3.2324 \(\int \frac{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=267 \[ -\frac{x \left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{e^5}-\frac{x^2 \left (2 A c e (c d-b e)-B \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{2 e^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}-\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{e^6}-\frac{c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac{B c^2 x^4}{4 e^2} \]

[Out]

-(((2*B*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e)) - A*e*(3*c^2*d^2 + b^2*e^2 - 2*c*e
*(2*b*d - a*e)))*x)/e^5) - ((2*A*c*e*(c*d - b*e) - B*(3*c^2*d^2 + b^2*e^2 - 2*c*
e*(2*b*d - a*e)))*x^2)/(2*e^4) - (c*(2*B*c*d - 2*b*B*e - A*c*e)*x^3)/(3*e^3) + (
B*c^2*x^4)/(4*e^2) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^6*(d + e*x)) - (
(c*d^2 - b*d*e + a*e^2)*(2*A*e*(2*c*d - b*e) - B*(5*c*d^2 - e*(3*b*d - a*e)))*Lo
g[d + e*x])/e^6

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Rubi [A]  time = 1.06937, antiderivative size = 264, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{x \left (2 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )-A e \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{e^5}-\frac{x^2 \left (2 A c e (c d-b e)-B \left (-2 c e (2 b d-a e)+b^2 e^2+3 c^2 d^2\right )\right )}{2 e^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{e^6 (d+e x)}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{e^6}-\frac{c x^3 (-A c e-2 b B e+2 B c d)}{3 e^3}+\frac{B c^2 x^4}{4 e^2} \]

Antiderivative was successfully verified.

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

[Out]

-(((2*B*(2*c*d - b*e)*(c*d^2 - e*(b*d - a*e)) - A*e*(3*c^2*d^2 + b^2*e^2 - 2*c*e
*(2*b*d - a*e)))*x)/e^5) - ((2*A*c*e*(c*d - b*e) - B*(3*c^2*d^2 + b^2*e^2 - 2*c*
e*(2*b*d - a*e)))*x^2)/(2*e^4) - (c*(2*B*c*d - 2*b*B*e - A*c*e)*x^3)/(3*e^3) + (
B*c^2*x^4)/(4*e^2) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(e^6*(d + e*x)) + (
(c*d^2 - b*d*e + a*e^2)*(5*B*c*d^2 - B*e*(3*b*d - a*e) - 2*A*e*(2*c*d - b*e))*Lo
g[d + e*x])/e^6

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.25056, size = 250, normalized size = 0.94 \[ \frac{6 e^2 x^2 \left (B \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )+2 A c e (b e-c d)\right )+12 e x \left (A e \left (2 c e (a e-2 b d)+b^2 e^2+3 c^2 d^2\right )-2 B (2 c d-b e) \left (e (a e-b d)+c d^2\right )\right )+\frac{12 (B d-A e) \left (e (a e-b d)+c d^2\right )^2}{d+e x}+12 \log (d+e x) \left (e (a e-b d)+c d^2\right ) \left (B e (a e-3 b d)+2 A e (b e-2 c d)+5 B c d^2\right )+4 c e^3 x^3 (A c e+2 b B e-2 B c d)+3 B c^2 e^4 x^4}{12 e^6} \]

Antiderivative was successfully verified.

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

[Out]

(12*e*(A*e*(3*c^2*d^2 + b^2*e^2 + 2*c*e*(-2*b*d + a*e)) - 2*B*(2*c*d - b*e)*(c*d
^2 + e*(-(b*d) + a*e)))*x + 6*e^2*(2*A*c*e*(-(c*d) + b*e) + B*(3*c^2*d^2 + b^2*e
^2 + 2*c*e*(-2*b*d + a*e)))*x^2 + 4*c*e^3*(-2*B*c*d + 2*b*B*e + A*c*e)*x^3 + 3*B
*c^2*e^4*x^4 + (12*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^2)/(d + e*x) + 12*(c*d
^2 + e*(-(b*d) + a*e))*(5*B*c*d^2 + B*e*(-3*b*d + a*e) + 2*A*e*(-2*c*d + b*e))*L
og[d + e*x])/(12*e^6)

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Maple [B]  time = 0.016, size = 609, normalized size = 2.3 \[ -{\frac{2\,B{c}^{2}{x}^{3}d}{3\,{e}^{3}}}+{\frac{2\,B{x}^{3}bc}{3\,{e}^{2}}}+{\frac{{b}^{2}B{x}^{2}}{2\,{e}^{2}}}+{\frac{B{c}^{2}{x}^{4}}{4\,{e}^{2}}}-4\,{\frac{\ln \left ( ex+d \right ) Babd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) Bac{d}^{2}}{{e}^{4}}}-8\,{\frac{\ln \left ( ex+d \right ) Bbc{d}^{3}}{{e}^{5}}}+2\,{\frac{Adab}{{e}^{2} \left ( ex+d \right ) }}-2\,{\frac{aAc{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{A{d}^{3}bc}{{e}^{4} \left ( ex+d \right ) }}-2\,{\frac{abB{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+2\,{\frac{aBc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-2\,{\frac{Bbc{d}^{4}}{{e}^{5} \left ( ex+d \right ) }}-2\,{\frac{B{x}^{2}bcd}{{e}^{3}}}-4\,{\frac{Abcdx}{{e}^{3}}}-4\,{\frac{aBcdx}{{e}^{3}}}+6\,{\frac{Bbc{d}^{2}x}{{e}^{4}}}-4\,{\frac{\ln \left ( ex+d \right ) Aacd}{{e}^{3}}}+6\,{\frac{\ln \left ( ex+d \right ) Abc{d}^{2}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) B{a}^{2}}{{e}^{2}}}-{\frac{A{a}^{2}}{e \left ( ex+d \right ) }}+{\frac{A{c}^{2}{x}^{3}}{3\,{e}^{2}}}+{\frac{A{b}^{2}x}{{e}^{2}}}+{\frac{3\,B{c}^{2}{x}^{2}{d}^{2}}{2\,{e}^{4}}}+2\,{\frac{aAcx}{{e}^{2}}}+3\,{\frac{A{c}^{2}{d}^{2}x}{{e}^{4}}}+2\,{\frac{abBx}{{e}^{2}}}-2\,{\frac{{b}^{2}Bdx}{{e}^{3}}}-4\,{\frac{\ln \left ( ex+d \right ) A{c}^{2}{d}^{3}}{{e}^{5}}}-4\,{\frac{B{c}^{2}{d}^{3}x}{{e}^{5}}}+{\frac{Ab{x}^{2}c}{{e}^{2}}}-{\frac{A{c}^{2}{x}^{2}d}{{e}^{3}}}+{\frac{aBc{x}^{2}}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) B{b}^{2}{d}^{2}}{{e}^{4}}}+5\,{\frac{\ln \left ( ex+d \right ) B{c}^{2}{d}^{4}}{{e}^{6}}}-{\frac{A{b}^{2}{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{A{d}^{4}{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+{\frac{Bd{a}^{2}}{{e}^{2} \left ( ex+d \right ) }}+{\frac{{b}^{2}B{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}+{\frac{B{c}^{2}{d}^{5}}{{e}^{6} \left ( ex+d \right ) }}+2\,{\frac{\ln \left ( ex+d \right ) Aab}{{e}^{2}}}-2\,{\frac{\ln \left ( ex+d \right ) A{b}^{2}d}{{e}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^2/(e*x+d)^2,x)

[Out]

-2/3/e^3*B*x^3*c^2*d+2/3/e^2*B*x^3*b*c+1/2*b^2*B*x^2/e^2+1/4*B*c^2*x^4/e^2-4/e^3
*ln(e*x+d)*B*a*b*d+6/e^4*ln(e*x+d)*B*a*c*d^2-8/e^5*ln(e*x+d)*B*b*c*d^3+2/e^2/(e*
x+d)*A*d*a*b-2/e^3/(e*x+d)*A*a*c*d^2+2/e^4/(e*x+d)*A*b*c*d^3-2/e^3/(e*x+d)*B*a*b
*d^2+2/e^4/(e*x+d)*B*a*c*d^3-2/e^5/(e*x+d)*B*b*c*d^4-2/e^3*B*x^2*b*c*d-4/e^3*A*b
*c*d*x-4/e^3*B*a*c*d*x+6/e^4*B*b*c*d^2*x-4/e^3*ln(e*x+d)*A*a*c*d+6/e^4*ln(e*x+d)
*A*b*c*d^2+1/e^2*ln(e*x+d)*B*a^2-1/e/(e*x+d)*A*a^2+1/3/e^2*A*x^3*c^2+1/e^2*A*b^2
*x+3/2/e^4*B*x^2*c^2*d^2+2/e^2*A*a*c*x+3/e^4*A*c^2*d^2*x+2/e^2*B*a*b*x-2/e^3*B*b
^2*d*x-4/e^5*ln(e*x+d)*A*c^2*d^3-4/e^5*B*c^2*d^3*x+1/e^2*A*x^2*b*c-1/e^3*A*x^2*c
^2*d+1/e^2*B*x^2*a*c+3/e^4*ln(e*x+d)*B*b^2*d^2+5/e^6*ln(e*x+d)*B*c^2*d^4-1/e^3/(
e*x+d)*A*b^2*d^2-1/e^5/(e*x+d)*A*c^2*d^4+1/e^2/(e*x+d)*B*d*a^2+1/e^4/(e*x+d)*B*b
^2*d^3+1/e^6/(e*x+d)*B*c^2*d^5+2/e^2*ln(e*x+d)*A*a*b-2/e^3*ln(e*x+d)*A*b^2*d

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Maxima [A]  time = 0.696336, size = 522, normalized size = 1.96 \[ \frac{B c^{2} d^{5} - A a^{2} e^{5} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (B a^{2} + 2 \, A a b\right )} d e^{4}}{e^{7} x + d e^{6}} + \frac{3 \, B c^{2} e^{3} x^{4} - 4 \,{\left (2 \, B c^{2} d e^{2} -{\left (2 \, B b c + A c^{2}\right )} e^{3}\right )} x^{3} + 6 \,{\left (3 \, B c^{2} d^{2} e - 2 \,{\left (2 \, B b c + A c^{2}\right )} d e^{2} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{3}\right )} x^{2} - 12 \,{\left (4 \, B c^{2} d^{3} - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e + 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{2} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{3}\right )} x}{12 \, e^{5}} + \frac{{\left (5 \, B c^{2} d^{4} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^2,x, algorithm="maxima")

[Out]

(B*c^2*d^5 - A*a^2*e^5 - (2*B*b*c + A*c^2)*d^4*e + (B*b^2 + 2*(B*a + A*b)*c)*d^3
*e^2 - (2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + (B*a^2 + 2*A*a*b)*d*e^4)/(e^7*x + d
*e^6) + 1/12*(3*B*c^2*e^3*x^4 - 4*(2*B*c^2*d*e^2 - (2*B*b*c + A*c^2)*e^3)*x^3 +
6*(3*B*c^2*d^2*e - 2*(2*B*b*c + A*c^2)*d*e^2 + (B*b^2 + 2*(B*a + A*b)*c)*e^3)*x^
2 - 12*(4*B*c^2*d^3 - 3*(2*B*b*c + A*c^2)*d^2*e + 2*(B*b^2 + 2*(B*a + A*b)*c)*d*
e^2 - (2*B*a*b + A*b^2 + 2*A*a*c)*e^3)*x)/e^5 + (5*B*c^2*d^4 - 4*(2*B*b*c + A*c^
2)*d^3*e + 3*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^2 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d
*e^3 + (B*a^2 + 2*A*a*b)*e^4)*log(e*x + d)/e^6

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Fricas [A]  time = 0.26764, size = 765, normalized size = 2.87 \[ \frac{3 \, B c^{2} e^{5} x^{5} + 12 \, B c^{2} d^{5} - 12 \, A a^{2} e^{5} - 12 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 12 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 12 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 12 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} -{\left (5 \, B c^{2} d e^{4} - 4 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 2 \,{\left (5 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{5}\right )} x^{3} - 6 \,{\left (5 \, B c^{2} d^{3} e^{2} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{4} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} - 12 \,{\left (4 \, B c^{2} d^{4} e - 3 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 2 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} -{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4}\right )} x + 12 \,{\left (5 \, B c^{2} d^{5} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e^{2} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} +{\left (B a^{2} + 2 \, A a b\right )} d e^{4} +{\left (5 \, B c^{2} d^{4} e - 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{3} - 2 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} +{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{7} x + d e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^2,x, algorithm="fricas")

[Out]

1/12*(3*B*c^2*e^5*x^5 + 12*B*c^2*d^5 - 12*A*a^2*e^5 - 12*(2*B*b*c + A*c^2)*d^4*e
+ 12*(B*b^2 + 2*(B*a + A*b)*c)*d^3*e^2 - 12*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3
+ 12*(B*a^2 + 2*A*a*b)*d*e^4 - (5*B*c^2*d*e^4 - 4*(2*B*b*c + A*c^2)*e^5)*x^4 +
2*(5*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + 3*(B*b^2 + 2*(B*a + A*b)*c)*e^5
)*x^3 - 6*(5*B*c^2*d^3*e^2 - 4*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*(B*a + A
*b)*c)*d*e^4 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*e^5)*x^2 - 12*(4*B*c^2*d^4*e - 3*(2
*B*b*c + A*c^2)*d^3*e^2 + 2*(B*b^2 + 2*(B*a + A*b)*c)*d^2*e^3 - (2*B*a*b + A*b^2
+ 2*A*a*c)*d*e^4)*x + 12*(5*B*c^2*d^5 - 4*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 +
2*(B*a + A*b)*c)*d^3*e^2 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + (B*a^2 + 2*A*
a*b)*d*e^4 + (5*B*c^2*d^4*e - 4*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*(B*a +
A*b)*c)*d^2*e^3 - 2*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + (B*a^2 + 2*A*a*b)*e^5)*x
)*log(e*x + d))/(e^7*x + d*e^6)

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Sympy [A]  time = 17.801, size = 432, normalized size = 1.62 \[ \frac{B c^{2} x^{4}}{4 e^{2}} + \frac{- A a^{2} e^{5} + 2 A a b d e^{4} - 2 A a c d^{2} e^{3} - A b^{2} d^{2} e^{3} + 2 A b c d^{3} e^{2} - A c^{2} d^{4} e + B a^{2} d e^{4} - 2 B a b d^{2} e^{3} + 2 B a c d^{3} e^{2} + B b^{2} d^{3} e^{2} - 2 B b c d^{4} e + B c^{2} d^{5}}{d e^{6} + e^{7} x} + \frac{x^{3} \left (A c^{2} e + 2 B b c e - 2 B c^{2} d\right )}{3 e^{3}} + \frac{x^{2} \left (2 A b c e^{2} - 2 A c^{2} d e + 2 B a c e^{2} + B b^{2} e^{2} - 4 B b c d e + 3 B c^{2} d^{2}\right )}{2 e^{4}} + \frac{x \left (2 A a c e^{3} + A b^{2} e^{3} - 4 A b c d e^{2} + 3 A c^{2} d^{2} e + 2 B a b e^{3} - 4 B a c d e^{2} - 2 B b^{2} d e^{2} + 6 B b c d^{2} e - 4 B c^{2} d^{3}\right )}{e^{5}} + \frac{\left (a e^{2} - b d e + c d^{2}\right ) \left (2 A b e^{2} - 4 A c d e + B a e^{2} - 3 B b d e + 5 B c d^{2}\right ) \log{\left (d + e x \right )}}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**2/(e*x+d)**2,x)

[Out]

B*c**2*x**4/(4*e**2) + (-A*a**2*e**5 + 2*A*a*b*d*e**4 - 2*A*a*c*d**2*e**3 - A*b*
*2*d**2*e**3 + 2*A*b*c*d**3*e**2 - A*c**2*d**4*e + B*a**2*d*e**4 - 2*B*a*b*d**2*
e**3 + 2*B*a*c*d**3*e**2 + B*b**2*d**3*e**2 - 2*B*b*c*d**4*e + B*c**2*d**5)/(d*e
**6 + e**7*x) + x**3*(A*c**2*e + 2*B*b*c*e - 2*B*c**2*d)/(3*e**3) + x**2*(2*A*b*
c*e**2 - 2*A*c**2*d*e + 2*B*a*c*e**2 + B*b**2*e**2 - 4*B*b*c*d*e + 3*B*c**2*d**2
)/(2*e**4) + x*(2*A*a*c*e**3 + A*b**2*e**3 - 4*A*b*c*d*e**2 + 3*A*c**2*d**2*e +
2*B*a*b*e**3 - 4*B*a*c*d*e**2 - 2*B*b**2*d*e**2 + 6*B*b*c*d**2*e - 4*B*c**2*d**3
)/e**5 + (a*e**2 - b*d*e + c*d**2)*(2*A*b*e**2 - 4*A*c*d*e + B*a*e**2 - 3*B*b*d*
e + 5*B*c*d**2)*log(d + e*x)/e**6

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GIAC/XCAS [A]  time = 0.267156, size = 744, normalized size = 2.79 \[ \frac{1}{12} \,{\left (3 \, B c^{2} - \frac{4 \,{\left (5 \, B c^{2} d e - 2 \, B b c e^{2} - A c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{6 \,{\left (10 \, B c^{2} d^{2} e^{2} - 8 \, B b c d e^{3} - 4 \, A c^{2} d e^{3} + B b^{2} e^{4} + 2 \, B a c e^{4} + 2 \, A b c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac{12 \,{\left (10 \, B c^{2} d^{3} e^{3} - 12 \, B b c d^{2} e^{4} - 6 \, A c^{2} d^{2} e^{4} + 3 \, B b^{2} d e^{5} + 6 \, B a c d e^{5} + 6 \, A b c d e^{5} - 2 \, B a b e^{6} - A b^{2} e^{6} - 2 \, A a c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}}\right )}{\left (x e + d\right )}^{4} e^{\left (-6\right )} -{\left (5 \, B c^{2} d^{4} - 8 \, B b c d^{3} e - 4 \, A c^{2} d^{3} e + 3 \, B b^{2} d^{2} e^{2} + 6 \, B a c d^{2} e^{2} + 6 \, A b c d^{2} e^{2} - 4 \, B a b d e^{3} - 2 \, A b^{2} d e^{3} - 4 \, A a c d e^{3} + B a^{2} e^{4} + 2 \, A a b e^{4}\right )} e^{\left (-6\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (\frac{B c^{2} d^{5} e^{4}}{x e + d} - \frac{2 \, B b c d^{4} e^{5}}{x e + d} - \frac{A c^{2} d^{4} e^{5}}{x e + d} + \frac{B b^{2} d^{3} e^{6}}{x e + d} + \frac{2 \, B a c d^{3} e^{6}}{x e + d} + \frac{2 \, A b c d^{3} e^{6}}{x e + d} - \frac{2 \, B a b d^{2} e^{7}}{x e + d} - \frac{A b^{2} d^{2} e^{7}}{x e + d} - \frac{2 \, A a c d^{2} e^{7}}{x e + d} + \frac{B a^{2} d e^{8}}{x e + d} + \frac{2 \, A a b d e^{8}}{x e + d} - \frac{A a^{2} e^{9}}{x e + d}\right )} e^{\left (-10\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2*(B*x + A)/(e*x + d)^2,x, algorithm="giac")

[Out]

1/12*(3*B*c^2 - 4*(5*B*c^2*d*e - 2*B*b*c*e^2 - A*c^2*e^2)*e^(-1)/(x*e + d) + 6*(
10*B*c^2*d^2*e^2 - 8*B*b*c*d*e^3 - 4*A*c^2*d*e^3 + B*b^2*e^4 + 2*B*a*c*e^4 + 2*A
*b*c*e^4)*e^(-2)/(x*e + d)^2 - 12*(10*B*c^2*d^3*e^3 - 12*B*b*c*d^2*e^4 - 6*A*c^2
*d^2*e^4 + 3*B*b^2*d*e^5 + 6*B*a*c*d*e^5 + 6*A*b*c*d*e^5 - 2*B*a*b*e^6 - A*b^2*e
^6 - 2*A*a*c*e^6)*e^(-3)/(x*e + d)^3)*(x*e + d)^4*e^(-6) - (5*B*c^2*d^4 - 8*B*b*
c*d^3*e - 4*A*c^2*d^3*e + 3*B*b^2*d^2*e^2 + 6*B*a*c*d^2*e^2 + 6*A*b*c*d^2*e^2 -
4*B*a*b*d*e^3 - 2*A*b^2*d*e^3 - 4*A*a*c*d*e^3 + B*a^2*e^4 + 2*A*a*b*e^4)*e^(-6)*
ln(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (B*c^2*d^5*e^4/(x*e + d) - 2*B*b*c*d^4*e^5
/(x*e + d) - A*c^2*d^4*e^5/(x*e + d) + B*b^2*d^3*e^6/(x*e + d) + 2*B*a*c*d^3*e^6
/(x*e + d) + 2*A*b*c*d^3*e^6/(x*e + d) - 2*B*a*b*d^2*e^7/(x*e + d) - A*b^2*d^2*e
^7/(x*e + d) - 2*A*a*c*d^2*e^7/(x*e + d) + B*a^2*d*e^8/(x*e + d) + 2*A*a*b*d*e^8
/(x*e + d) - A*a^2*e^9/(x*e + d))*e^(-10)