3.2341 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=531 \[ -\frac{x \left (A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac{c x^3 \left (A c e (c d-b e)-B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{e^5}-\frac{x^2 \left (B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{2 e^6}-\frac{3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}+\frac{\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^8 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac{c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac{B c^3 x^5}{5 e^3} \]

[Out]

-(((A*e*(10*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*B*(5*c^3*d^4 - 2*c^2*
d^2*e*(5*b*d - 3*a*e) - b^2*e^3*(b*d - a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x)/e^7) - ((B*(10*c^3*
d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*A*c*e*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b
*d - a*e)))*x^2)/(2*e^6) - (c*(A*c*e*(c*d - b*e) - B*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*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) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(2*
e^8*(d + e*x)^2) + ((c*d^2 - b*d*e + a*e^2)^2*(3*A*e*(2*c*d - b*e) - B*(7*c*d^2 - e*(4*b*d - a*e))))/(e^8*(d +
e*x)) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*
d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*Log[d + e*x])/e^8

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Rubi [A]  time = 1.08488, antiderivative size = 530, normalized size of antiderivative = 1., 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} \[ -\frac{x \left (A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac{c x^3 \left (A c e (c d-b e)-B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{e^5}-\frac{x^2 \left (B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{2 e^6}-\frac{3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^8 (d+e x)}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac{c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac{B c^3 x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((A*e*(10*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*B*(5*c^3*d^4 - 2*c^2*
d^2*e*(5*b*d - 3*a*e) - b^2*e^3*(b*d - a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x)/e^7) - ((B*(10*c^3*
d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*A*c*e*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b
*d - a*e)))*x^2)/(2*e^6) - (c*(A*c*e*(c*d - b*e) - B*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*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) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(2*
e^8*(d + e*x)^2) - ((c*d^2 - b*d*e + a*e^2)^2*(7*B*c*d^2 - B*e*(4*b*d - a*e) - 3*A*e*(2*c*d - b*e)))/(e^8*(d +
e*x)) - (3*(c*d^2 - b*d*e + a*e^2)*(B*(7*c^2*d^3 - c*d*e*(8*b*d - 3*a*e) + b*e^2*(2*b*d - a*e)) - A*e*(5*c^2*
d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*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 (a+b x+c x^2\right )^3}{(d+e x)^3} \, dx &=\int \left (\frac{-A e \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )+3 B \left (5 c^3 d^4-2 c^2 d^2 e (5 b d-3 a e)-b^2 e^3 (b d-a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )}{e^7}+\frac{\left (-B \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )+3 A c e \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x}{e^6}+\frac{3 c \left (-A c e (c d-b e)+B \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) 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{(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^3}+\frac{\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7 (d+e x)^2}+\frac{3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac{\left (A e \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 B \left (5 c^3 d^4-2 c^2 d^2 e (5 b d-3 a e)-b^2 e^3 (b d-a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )\right ) x}{e^7}-\frac{\left (B \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 A c e \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^2}{2 e^6}-\frac{c \left (A c e (c d-b e)-B \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) 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{(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{2 e^8 (d+e x)^2}-\frac{\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^8 (d+e x)}-\frac{3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.340427, size = 503, normalized size = 0.95 \[ \frac{20 e x \left (3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )+A e \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )\right )+20 c e^3 x^3 \left (B \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+A c e (b e-c d)\right )+10 e^2 x^2 \left (3 A c e \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+B \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )\right )-60 \log (d+e x) \left (e (a e-b d)+c d^2\right ) \left (B \left (c d e (3 a e-8 b d)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )\right )-\frac{20 \left (e (a e-b d)+c d^2\right )^2 \left (B e (a e-4 b d)+3 A e (b e-2 c d)+7 B c d^2\right )}{d+e x}+\frac{10 (B d-A e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}+5 c^2 e^4 x^4 (A c e+3 b B e-3 B c d)+4 B c^3 e^5 x^5}{20 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e*(A*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e)) + 3*B*(5*c^3*d^4 + b
^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x + 10*e^2*(3
*A*c*e*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)) + B*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c
*e^2*(-3*b*d + 2*a*e)))*x^2 + 20*c*e^3*(A*c*e*(-(c*d) + b*e) + B*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*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*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))
^3)/(d + e*x)^2 - (20*(c*d^2 + e*(-(b*d) + a*e))^2*(7*B*c*d^2 + B*e*(-4*b*d + a*e) + 3*A*e*(-2*c*d + b*e)))/(d
+ e*x) - 60*(c*d^2 + e*(-(b*d) + a*e))*(-(A*e*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))) + B*(7*c^2*d^3 + b*
e^2*(2*b*d - a*e) + c*d*e*(-8*b*d + 3*a*e)))*Log[d + e*x])/(20*e^8)

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Maple [B]  time = 0.021, size = 1483, normalized size = 2.8 \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+a)^3/(e*x+d)^3,x)

[Out]

3/e^3*B*x^2*a*b*c+1/5*B*c^3*x^5/e^3+18/e^5*B*a*c^2*d^2*x+18/e^5*ln(e*x+d)*A*b^2*c*d^2-30/e^6*ln(e*x+d)*A*b*c^2
*d^3+18/e^5*ln(e*x+d)*A*a*c^2*d^2-9/e^4*ln(e*x+d)*B*a*b^2*d-30/e^6*ln(e*x+d)*B*a*c^2*d^3-30/e^6*ln(e*x+d)*B*b^
2*c*d^3+45/e^7*ln(e*x+d)*B*b*c^2*d^4+3/2/e^2/(e*x+d)^2*A*d*a^2*b-3/2/e^3/(e*x+d)^2*A*a^2*c*d^2-3/2/e^3/(e*x+d)
^2*A*d^2*a*b^2-3/2/e^5/(e*x+d)^2*A*a*c^2*d^4-3/2/e^5/(e*x+d)^2*A*b^2*c*d^4+3/2/e^6/(e*x+d)^2*A*b*c^2*d^5-3/2/e
^3/(e*x+d)^2*B*d^2*a^2*b+3/2/e^4/(e*x+d)^2*B*a^2*c*d^3+3/2/e^4/(e*x+d)^2*B*a*b^2*d^3+3/2/e^6/(e*x+d)^2*B*a*c^2
*d^5+3/2/e^6/(e*x+d)^2*B*b^2*c*d^5-9/2/e^4*B*x^2*a*c^2*d+6/e^3*A*a*b*c*x-9/e^4*A*a*c^2*d*x-9/e^4*A*b^2*c*d*x-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+18/e^5*A*x*b*c^2*d^2+18/e
^5*B*x*b^2*c*d^2-30/e^6*B*x*b*c^2*d^3+18/e^7/(e*x+d)*B*b*c^2*d^5-9/e^4/(e*x+d)*B*a^2*c*d^2-9/e^4/(e*x+d)*B*a*b
^2*d^2-15/e^6/(e*x+d)*B*a*c^2*d^4-15/e^6/(e*x+d)*B*b^2*c*d^4+12/e^5/(e*x+d)*A*a*c^2*d^3+12/e^5/(e*x+d)*A*b^2*c
*d^3-15/e^6/(e*x+d)*A*b*c^2*d^4+6/e^3/(e*x+d)*B*a^2*b*d+6/e^3/(e*x+d)*A*a*b^2*d+6/e^3/(e*x+d)*A*a^2*c*d-3/2/e^
7/(e*x+d)^2*B*b*c^2*d^6-9/e^4*ln(e*x+d)*B*a^2*c*d-1/2/e/(e*x+d)^2*A*a^3-1/e^2/(e*x+d)*B*a^3+1/e^3*A*b^3*x+1/2/
e^3*B*x^2*b^3+1/4/e^3*A*x^4*c^3+1/e^3*B*x^3*a*c^2+1/e^3*B*x^3*b^2*c+3/2/e^3*A*x^2*a*c^2+3/4/e^3*B*x^4*b*c^2+3/
e^3*ln(e*x+d)*A*a^2*c+3/e^3*ln(e*x+d)*A*a*b^2-3/e^4*ln(e*x+d)*A*b^3*d+15/e^7*ln(e*x+d)*A*c^3*d^4+3/e^3*ln(e*x+
d)*B*a^2*b+6/e^5*ln(e*x+d)*B*b^3*d^2-21/e^8*ln(e*x+d)*B*c^3*d^5+1/2/e^4/(e*x+d)^2*A*b^3*d^3-1/2/e^7/(e*x+d)^2*
A*c^3*d^6+1/2/e^2/(e*x+d)^2*B*a^3*d-1/2/e^5/(e*x+d)^2*B*b^3*d^4+1/2/e^8/(e*x+d)^2*B*c^3*d^7-3/e^2/(e*x+d)*A*a^
2*b-3/e^4/(e*x+d)*A*b^3*d^2+6/e^7/(e*x+d)*A*c^3*d^5+4/e^5/(e*x+d)*B*b^3*d^3-7/e^8/(e*x+d)*B*c^3*d^6+15/e^7*B*c
^3*d^4*x+1/e^3*A*x^3*b*c^2-1/e^4*A*x^3*c^3*d+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/e^4*B*b^3*d*x-10/e^6*A*c^3*d^3*x+3/e^3*B*a^2*c*x+3/e^3*B*a*b^2*x-3/4/e^4*B*x^4*c^3*d+36/e^5*l
n(e*x+d)*B*a*b*c*d^2+3/e^4/(e*x+d)^2*A*a*b*c*d^3-3/e^5/(e*x+d)^2*B*a*b*c*d^4-18/e^4/(e*x+d)*A*a*b*c*d^2+24/e^5
/(e*x+d)*B*a*b*c*d^3-18/e^4*B*a*b*c*d*x-18/e^4*ln(e*x+d)*A*a*b*c*d

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Maxima [A]  time = 1.07164, size = 1162, normalized size = 2.19 \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+a)^3/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.10852, size = 2790, normalized size = 5.25 \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+a)^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 - 10*A*a^3*e^7 + 110*(3*B*b*c^2 + A*c^3)*d^6*e - 270*(B*b^2*c + (B*a + A
*b)*c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 - 50*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 +
2*A*a*b)*c)*d^3*e^4 + 90*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - 10*(B*a^3 + 3*A*a^2*b)*d*e^6 - (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 + (B*a + 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 + (B*a + A*b)*c^2)*d*e^6
- 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 - 5*(3*B*b*c^2 + A*c^3)*d^3*e^4
+ 10*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + (3*B*a*b^2 +
A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 10*(50*B*c^3*d^5*e^2 - 34*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c
+ (B*a + A*b)*c^2)*d^3*e^4 - 11*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + 4*(3*B*a*b^2 + A*b^3 +
3*(B*a^2 + 2*A*a*b)*c)*d*e^6)*x^2 + 20*(8*B*c^3*d^6*e - 4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c + (B*a + A*
b)*c^2)*d^4*e^3 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 - 2*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*
a*b)*c)*d^2*e^5 + 6*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 - (B*a^3 + 3*A*a^2*b)*e^7)*x - 60*(7*B*c^3*d^7 - 5*(3*
B*b*c^2 + A*c^3)*d^6*e + 10*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c
)*d^4*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - (B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (7*B
*c^3*d^5*e^2 - 5*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 10*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 2*(B*b^3 + 3*A*a*c^2 +
3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - (B*a^2*b + A*a*b^2 + A*a
^2*c)*e^7)*x^2 + 2*(7*B*c^3*d^6*e - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 10*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 2
*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^5 - (
B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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Sympy [B]  time = 165.701, size = 1124, normalized size = 2.12 \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+a)**3/(e*x+d)**3,x)

[Out]

B*c**3*x**5/(5*e**3) - (A*a**3*e**7 + 3*A*a**2*b*d*e**6 - 9*A*a**2*c*d**2*e**5 - 9*A*a*b**2*d**2*e**5 + 30*A*a
*b*c*d**3*e**4 - 21*A*a*c**2*d**4*e**3 + 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 + B*a**3*d*e**6 - 9*B*a**2*b*d**2*e**5 + 15*B*a**2*c*d**3*e**4 + 15*B*a*b**2*d**3*e**4 - 42*B
*a*b*c*d**4*e**3 + 27*B*a*c**2*d**5*e**2 - 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 + 1
3*B*c**3*d**7 + x*(6*A*a**2*b*e**7 - 12*A*a**2*c*d*e**6 - 12*A*a*b**2*d*e**6 + 36*A*a*b*c*d**2*e**5 - 24*A*a*c
**2*d**3*e**4 + 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 + 2*B
*a**3*e**7 - 12*B*a**2*b*d*e**6 + 18*B*a**2*c*d**2*e**5 + 18*B*a*b**2*d**2*e**5 - 48*B*a*b*c*d**3*e**4 + 30*B*
a*c**2*d**4*e**3 - 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**10*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*a*c**2*e**2 + 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*a*c**2*e
**3 + 3*A*b**2*c*e**3 - 9*A*b*c**2*d*e**2 + 6*A*c**3*d**2*e + 6*B*a*b*c*e**3 - 9*B*a*c**2*d*e**2 + 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*(6*A*a*b*c*e**4 - 9*A*a*c**2*d*e**3 +
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*a**2*c*e**4 + 3*B*a*b**2*e**
4 - 18*B*a*b*c*d*e**3 + 18*B*a*c**2*d**2*e**2 - 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 + 3*(a*e**2 - b*d*e + c*d**2)*(A*a*c*e**3 + A*b**2*e**3 - 5*A*b*c*d*e**2 + 5*A*c**2*d**2
*e + B*a*b*e**3 - 3*B*a*c*d*e**2 - 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

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Giac [B]  time = 1.13003, size = 1415, normalized size = 2.66 \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+a)^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*B*a*c^2*d^3*e^2 + 10*A*b*c^2*d^3*
e^2 - 2*B*b^3*d^2*e^3 - 12*B*a*b*c*d^2*e^3 - 6*A*b^2*c*d^2*e^3 - 6*A*a*c^2*d^2*e^3 + 3*B*a*b^2*d*e^4 + A*b^3*d
*e^4 + 3*B*a^2*c*d*e^4 + 6*A*a*b*c*d*e^4 - B*a^2*b*e^5 - A*a*b^2*e^5 - A*a^2*c*e^5)*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*B*a
*c^2*x^3*e^12 + 20*A*b*c^2*x^3*e^12 - 90*B*b^2*c*d*x^2*e^11 - 90*B*a*c^2*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*B*a*c^2*d^2*x*e^10 + 360*A*b*c^2*d^2*x*e^10 + 10*B*b^3*x^2*e^12 + 60*B*a*b*c*x^2*
e^12 + 30*A*b^2*c*x^2*e^12 + 30*A*a*c^2*x^2*e^12 - 60*B*b^3*d*x*e^11 - 360*B*a*b*c*d*x*e^11 - 180*A*b^2*c*d*x*
e^11 - 180*A*a*c^2*d*x*e^11 + 60*B*a*b^2*x*e^12 + 20*A*b^3*x*e^12 + 60*B*a^2*c*x*e^12 + 120*A*a*b*c*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*B*a*c^2*d^5*e^2 + 27*A
*b*c^2*d^5*e^2 - 7*B*b^3*d^4*e^3 - 42*B*a*b*c*d^4*e^3 - 21*A*b^2*c*d^4*e^3 - 21*A*a*c^2*d^4*e^3 + 15*B*a*b^2*d
^3*e^4 + 5*A*b^3*d^3*e^4 + 15*B*a^2*c*d^3*e^4 + 30*A*a*b*c*d^3*e^4 - 9*B*a^2*b*d^2*e^5 - 9*A*a*b^2*d^2*e^5 - 9
*A*a^2*c*d^2*e^5 + B*a^3*d*e^6 + 3*A*a^2*b*d*e^6 + A*a^3*e^7 + 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*B*a*c^2*d^4*e^3 + 15*A*b*c^2*d^4*e^3 - 4*B*b^3*d^3*e^4 - 24*B*a*b*c*d^3*e^4
- 12*A*b^2*c*d^3*e^4 - 12*A*a*c^2*d^3*e^4 + 9*B*a*b^2*d^2*e^5 + 3*A*b^3*d^2*e^5 + 9*B*a^2*c*d^2*e^5 + 18*A*a*
b*c*d^2*e^5 - 6*B*a^2*b*d*e^6 - 6*A*a*b^2*d*e^6 - 6*A*a^2*c*d*e^6 + B*a^3*e^7 + 3*A*a^2*b*e^7)*x)*e^(-8)/(x*e
+ d)^2