3.1135 \(\int \frac{(A+B x) (b x+c x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=359 \[ \frac{x^2 (c d-b e) \left (3 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{2 e^6}-\frac{x (c d-b e) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{e^7}+\frac{3 d (c d-b e) \log (d+e x) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8}-\frac{c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}-\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (d+e x)}+\frac{d^3 (B d-A e) (c d-b e)^3}{2 e^8 (d+e x)^2}+\frac{c x^3 (c d-b e) (-A c e-b B e+2 B c d)}{e^5}+\frac{B c^3 x^5}{5 e^3} \]

[Out]

-(((c*d - b*e)*(A*e*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - 3*B*d*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))*x)/e^7) + ((
c*d - b*e)*(3*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*x^2)/(2*e^6) + (c*(c*d - b*e)*(2*B*c
*d - b*B*e - A*c*e)*x^3)/e^5 - (c^2*(3*B*c*d - 3*b*B*e - A*c*e)*x^4)/(4*e^4) + (B*c^3*x^5)/(5*e^3) + (d^3*(B*d
- A*e)*(c*d - b*e)^3)/(2*e^8*(d + e*x)^2) - (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) - 3*A*e*(2*c*d - b*e)))/(
e^8*(d + e*x)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*
e^2))*Log[d + e*x])/e^8

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Rubi [A]  time = 0.582801, antiderivative size = 359, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {771} \[ \frac{x^2 (c d-b e) \left (3 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{2 e^6}-\frac{x (c d-b e) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{e^7}+\frac{3 d (c d-b e) \log (d+e x) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8}-\frac{c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}-\frac{d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (d+e x)}+\frac{d^3 (B d-A e) (c d-b e)^3}{2 e^8 (d+e x)^2}+\frac{c x^3 (c d-b e) (-A c e-b B e+2 B c d)}{e^5}+\frac{B c^3 x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^3,x]

[Out]

-(((c*d - b*e)*(A*e*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - 3*B*d*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))*x)/e^7) + ((
c*d - b*e)*(3*A*c*e*(2*c*d - b*e) - B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*x^2)/(2*e^6) + (c*(c*d - b*e)*(2*B*c
*d - b*B*e - A*c*e)*x^3)/e^5 - (c^2*(3*B*c*d - 3*b*B*e - A*c*e)*x^4)/(4*e^4) + (B*c^3*x^5)/(5*e^3) + (d^3*(B*d
- A*e)*(c*d - b*e)^3)/(2*e^8*(d + e*x)^2) - (d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) - 3*A*e*(2*c*d - b*e)))/(
e^8*(d + e*x)) + (3*d*(c*d - b*e)*(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2) - B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*
e^2))*Log[d + e*x])/e^8

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

Mathematica [A]  time = 0.167004, size = 342, normalized size = 0.95 \[ \frac{10 e^2 x^2 (b e-c d) \left (3 A c e (b e-2 c d)+B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )+20 e x (b e-c d) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )-60 d (c d-b e) \log (d+e x) \left (B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )-A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )+5 c^2 e^4 x^4 (A c e+3 b B e-3 B c d)-\frac{20 d^2 (c d-b e)^2 (3 A e (b e-2 c d)+B d (7 c d-4 b e))}{d+e x}+\frac{10 d^3 (B d-A e) (c d-b e)^3}{(d+e x)^2}-20 c e^3 x^3 (c d-b e) (A c e+b B e-2 B c d)+4 B c^3 e^5 x^5}{20 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(b*x + c*x^2)^3)/(d + e*x)^3,x]

[Out]

(20*e*(-(c*d) + b*e)*(A*e*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - 3*B*d*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))*x + 10
*e^2*(-(c*d) + b*e)*(3*A*c*e*(-2*c*d + b*e) + B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*x^2 - 20*c*e^3*(c*d - b*e)
*(-2*B*c*d + b*B*e + A*c*e)*x^3 + 5*c^2*e^4*(-3*B*c*d + 3*b*B*e + A*c*e)*x^4 + 4*B*c^3*e^5*x^5 + (10*d^3*(B*d
- A*e)*(c*d - b*e)^3)/(d + e*x)^2 - (20*d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) + 3*A*e*(-2*c*d + b*e)))/(d + e
*x) - 60*d*(c*d - b*e)*(-(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)) + B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*Lo
g[d + e*x])/(20*e^8)

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Maple [B]  time = 0.014, size = 775, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x)

[Out]

1/5*B*c^3*x^5/e^3+45*d^4/e^7*ln(e*x+d)*B*b*c^2-3/2*d^4/e^5/(e*x+d)^2*A*b^2*c+3/2*d^5/e^6/(e*x+d)^2*A*b*c^2+3/2
*d^5/e^6/(e*x+d)^2*B*b^2*c-3/2*d^6/e^7/(e*x+d)^2*B*b*c^2+12*d^3/e^5/(e*x+d)*A*b^2*c-15*d^4/e^6/(e*x+d)*A*b*c^2
-15*d^4/e^6/(e*x+d)*B*b^2*c+18*d^5/e^7/(e*x+d)*B*b*c^2-3/e^4*B*x^3*b*c^2*d-9/2/e^4*A*x^2*b*c^2*d-9/2/e^4*B*x^2
*b^2*c*d+9/e^5*B*x^2*b*c^2*d^2-9/e^4*A*b^2*c*d*x+18/e^5*A*b*c^2*d^2*x+18/e^5*B*b^2*c*d^2*x-30/e^6*B*b*c^2*d^3*
x+18*d^2/e^5*ln(e*x+d)*A*b^2*c-30*d^3/e^6*ln(e*x+d)*A*b*c^2-30*d^3/e^6*ln(e*x+d)*B*b^2*c+1/4/e^3*A*x^4*c^3+1/2
/e^3*B*x^2*b^3+1/e^3*A*b^3*x-10/e^6*A*c^3*d^3*x-3/e^4*B*b^3*d*x+15/e^7*B*c^3*d^4*x-3*d/e^4*ln(e*x+d)*A*b^3+15*
d^4/e^7*ln(e*x+d)*A*c^3+1/e^3*A*x^3*b*c^2-1/e^4*A*x^3*c^3*d+1/e^3*B*x^3*b^2*c-7*d^6/e^8/(e*x+d)*B*c^3+6*d^2/e^
5*ln(e*x+d)*B*b^3-21*d^5/e^8*ln(e*x+d)*B*c^3+1/2*d^3/e^4/(e*x+d)^2*A*b^3-1/2*d^6/e^7/(e*x+d)^2*A*c^3-1/2*d^4/e
^5/(e*x+d)^2*B*b^3+1/2*d^7/e^8/(e*x+d)^2*B*c^3+2/e^5*B*x^3*c^3*d^2+3/2/e^3*A*x^2*b^2*c+3/e^5*A*x^2*c^3*d^2-5/e
^6*B*x^2*c^3*d^3+3/4/e^3*B*x^4*b*c^2-3/4/e^4*B*x^4*c^3*d-3*d^2/e^4/(e*x+d)*A*b^3+6*d^5/e^7/(e*x+d)*A*c^3+4*d^3
/e^5/(e*x+d)*B*b^3

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Maxima [A]  time = 1.10682, size = 743, normalized size = 2.07 \begin{align*} -\frac{13 \, B c^{3} d^{7} + 5 \, A b^{3} d^{3} e^{4} - 11 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 27 \,{\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 7 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + 2 \,{\left (7 \, B c^{3} d^{6} e + 3 \, A b^{3} d^{2} e^{5} - 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \,{\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 4 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x}{2 \,{\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac{4 \, B c^{3} e^{4} x^{5} - 5 \,{\left (3 \, B c^{3} d e^{3} -{\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{4} + 20 \,{\left (2 \, B c^{3} d^{2} e^{2} -{\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} +{\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{3} - 10 \,{\left (10 \, B c^{3} d^{3} e - 6 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 9 \,{\left (B b^{2} c + A b c^{2}\right )} d e^{3} -{\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{2} + 20 \,{\left (15 \, B c^{3} d^{4} + A b^{3} e^{4} - 10 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \,{\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} - 3 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x}{20 \, e^{7}} - \frac{3 \,{\left (7 \, B c^{3} d^{5} + A b^{3} d e^{4} - 5 \,{\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 10 \,{\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{2} - 2 \,{\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="maxima")

[Out]

-1/2*(13*B*c^3*d^7 + 5*A*b^3*d^3*e^4 - 11*(3*B*b*c^2 + A*c^3)*d^6*e + 27*(B*b^2*c + A*b*c^2)*d^5*e^2 - 7*(B*b^
3 + 3*A*b^2*c)*d^4*e^3 + 2*(7*B*c^3*d^6*e + 3*A*b^3*d^2*e^5 - 6*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 15*(B*b^2*c + A*
b*c^2)*d^4*e^3 - 4*(B*b^3 + 3*A*b^2*c)*d^3*e^4)*x)/(e^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*B*c^3*e^4*x^5 -
5*(3*B*c^3*d*e^3 - (3*B*b*c^2 + A*c^3)*e^4)*x^4 + 20*(2*B*c^3*d^2*e^2 - (3*B*b*c^2 + A*c^3)*d*e^3 + (B*b^2*c +
A*b*c^2)*e^4)*x^3 - 10*(10*B*c^3*d^3*e - 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 9*(B*b^2*c + A*b*c^2)*d*e^3 - (B*b^3
+ 3*A*b^2*c)*e^4)*x^2 + 20*(15*B*c^3*d^4 + A*b^3*e^4 - 10*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + A*b*c^2)*
d^2*e^2 - 3*(B*b^3 + 3*A*b^2*c)*d*e^3)*x)/e^7 - 3*(7*B*c^3*d^5 + A*b^3*d*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 1
0*(B*b^2*c + A*b*c^2)*d^3*e^2 - 2*(B*b^3 + 3*A*b^2*c)*d^2*e^3)*log(e*x + d)/e^8

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Fricas [B]  time = 1.63908, size = 1712, normalized size = 4.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="fricas")

[Out]

1/20*(4*B*c^3*e^7*x^7 - 130*B*c^3*d^7 - 50*A*b^3*d^3*e^4 + 110*(3*B*b*c^2 + A*c^3)*d^6*e - 270*(B*b^2*c + A*b*
c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*b^2*c)*d^4*e^3 - (7*B*c^3*d*e^6 - 5*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 2*(7*B*c^3*d
^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d*e^6 + 10*(B*b^2*c + A*b*c^2)*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 5*(3*B*b*c^2 + A
*c^3)*d^2*e^5 + 10*(B*b^2*c + A*b*c^2)*d*e^6 - 2*(B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 + A*b^3*e^
7 - 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 10*(B*b^2*c + A*b*c^2)*d^2*e^5 - 2*(B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 + 10*(50
*B*c^3*d^5*e^2 + 4*A*b^3*d*e^6 - 34*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c + A*b*c^2)*d^3*e^4 - 11*(B*b^3 +
3*A*b^2*c)*d^2*e^5)*x^2 + 20*(8*B*c^3*d^6*e - 2*A*b^3*d^2*e^5 - 4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c +
A*b*c^2)*d^4*e^3 + (B*b^3 + 3*A*b^2*c)*d^3*e^4)*x - 60*(7*B*c^3*d^7 + A*b^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^
6*e + 10*(B*b^2*c + A*b*c^2)*d^5*e^2 - 2*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + (7*B*c^3*d^5*e^2 + A*b^3*d*e^6 - 5*(3*B
*b*c^2 + A*c^3)*d^4*e^3 + 10*(B*b^2*c + A*b*c^2)*d^3*e^4 - 2*(B*b^3 + 3*A*b^2*c)*d^2*e^5)*x^2 + 2*(7*B*c^3*d^6
*e + A*b^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 10*(B*b^2*c + A*b*c^2)*d^4*e^3 - 2*(B*b^3 + 3*A*b^2*c)*d^
3*e^4)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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Sympy [A]  time = 9.66994, size = 638, normalized size = 1.78 \begin{align*} \frac{B c^{3} x^{5}}{5 e^{3}} + \frac{3 d \left (b e - c d\right ) \left (- A b^{2} e^{3} + 5 A b c d e^{2} - 5 A c^{2} d^{2} e + 2 B b^{2} d e^{2} - 8 B b c d^{2} e + 7 B c^{2} d^{3}\right ) \log{\left (d + e x \right )}}{e^{8}} + \frac{- 5 A b^{3} d^{3} e^{4} + 21 A b^{2} c d^{4} e^{3} - 27 A b c^{2} d^{5} e^{2} + 11 A c^{3} d^{6} e + 7 B b^{3} d^{4} e^{3} - 27 B b^{2} c d^{5} e^{2} + 33 B b c^{2} d^{6} e - 13 B c^{3} d^{7} + x \left (- 6 A b^{3} d^{2} e^{5} + 24 A b^{2} c d^{3} e^{4} - 30 A b c^{2} d^{4} e^{3} + 12 A c^{3} d^{5} e^{2} + 8 B b^{3} d^{3} e^{4} - 30 B b^{2} c d^{4} e^{3} + 36 B b c^{2} d^{5} e^{2} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac{x^{4} \left (A c^{3} e + 3 B b c^{2} e - 3 B c^{3} d\right )}{4 e^{4}} + \frac{x^{3} \left (A b c^{2} e^{2} - A c^{3} d e + B b^{2} c e^{2} - 3 B b c^{2} d e + 2 B c^{3} d^{2}\right )}{e^{5}} + \frac{x^{2} \left (3 A b^{2} c e^{3} - 9 A b c^{2} d e^{2} + 6 A c^{3} d^{2} e + B b^{3} e^{3} - 9 B b^{2} c d e^{2} + 18 B b c^{2} d^{2} e - 10 B c^{3} d^{3}\right )}{2 e^{6}} - \frac{x \left (- A b^{3} e^{4} + 9 A b^{2} c d e^{3} - 18 A b c^{2} d^{2} e^{2} + 10 A c^{3} d^{3} e + 3 B b^{3} d e^{3} - 18 B b^{2} c d^{2} e^{2} + 30 B b c^{2} d^{3} e - 15 B c^{3} d^{4}\right )}{e^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/(e*x+d)**3,x)

[Out]

B*c**3*x**5/(5*e**3) + 3*d*(b*e - c*d)*(-A*b**2*e**3 + 5*A*b*c*d*e**2 - 5*A*c**2*d**2*e + 2*B*b**2*d*e**2 - 8*
B*b*c*d**2*e + 7*B*c**2*d**3)*log(d + e*x)/e**8 + (-5*A*b**3*d**3*e**4 + 21*A*b**2*c*d**4*e**3 - 27*A*b*c**2*d
**5*e**2 + 11*A*c**3*d**6*e + 7*B*b**3*d**4*e**3 - 27*B*b**2*c*d**5*e**2 + 33*B*b*c**2*d**6*e - 13*B*c**3*d**7
+ x*(-6*A*b**3*d**2*e**5 + 24*A*b**2*c*d**3*e**4 - 30*A*b*c**2*d**4*e**3 + 12*A*c**3*d**5*e**2 + 8*B*b**3*d**
3*e**4 - 30*B*b**2*c*d**4*e**3 + 36*B*b*c**2*d**5*e**2 - 14*B*c**3*d**6*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**1
0*x**2) + x**4*(A*c**3*e + 3*B*b*c**2*e - 3*B*c**3*d)/(4*e**4) + x**3*(A*b*c**2*e**2 - A*c**3*d*e + B*b**2*c*e
**2 - 3*B*b*c**2*d*e + 2*B*c**3*d**2)/e**5 + x**2*(3*A*b**2*c*e**3 - 9*A*b*c**2*d*e**2 + 6*A*c**3*d**2*e + B*b
**3*e**3 - 9*B*b**2*c*d*e**2 + 18*B*b*c**2*d**2*e - 10*B*c**3*d**3)/(2*e**6) - x*(-A*b**3*e**4 + 9*A*b**2*c*d*
e**3 - 18*A*b*c**2*d**2*e**2 + 10*A*c**3*d**3*e + 3*B*b**3*d*e**3 - 18*B*b**2*c*d**2*e**2 + 30*B*b*c**2*d**3*e
- 15*B*c**3*d**4)/e**7

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Giac [A]  time = 1.22162, size = 809, normalized size = 2.25 \begin{align*} -3 \,{\left (7 \, B c^{3} d^{5} - 15 \, B b c^{2} d^{4} e - 5 \, A c^{3} d^{4} e + 10 \, B b^{2} c d^{3} e^{2} + 10 \, A b c^{2} d^{3} e^{2} - 2 \, B b^{3} d^{2} e^{3} - 6 \, A b^{2} c d^{2} e^{3} + A b^{3} d e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{20} \,{\left (4 \, B c^{3} x^{5} e^{12} - 15 \, B c^{3} d x^{4} e^{11} + 40 \, B c^{3} d^{2} x^{3} e^{10} - 100 \, B c^{3} d^{3} x^{2} e^{9} + 300 \, B c^{3} d^{4} x e^{8} + 15 \, B b c^{2} x^{4} e^{12} + 5 \, A c^{3} x^{4} e^{12} - 60 \, B b c^{2} d x^{3} e^{11} - 20 \, A c^{3} d x^{3} e^{11} + 180 \, B b c^{2} d^{2} x^{2} e^{10} + 60 \, A c^{3} d^{2} x^{2} e^{10} - 600 \, B b c^{2} d^{3} x e^{9} - 200 \, A c^{3} d^{3} x e^{9} + 20 \, B b^{2} c x^{3} e^{12} + 20 \, A b c^{2} x^{3} e^{12} - 90 \, B b^{2} c d x^{2} e^{11} - 90 \, A b c^{2} d x^{2} e^{11} + 360 \, B b^{2} c d^{2} x e^{10} + 360 \, A b c^{2} d^{2} x e^{10} + 10 \, B b^{3} x^{2} e^{12} + 30 \, A b^{2} c x^{2} e^{12} - 60 \, B b^{3} d x e^{11} - 180 \, A b^{2} c d x e^{11} + 20 \, A b^{3} x e^{12}\right )} e^{\left (-15\right )} - \frac{{\left (13 \, B c^{3} d^{7} - 33 \, B b c^{2} d^{6} e - 11 \, A c^{3} d^{6} e + 27 \, B b^{2} c d^{5} e^{2} + 27 \, A b c^{2} d^{5} e^{2} - 7 \, B b^{3} d^{4} e^{3} - 21 \, A b^{2} c d^{4} e^{3} + 5 \, A b^{3} d^{3} e^{4} + 2 \,{\left (7 \, B c^{3} d^{6} e - 18 \, B b c^{2} d^{5} e^{2} - 6 \, A c^{3} d^{5} e^{2} + 15 \, B b^{2} c d^{4} e^{3} + 15 \, A b c^{2} d^{4} e^{3} - 4 \, B b^{3} d^{3} e^{4} - 12 \, A b^{2} c d^{3} e^{4} + 3 \, A b^{3} d^{2} e^{5}\right )} x\right )} e^{\left (-8\right )}}{2 \,{\left (x e + d\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="giac")

[Out]

-3*(7*B*c^3*d^5 - 15*B*b*c^2*d^4*e - 5*A*c^3*d^4*e + 10*B*b^2*c*d^3*e^2 + 10*A*b*c^2*d^3*e^2 - 2*B*b^3*d^2*e^3
- 6*A*b^2*c*d^2*e^3 + A*b^3*d*e^4)*e^(-8)*log(abs(x*e + d)) + 1/20*(4*B*c^3*x^5*e^12 - 15*B*c^3*d*x^4*e^11 +
40*B*c^3*d^2*x^3*e^10 - 100*B*c^3*d^3*x^2*e^9 + 300*B*c^3*d^4*x*e^8 + 15*B*b*c^2*x^4*e^12 + 5*A*c^3*x^4*e^12 -
60*B*b*c^2*d*x^3*e^11 - 20*A*c^3*d*x^3*e^11 + 180*B*b*c^2*d^2*x^2*e^10 + 60*A*c^3*d^2*x^2*e^10 - 600*B*b*c^2*
d^3*x*e^9 - 200*A*c^3*d^3*x*e^9 + 20*B*b^2*c*x^3*e^12 + 20*A*b*c^2*x^3*e^12 - 90*B*b^2*c*d*x^2*e^11 - 90*A*b*c
^2*d*x^2*e^11 + 360*B*b^2*c*d^2*x*e^10 + 360*A*b*c^2*d^2*x*e^10 + 10*B*b^3*x^2*e^12 + 30*A*b^2*c*x^2*e^12 - 60
*B*b^3*d*x*e^11 - 180*A*b^2*c*d*x*e^11 + 20*A*b^3*x*e^12)*e^(-15) - 1/2*(13*B*c^3*d^7 - 33*B*b*c^2*d^6*e - 11*
A*c^3*d^6*e + 27*B*b^2*c*d^5*e^2 + 27*A*b*c^2*d^5*e^2 - 7*B*b^3*d^4*e^3 - 21*A*b^2*c*d^4*e^3 + 5*A*b^3*d^3*e^4
+ 2*(7*B*c^3*d^6*e - 18*B*b*c^2*d^5*e^2 - 6*A*c^3*d^5*e^2 + 15*B*b^2*c*d^4*e^3 + 15*A*b*c^2*d^4*e^3 - 4*B*b^3
*d^3*e^4 - 12*A*b^2*c*d^3*e^4 + 3*A*b^3*d^2*e^5)*x)*e^(-8)/(x*e + d)^2