3.2331 \(\int \frac{(A+B x) (a+b x+c x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=304 \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^6 (d+e x)^5}+\frac{\left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{6 e^6 (d+e x)^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6 (d+e x)^7}+\frac{c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2} \]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(7*e^6*(d + e*x)^7) + ((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))))/(6*e^6*(d + e*x)^6) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d
- a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(5*e^6*(d + e*x)^5) + (2*A*c*e*(2*c*d - b*e) - B*(1
0*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(4*e^6*(d + e*x)^4) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(3*e^6*(d +
e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)

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Rubi [A]  time = 0.31928, antiderivative size = 302, 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, Rules used = {771} \[ \frac{2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )}{4 e^6 (d+e x)^4}+\frac{B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )}{5 e^6 (d+e x)^5}-\frac{\left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{6 e^6 (d+e x)^6}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^2}{7 e^6 (d+e x)^7}+\frac{c (-A c e-2 b B e+5 B c d)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^8,x]

[Out]

((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^2)/(7*e^6*(d + e*x)^7) - ((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)))/(6*e^6*(d + e*x)^6) + (B*(10*c^2*d^3 + b*e^2*(3*b*d - 2*a*e) - 6*c*d*e*(2*b*d
- a*e)) - A*e*(6*c^2*d^2 + b^2*e^2 - 2*c*e*(3*b*d - a*e)))/(5*e^6*(d + e*x)^5) + (2*A*c*e*(2*c*d - b*e) - B*(1
0*c^2*d^2 + b^2*e^2 - 2*c*e*(4*b*d - a*e)))/(4*e^6*(d + e*x)^4) + (c*(5*B*c*d - 2*b*B*e - A*c*e))/(3*e^6*(d +
e*x)^3) - (B*c^2)/(2*e^6*(d + e*x)^2)

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{(A+B x) \left (a+b x+c x^2\right )^2}{(d+e x)^8} \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^2}{e^5 (d+e x)^8}+\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{e^5 (d+e x)^7}+\frac{-B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )+A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{e^5 (d+e x)^6}+\frac{-2 A c e (2 c d-b e)+B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{e^5 (d+e x)^5}+\frac{c (-5 B c d+2 b B e+A c e)}{e^5 (d+e x)^4}+\frac{B c^2}{e^5 (d+e x)^3}\right ) \, dx\\ &=\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^2}{7 e^6 (d+e x)^7}-\frac{\left (c d^2-b d e+a e^2\right ) \left (5 B c d^2-B e (3 b d-a e)-2 A e (2 c d-b e)\right )}{6 e^6 (d+e x)^6}+\frac{B \left (10 c^2 d^3+b e^2 (3 b d-2 a e)-6 c d e (2 b d-a e)\right )-A e \left (6 c^2 d^2+b^2 e^2-2 c e (3 b d-a e)\right )}{5 e^6 (d+e x)^5}+\frac{2 A c e (2 c d-b e)-B \left (10 c^2 d^2+b^2 e^2-2 c e (4 b d-a e)\right )}{4 e^6 (d+e x)^4}+\frac{c (5 B c d-2 b B e-A c e)}{3 e^6 (d+e x)^3}-\frac{B c^2}{2 e^6 (d+e x)^2}\\ \end{align*}

Mathematica [A]  time = 0.229766, size = 377, normalized size = 1.24 \[ -\frac{2 A e \left (2 e^2 \left (15 a^2 e^2+5 a b e (d+7 e x)+b^2 \left (d^2+7 d e x+21 e^2 x^2\right )\right )+c e \left (4 a e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+B \left (e^2 \left (10 a^2 e^2 (d+7 e x)+8 a b e \left (d^2+7 d e x+21 e^2 x^2\right )+3 b^2 \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )\right )+2 c e \left (3 a e \left (7 d^2 e x+d^3+21 d e^2 x^2+35 e^3 x^3\right )+4 b \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )\right )+10 c^2 \left (21 d^3 e^2 x^2+35 d^2 e^3 x^3+7 d^4 e x+d^5+35 d e^4 x^4+21 e^5 x^5\right )\right )}{420 e^6 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^2)/(d + e*x)^8,x]

[Out]

-(2*A*e*(2*c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4) + 2*e^2*(15*a^2*e^2 + 5*a*b*e*(d
+ 7*e*x) + b^2*(d^2 + 7*d*e*x + 21*e^2*x^2)) + c*e*(4*a*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b*(d^3 + 7*d^2*e*x
+ 21*d*e^2*x^2 + 35*e^3*x^3))) + B*(10*c^2*(d^5 + 7*d^4*e*x + 21*d^3*e^2*x^2 + 35*d^2*e^3*x^3 + 35*d*e^4*x^4
+ 21*e^5*x^5) + e^2*(10*a^2*e^2*(d + 7*e*x) + 8*a*b*e*(d^2 + 7*d*e*x + 21*e^2*x^2) + 3*b^2*(d^3 + 7*d^2*e*x +
21*d*e^2*x^2 + 35*e^3*x^3)) + 2*c*e*(3*a*e*(d^3 + 7*d^2*e*x + 21*d*e^2*x^2 + 35*e^3*x^3) + 4*b*(d^4 + 7*d^3*e*
x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 + 35*e^4*x^4))))/(420*e^6*(d + e*x)^7)

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Maple [A]  time = 0.007, size = 453, normalized size = 1.5 \begin{align*} -{\frac{2\,Aab{e}^{4}-4\,Aacd{e}^{3}-2\,Ad{b}^{2}{e}^{3}+6\,A{d}^{2}bc{e}^{2}-4\,A{c}^{2}{d}^{3}e+B{a}^{2}{e}^{4}-4\,Bdab{e}^{3}+6\,Bac{d}^{2}{e}^{2}+3\,B{b}^{2}{d}^{2}{e}^{2}-8\,B{d}^{3}bce+5\,B{c}^{2}{d}^{4}}{6\,{e}^{6} \left ( ex+d \right ) ^{6}}}-{\frac{2\,aA{e}^{3}c+A{b}^{2}{e}^{3}-6\,Abcd{e}^{2}+6\,A{c}^{2}{d}^{2}e+2\,B{e}^{3}ab-6\,aBd{e}^{2}c-3\,B{b}^{2}d{e}^{2}+12\,Bbc{d}^{2}e-10\,B{c}^{2}{d}^{3}}{5\,{e}^{6} \left ( ex+d \right ) ^{5}}}-{\frac{A{a}^{2}{e}^{5}-2\,Adab{e}^{4}+2\,A{d}^{2}ac{e}^{3}+A{d}^{2}{b}^{2}{e}^{3}-2\,A{d}^{3}bc{e}^{2}+A{d}^{4}{c}^{2}e-B{a}^{2}d{e}^{4}+2\,B{d}^{2}ab{e}^{3}-2\,aBc{d}^{3}{e}^{2}-B{d}^{3}{b}^{2}{e}^{2}+2\,B{d}^{4}bce-B{c}^{2}{d}^{5}}{7\,{e}^{6} \left ( ex+d \right ) ^{7}}}-{\frac{c \left ( Ace+2\,bBe-5\,Bcd \right ) }{3\,{e}^{6} \left ( ex+d \right ) ^{3}}}-{\frac{B{c}^{2}}{2\,{e}^{6} \left ( ex+d \right ) ^{2}}}-{\frac{2\,Abc{e}^{2}-4\,A{c}^{2}de+2\,aBc{e}^{2}+B{e}^{2}{b}^{2}-8\,Bbcde+10\,B{c}^{2}{d}^{2}}{4\,{e}^{6} \left ( ex+d \right ) ^{4}}} \end{align*}

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)^8,x)

[Out]

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

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Maxima [A]  time = 1.07659, size = 617, normalized size = 2.03 \begin{align*} -\frac{210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 60 \, A a^{2} e^{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} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \,{\left (5 \, B c^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \,{\left (10 \, 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} + 21 \,{\left (10 \, 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} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 7 \,{\left (10 \, 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} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{420 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}

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)^8,x, algorithm="maxima")

[Out]

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 60*A*a^2*e^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 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + 10*(B*a^2 + 2*A*a*b)*d*e^4 + 70*(5*B*c^2*d*e^4 + 2*(2*B*
b*c + A*c^2)*e^5)*x^4 + 35*(10*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 + 21*(10*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 + 4*(2*B*a*b + A*b^
2 + 2*A*a*c)*e^5)*x^2 + 7*(10*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
+ 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + 10*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^
5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

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Fricas [A]  time = 1.02406, size = 999, normalized size = 3.29 \begin{align*} -\frac{210 \, B c^{2} e^{5} x^{5} + 10 \, B c^{2} d^{5} + 60 \, A a^{2} e^{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} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{3} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{4} + 70 \,{\left (5 \, B c^{2} d e^{4} + 2 \,{\left (2 \, B b c + A c^{2}\right )} e^{5}\right )} x^{4} + 35 \,{\left (10 \, 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} + 21 \,{\left (10 \, 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} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{5}\right )} x^{2} + 7 \,{\left (10 \, 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} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{4} + 10 \,{\left (B a^{2} + 2 \, A a b\right )} e^{5}\right )} x}{420 \,{\left (e^{13} x^{7} + 7 \, d e^{12} x^{6} + 21 \, d^{2} e^{11} x^{5} + 35 \, d^{3} e^{10} x^{4} + 35 \, d^{4} e^{9} x^{3} + 21 \, d^{5} e^{8} x^{2} + 7 \, d^{6} e^{7} x + d^{7} e^{6}\right )}} \end{align*}

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)^8,x, algorithm="fricas")

[Out]

-1/420*(210*B*c^2*e^5*x^5 + 10*B*c^2*d^5 + 60*A*a^2*e^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 + 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d^2*e^3 + 10*(B*a^2 + 2*A*a*b)*d*e^4 + 70*(5*B*c^2*d*e^4 + 2*(2*B*
b*c + A*c^2)*e^5)*x^4 + 35*(10*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 + 21*(10*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 + 4*(2*B*a*b + A*b^
2 + 2*A*a*c)*e^5)*x^2 + 7*(10*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
+ 4*(2*B*a*b + A*b^2 + 2*A*a*c)*d*e^4 + 10*(B*a^2 + 2*A*a*b)*e^5)*x)/(e^13*x^7 + 7*d*e^12*x^6 + 21*d^2*e^11*x^
5 + 35*d^3*e^10*x^4 + 35*d^4*e^9*x^3 + 21*d^5*e^8*x^2 + 7*d^6*e^7*x + d^7*e^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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)**8,x)

[Out]

Timed out

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Giac [A]  time = 1.11709, size = 624, normalized size = 2.05 \begin{align*} -\frac{{\left (210 \, B c^{2} x^{5} e^{5} + 350 \, B c^{2} d x^{4} e^{4} + 350 \, B c^{2} d^{2} x^{3} e^{3} + 210 \, B c^{2} d^{3} x^{2} e^{2} + 70 \, B c^{2} d^{4} x e + 10 \, B c^{2} d^{5} + 280 \, B b c x^{4} e^{5} + 140 \, A c^{2} x^{4} e^{5} + 280 \, B b c d x^{3} e^{4} + 140 \, A c^{2} d x^{3} e^{4} + 168 \, B b c d^{2} x^{2} e^{3} + 84 \, A c^{2} d^{2} x^{2} e^{3} + 56 \, B b c d^{3} x e^{2} + 28 \, A c^{2} d^{3} x e^{2} + 8 \, B b c d^{4} e + 4 \, A c^{2} d^{4} e + 105 \, B b^{2} x^{3} e^{5} + 210 \, B a c x^{3} e^{5} + 210 \, A b c x^{3} e^{5} + 63 \, B b^{2} d x^{2} e^{4} + 126 \, B a c d x^{2} e^{4} + 126 \, A b c d x^{2} e^{4} + 21 \, B b^{2} d^{2} x e^{3} + 42 \, B a c d^{2} x e^{3} + 42 \, A b c d^{2} x e^{3} + 3 \, B b^{2} d^{3} e^{2} + 6 \, B a c d^{3} e^{2} + 6 \, A b c d^{3} e^{2} + 168 \, B a b x^{2} e^{5} + 84 \, A b^{2} x^{2} e^{5} + 168 \, A a c x^{2} e^{5} + 56 \, B a b d x e^{4} + 28 \, A b^{2} d x e^{4} + 56 \, A a c d x e^{4} + 8 \, B a b d^{2} e^{3} + 4 \, A b^{2} d^{2} e^{3} + 8 \, A a c d^{2} e^{3} + 70 \, B a^{2} x e^{5} + 140 \, A a b x e^{5} + 10 \, B a^{2} d e^{4} + 20 \, A a b d e^{4} + 60 \, A a^{2} e^{5}\right )} e^{\left (-6\right )}}{420 \,{\left (x e + d\right )}^{7}} \end{align*}

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)^8,x, algorithm="giac")

[Out]

-1/420*(210*B*c^2*x^5*e^5 + 350*B*c^2*d*x^4*e^4 + 350*B*c^2*d^2*x^3*e^3 + 210*B*c^2*d^3*x^2*e^2 + 70*B*c^2*d^4
*x*e + 10*B*c^2*d^5 + 280*B*b*c*x^4*e^5 + 140*A*c^2*x^4*e^5 + 280*B*b*c*d*x^3*e^4 + 140*A*c^2*d*x^3*e^4 + 168*
B*b*c*d^2*x^2*e^3 + 84*A*c^2*d^2*x^2*e^3 + 56*B*b*c*d^3*x*e^2 + 28*A*c^2*d^3*x*e^2 + 8*B*b*c*d^4*e + 4*A*c^2*d
^4*e + 105*B*b^2*x^3*e^5 + 210*B*a*c*x^3*e^5 + 210*A*b*c*x^3*e^5 + 63*B*b^2*d*x^2*e^4 + 126*B*a*c*d*x^2*e^4 +
126*A*b*c*d*x^2*e^4 + 21*B*b^2*d^2*x*e^3 + 42*B*a*c*d^2*x*e^3 + 42*A*b*c*d^2*x*e^3 + 3*B*b^2*d^3*e^2 + 6*B*a*c
*d^3*e^2 + 6*A*b*c*d^3*e^2 + 168*B*a*b*x^2*e^5 + 84*A*b^2*x^2*e^5 + 168*A*a*c*x^2*e^5 + 56*B*a*b*d*x*e^4 + 28*
A*b^2*d*x*e^4 + 56*A*a*c*d*x*e^4 + 8*B*a*b*d^2*e^3 + 4*A*b^2*d^2*e^3 + 8*A*a*c*d^2*e^3 + 70*B*a^2*x*e^5 + 140*
A*a*b*x*e^5 + 10*B*a^2*d*e^4 + 20*A*a*b*d*e^4 + 60*A*a^2*e^5)*e^(-6)/(x*e + d)^7