3.1123 \(\int \frac{(A+B x) \left (b x+c x^2\right )^2}{(d+e x)^5} \, dx\)

Optimal. Leaf size=240 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac{c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac{B c^2 x}{e^5} \]

[Out]

(B*c^2*x)/e^5 + (d^2*(B*d - A*e)*(c*d - b*e)^2)/(4*e^6*(d + e*x)^4) - (d*(c*d -
b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(3*e^6*(d + e*x)^3) - (A*e*(6*
c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/(2*e
^6*(d + e*x)^2) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))
/(e^6*(d + e*x)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*Log[d + e*x])/e^6

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Rubi [A]  time = 0.735923, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{2 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )}{e^6 (d+e x)}-\frac{A e \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )-B d \left (3 b^2 e^2-12 b c d e+10 c^2 d^2\right )}{2 e^6 (d+e x)^2}+\frac{d^2 (B d-A e) (c d-b e)^2}{4 e^6 (d+e x)^4}-\frac{d (c d-b e) (B d (5 c d-3 b e)-2 A e (2 c d-b e))}{3 e^6 (d+e x)^3}-\frac{c \log (d+e x) (-A c e-2 b B e+5 B c d)}{e^6}+\frac{B c^2 x}{e^5} \]

Antiderivative was successfully verified.

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

[Out]

(B*c^2*x)/e^5 + (d^2*(B*d - A*e)*(c*d - b*e)^2)/(4*e^6*(d + e*x)^4) - (d*(c*d -
b*e)*(B*d*(5*c*d - 3*b*e) - 2*A*e*(2*c*d - b*e)))/(3*e^6*(d + e*x)^3) - (A*e*(6*
c^2*d^2 - 6*b*c*d*e + b^2*e^2) - B*d*(10*c^2*d^2 - 12*b*c*d*e + 3*b^2*e^2))/(2*e
^6*(d + e*x)^2) + (2*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))
/(e^6*(d + e*x)) - (c*(5*B*c*d - 2*b*B*e - A*c*e)*Log[d + e*x])/e^6

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{c^{2} \int B\, dx}{e^{5}} + \frac{c \left (A c e + 2 B b e - 5 B c d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{d^{2} \left (A e - B d\right ) \left (b e - c d\right )^{2}}{4 e^{6} \left (d + e x\right )^{4}} + \frac{d \left (b e - c d\right ) \left (2 A b e^{2} - 4 A c d e - 3 B b d e + 5 B c d^{2}\right )}{3 e^{6} \left (d + e x\right )^{3}} - \frac{2 A b c e^{2} - 4 A c^{2} d e + B b^{2} e^{2} - 8 B b c d e + 10 B c^{2} d^{2}}{e^{6} \left (d + e x\right )} - \frac{A b^{2} e^{3} - 6 A b c d e^{2} + 6 A c^{2} d^{2} e - 3 B b^{2} d e^{2} + 12 B b c d^{2} e - 10 B c^{2} d^{3}}{2 e^{6} \left (d + e x\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**5,x)

[Out]

c**2*Integral(B, x)/e**5 + c*(A*c*e + 2*B*b*e - 5*B*c*d)*log(d + e*x)/e**6 - d**
2*(A*e - B*d)*(b*e - c*d)**2/(4*e**6*(d + e*x)**4) + d*(b*e - c*d)*(2*A*b*e**2 -
4*A*c*d*e - 3*B*b*d*e + 5*B*c*d**2)/(3*e**6*(d + e*x)**3) - (2*A*b*c*e**2 - 4*A
*c**2*d*e + B*b**2*e**2 - 8*B*b*c*d*e + 10*B*c**2*d**2)/(e**6*(d + e*x)) - (A*b*
*2*e**3 - 6*A*b*c*d*e**2 + 6*A*c**2*d**2*e - 3*B*b**2*d*e**2 + 12*B*b*c*d**2*e -
10*B*c**2*d**3)/(2*e**6*(d + e*x)**2)

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Mathematica [A]  time = 0.273999, size = 275, normalized size = 1.15 \[ -\frac{A e \left (b^2 e^2 \left (d^2+4 d e x+6 e^2 x^2\right )+6 b c e \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )+c^2 (-d) \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )\right )+12 c (d+e x)^4 \log (d+e x) (-A c e-2 b B e+5 B c d)+B \left (3 b^2 e^2 \left (d^3+4 d^2 e x+6 d e^2 x^2+4 e^3 x^3\right )-2 b c d e \left (25 d^3+88 d^2 e x+108 d e^2 x^2+48 e^3 x^3\right )+c^2 \left (77 d^5+248 d^4 e x+252 d^3 e^2 x^2+48 d^2 e^3 x^3-48 d e^4 x^4-12 e^5 x^5\right )\right )}{12 e^6 (d+e x)^4} \]

Antiderivative was successfully verified.

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

[Out]

-(A*e*(b^2*e^2*(d^2 + 4*d*e*x + 6*e^2*x^2) + 6*b*c*e*(d^3 + 4*d^2*e*x + 6*d*e^2*
x^2 + 4*e^3*x^3) - c^2*d*(25*d^3 + 88*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3)) + B
*(3*b^2*e^2*(d^3 + 4*d^2*e*x + 6*d*e^2*x^2 + 4*e^3*x^3) - 2*b*c*d*e*(25*d^3 + 88
*d^2*e*x + 108*d*e^2*x^2 + 48*e^3*x^3) + c^2*(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)) + 12*c*(5*B*c*d - 2*b*B*e -
A*c*e)*(d + e*x)^4*Log[d + e*x])/(12*e^6*(d + e*x)^4)

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Maple [A]  time = 0.016, size = 465, normalized size = 1.9 \[{\frac{{c}^{2}\ln \left ( ex+d \right ) A}{{e}^{5}}}-{\frac{{b}^{2}B}{{e}^{4} \left ( ex+d \right ) }}-{\frac{{b}^{2}A}{2\,{e}^{3} \left ( ex+d \right ) ^{2}}}-2\,{\frac{Abc}{{e}^{4} \left ( ex+d \right ) }}+2\,{\frac{c\ln \left ( ex+d \right ) bB}{{e}^{5}}}-{\frac{B{d}^{4}bc}{2\,{e}^{5} \left ( ex+d \right ) ^{4}}}-2\,{\frac{A{d}^{2}bc}{{e}^{4} \left ( ex+d \right ) ^{3}}}-6\,{\frac{Bbc{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+8\,{\frac{Bbcd}{{e}^{5} \left ( ex+d \right ) }}-{\frac{{d}^{2}A{b}^{2}}{4\,{e}^{3} \left ( ex+d \right ) ^{4}}}-{\frac{{d}^{4}A{c}^{2}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{B{d}^{3}{b}^{2}}{4\,{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{B{c}^{2}{d}^{5}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}}-10\,{\frac{B{c}^{2}{d}^{2}}{{e}^{6} \left ( ex+d \right ) }}-3\,{\frac{A{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) ^{2}}}+{\frac{3\,{b}^{2}Bd}{2\,{e}^{4} \left ( ex+d \right ) ^{2}}}+5\,{\frac{B{c}^{2}{d}^{3}}{{e}^{6} \left ( ex+d \right ) ^{2}}}+{\frac{2\,dA{b}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-5\,{\frac{{c}^{2}\ln \left ( ex+d \right ) Bd}{{e}^{6}}}+{\frac{8\,B{d}^{3}bc}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}+3\,{\frac{Abcd}{{e}^{4} \left ( ex+d \right ) ^{2}}}+{\frac{A{d}^{3}bc}{2\,{e}^{4} \left ( ex+d \right ) ^{4}}}+{\frac{4\,A{d}^{3}{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{B{d}^{2}{b}^{2}}{{e}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{5\,B{c}^{2}{d}^{4}}{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}+{\frac{B{c}^{2}x}{{e}^{5}}}+4\,{\frac{A{c}^{2}d}{{e}^{5} \left ( ex+d \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x)^2/(e*x+d)^5,x)

[Out]

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

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Maxima [A]  time = 0.697824, size = 433, normalized size = 1.8 \[ -\frac{77 \, B c^{2} d^{5} + A b^{2} d^{2} e^{3} - 25 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} + 12 \,{\left (10 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} + 6 \,{\left (50 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} + 4 \,{\left (65 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} + \frac{B c^{2} x}{e^{5}} - \frac{{\left (5 \, B c^{2} d -{\left (2 \, B b c + A c^{2}\right )} e\right )} \log \left (e x + d\right )}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^5,x, algorithm="maxima")

[Out]

-1/12*(77*B*c^2*d^5 + A*b^2*d^2*e^3 - 25*(2*B*b*c + A*c^2)*d^4*e + 3*(B*b^2 + 2*
A*b*c)*d^3*e^2 + 12*(10*B*c^2*d^2*e^3 - 4*(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*A
*b*c)*e^5)*x^3 + 6*(50*B*c^2*d^3*e^2 + A*b^2*e^5 - 18*(2*B*b*c + A*c^2)*d^2*e^3
+ 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 + 4*(65*B*c^2*d^4*e + A*b^2*d*e^4 - 22*(2*B*b*c
+ A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^2*e^3)*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) + B*c^2*x/e^5 - (5*B*c^2*d - (2*B*b*c + A*c
^2)*e)*log(e*x + d)/e^6

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Fricas [A]  time = 0.30746, size = 641, normalized size = 2.67 \[ \frac{12 \, B c^{2} e^{5} x^{5} + 48 \, B c^{2} d e^{4} x^{4} - 77 \, B c^{2} d^{5} - A b^{2} d^{2} e^{3} + 25 \,{\left (2 \, B b c + A c^{2}\right )} d^{4} e - 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{3} e^{2} - 12 \,{\left (4 \, B c^{2} d^{2} e^{3} - 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{4} +{\left (B b^{2} + 2 \, A b c\right )} e^{5}\right )} x^{3} - 6 \,{\left (42 \, B c^{2} d^{3} e^{2} + A b^{2} e^{5} - 18 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d e^{4}\right )} x^{2} - 4 \,{\left (62 \, B c^{2} d^{4} e + A b^{2} d e^{4} - 22 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2} + 3 \,{\left (B b^{2} + 2 \, A b c\right )} d^{2} e^{3}\right )} x - 12 \,{\left (5 \, B c^{2} d^{5} -{\left (2 \, B b c + A c^{2}\right )} d^{4} e +{\left (5 \, B c^{2} d e^{4} -{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 4 \,{\left (5 \, B c^{2} d^{2} e^{3} -{\left (2 \, B b c + A c^{2}\right )} d e^{4}\right )} x^{3} + 6 \,{\left (5 \, B c^{2} d^{3} e^{2} -{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{3}\right )} x^{2} + 4 \,{\left (5 \, B c^{2} d^{4} e -{\left (2 \, B b c + A c^{2}\right )} d^{3} e^{2}\right )} x\right )} \log \left (e x + d\right )}{12 \,{\left (e^{10} x^{4} + 4 \, d e^{9} x^{3} + 6 \, d^{2} e^{8} x^{2} + 4 \, d^{3} e^{7} x + d^{4} e^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^5,x, algorithm="fricas")

[Out]

1/12*(12*B*c^2*e^5*x^5 + 48*B*c^2*d*e^4*x^4 - 77*B*c^2*d^5 - A*b^2*d^2*e^3 + 25*
(2*B*b*c + A*c^2)*d^4*e - 3*(B*b^2 + 2*A*b*c)*d^3*e^2 - 12*(4*B*c^2*d^2*e^3 - 4*
(2*B*b*c + A*c^2)*d*e^4 + (B*b^2 + 2*A*b*c)*e^5)*x^3 - 6*(42*B*c^2*d^3*e^2 + A*b
^2*e^5 - 18*(2*B*b*c + A*c^2)*d^2*e^3 + 3*(B*b^2 + 2*A*b*c)*d*e^4)*x^2 - 4*(62*B
*c^2*d^4*e + A*b^2*d*e^4 - 22*(2*B*b*c + A*c^2)*d^3*e^2 + 3*(B*b^2 + 2*A*b*c)*d^
2*e^3)*x - 12*(5*B*c^2*d^5 - (2*B*b*c + A*c^2)*d^4*e + (5*B*c^2*d*e^4 - (2*B*b*c
+ A*c^2)*e^5)*x^4 + 4*(5*B*c^2*d^2*e^3 - (2*B*b*c + A*c^2)*d*e^4)*x^3 + 6*(5*B*
c^2*d^3*e^2 - (2*B*b*c + A*c^2)*d^2*e^3)*x^2 + 4*(5*B*c^2*d^4*e - (2*B*b*c + A*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 = 162.35, size = 381, normalized size = 1.59 \[ \frac{B c^{2} x}{e^{5}} + \frac{c \left (A c e + 2 B b e - 5 B c d\right ) \log{\left (d + e x \right )}}{e^{6}} - \frac{A b^{2} d^{2} e^{3} + 6 A b c d^{3} e^{2} - 25 A c^{2} d^{4} e + 3 B b^{2} d^{3} e^{2} - 50 B b c d^{4} e + 77 B c^{2} d^{5} + x^{3} \left (24 A b c e^{5} - 48 A c^{2} d e^{4} + 12 B b^{2} e^{5} - 96 B b c d e^{4} + 120 B c^{2} d^{2} e^{3}\right ) + x^{2} \left (6 A b^{2} e^{5} + 36 A b c d e^{4} - 108 A c^{2} d^{2} e^{3} + 18 B b^{2} d e^{4} - 216 B b c d^{2} e^{3} + 300 B c^{2} d^{3} e^{2}\right ) + x \left (4 A b^{2} d e^{4} + 24 A b c d^{2} e^{3} - 88 A c^{2} d^{3} e^{2} + 12 B b^{2} d^{2} e^{3} - 176 B b c d^{3} e^{2} + 260 B c^{2} d^{4} e\right )}{12 d^{4} e^{6} + 48 d^{3} e^{7} x + 72 d^{2} e^{8} x^{2} + 48 d e^{9} x^{3} + 12 e^{10} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x)**2/(e*x+d)**5,x)

[Out]

B*c**2*x/e**5 + c*(A*c*e + 2*B*b*e - 5*B*c*d)*log(d + e*x)/e**6 - (A*b**2*d**2*e
**3 + 6*A*b*c*d**3*e**2 - 25*A*c**2*d**4*e + 3*B*b**2*d**3*e**2 - 50*B*b*c*d**4*
e + 77*B*c**2*d**5 + x**3*(24*A*b*c*e**5 - 48*A*c**2*d*e**4 + 12*B*b**2*e**5 - 9
6*B*b*c*d*e**4 + 120*B*c**2*d**2*e**3) + x**2*(6*A*b**2*e**5 + 36*A*b*c*d*e**4 -
108*A*c**2*d**2*e**3 + 18*B*b**2*d*e**4 - 216*B*b*c*d**2*e**3 + 300*B*c**2*d**3
*e**2) + x*(4*A*b**2*d*e**4 + 24*A*b*c*d**2*e**3 - 88*A*c**2*d**3*e**2 + 12*B*b*
*2*d**2*e**3 - 176*B*b*c*d**3*e**2 + 260*B*c**2*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/XCAS [A]  time = 0.285446, size = 640, normalized size = 2.67 \[{\left (x e + d\right )} B c^{2} e^{\left (-6\right )} +{\left (5 \, B c^{2} d - 2 \, B b c e - A c^{2} e\right )} e^{\left (-6\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{1}{12} \,{\left (\frac{120 \, B c^{2} d^{2} e^{22}}{x e + d} - \frac{60 \, B c^{2} d^{3} e^{22}}{{\left (x e + d\right )}^{2}} + \frac{20 \, B c^{2} d^{4} e^{22}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B c^{2} d^{5} e^{22}}{{\left (x e + d\right )}^{4}} - \frac{96 \, B b c d e^{23}}{x e + d} - \frac{48 \, A c^{2} d e^{23}}{x e + d} + \frac{72 \, B b c d^{2} e^{23}}{{\left (x e + d\right )}^{2}} + \frac{36 \, A c^{2} d^{2} e^{23}}{{\left (x e + d\right )}^{2}} - \frac{32 \, B b c d^{3} e^{23}}{{\left (x e + d\right )}^{3}} - \frac{16 \, A c^{2} d^{3} e^{23}}{{\left (x e + d\right )}^{3}} + \frac{6 \, B b c d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{3 \, A c^{2} d^{4} e^{23}}{{\left (x e + d\right )}^{4}} + \frac{12 \, B b^{2} e^{24}}{x e + d} + \frac{24 \, A b c e^{24}}{x e + d} - \frac{18 \, B b^{2} d e^{24}}{{\left (x e + d\right )}^{2}} - \frac{36 \, A b c d e^{24}}{{\left (x e + d\right )}^{2}} + \frac{12 \, B b^{2} d^{2} e^{24}}{{\left (x e + d\right )}^{3}} + \frac{24 \, A b c d^{2} e^{24}}{{\left (x e + d\right )}^{3}} - \frac{3 \, B b^{2} d^{3} e^{24}}{{\left (x e + d\right )}^{4}} - \frac{6 \, A b c d^{3} e^{24}}{{\left (x e + d\right )}^{4}} + \frac{6 \, A b^{2} e^{25}}{{\left (x e + d\right )}^{2}} - \frac{8 \, A b^{2} d e^{25}}{{\left (x e + d\right )}^{3}} + \frac{3 \, A b^{2} d^{2} e^{25}}{{\left (x e + d\right )}^{4}}\right )} e^{\left (-28\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x)^2*(B*x + A)/(e*x + d)^5,x, algorithm="giac")

[Out]

(x*e + d)*B*c^2*e^(-6) + (5*B*c^2*d - 2*B*b*c*e - A*c^2*e)*e^(-6)*ln(abs(x*e + d
)*e^(-1)/(x*e + d)^2) - 1/12*(120*B*c^2*d^2*e^22/(x*e + d) - 60*B*c^2*d^3*e^22/(
x*e + d)^2 + 20*B*c^2*d^4*e^22/(x*e + d)^3 - 3*B*c^2*d^5*e^22/(x*e + d)^4 - 96*B
*b*c*d*e^23/(x*e + d) - 48*A*c^2*d*e^23/(x*e + d) + 72*B*b*c*d^2*e^23/(x*e + d)^
2 + 36*A*c^2*d^2*e^23/(x*e + d)^2 - 32*B*b*c*d^3*e^23/(x*e + d)^3 - 16*A*c^2*d^3
*e^23/(x*e + d)^3 + 6*B*b*c*d^4*e^23/(x*e + d)^4 + 3*A*c^2*d^4*e^23/(x*e + d)^4
+ 12*B*b^2*e^24/(x*e + d) + 24*A*b*c*e^24/(x*e + d) - 18*B*b^2*d*e^24/(x*e + d)^
2 - 36*A*b*c*d*e^24/(x*e + d)^2 + 12*B*b^2*d^2*e^24/(x*e + d)^3 + 24*A*b*c*d^2*e
^24/(x*e + d)^3 - 3*B*b^2*d^3*e^24/(x*e + d)^4 - 6*A*b*c*d^3*e^24/(x*e + d)^4 +
6*A*b^2*e^25/(x*e + d)^2 - 8*A*b^2*d*e^25/(x*e + d)^3 + 3*A*b^2*d^2*e^25/(x*e +
d)^4)*e^(-28)