c^{2} d^{4} e + 22 \, b^{2} c d^{3} e^{2} + 22 \, a c^{2} d^{3} e^{2} - b^{3} d^{2} e^{3} - 6 \, a b c d^{2} e^{3} - a b^{2} d e^{4} - a^{2} c d e^{4} - a^{2} b e^{5} + 3 \,{\left (20 \, c^{3} d^{3} e^{2} - 30 \, b c^{2} d^{2} e^{3} + 12 \, b^{2} c d e^{4} + 12 \, a c^{2} d e^{4} - b^{3} e^{5} - 6 \, a b c e^{5}\right )} x^{2} + 3 \,{\left (35 \, c^{3} d^{4} e - 50 \, b c^{2} d^{3} e^{2} + 18 \, b^{2} c d^{2} e^{3} + 18 \, a c^{2} d^{2} e^{3} - b^{3} d e^{4} - 6 \, a b c d e^{4} - a b^{2} e^{5} - a^{2} c e^{5}\right )} x\right )} e^{\left (-6\right )}}{3 \,{\left (x e + d\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(2*c*x + b)/(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)*ln(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