3.2340 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{(d+e x)^2} \, dx\)

Optimal. Leaf size=525 \[ -\frac{x^2 \left (A e (c d-b e) \left (-c e (5 b d-6 a e)+b^2 e^2+4 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-b^2 e^3 (2 b d-3 a e)-3 c^2 d^2 e (4 b d-3 a e)+5 c^3 d^4\right )\right )}{2 e^6}-\frac{x \left (3 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-A e \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-b^2 e^3 (2 b d-3 a e)-3 c^2 d^2 e (4 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac{c x^4 \left (A c e (2 c d-3 b e)-3 B \left (-c e (2 b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{4 e^4}-\frac{x^3 \left (B (c d-b e) \left (-c e (5 b d-6 a e)+b^2 e^2+4 c^2 d^2\right )-3 A c e \left (-c e (2 b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{3 e^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)}-\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^8}-\frac{c^2 x^5 (-A c e-3 b B e+2 B c d)}{5 e^3}+\frac{B c^3 x^6}{6 e^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.09344, antiderivative size = 522, 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} \[ -\frac{x^2 \left (A e (c d-b e) \left (-c e (5 b d-6 a e)+b^2 e^2+4 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-b^2 e^3 (2 b d-3 a e)-3 c^2 d^2 e (4 b d-3 a e)+5 c^3 d^4\right )\right )}{2 e^6}-\frac{x \left (3 B (2 c d-b e) \left (c d^2-e (b d-a e)\right )^2-A e \left (3 c e^2 \left (a^2 e^2-4 a b d e+3 b^2 d^2\right )-b^2 e^3 (2 b d-3 a e)-3 c^2 d^2 e (4 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac{c x^4 \left (A c e (2 c d-3 b e)-3 B \left (-c e (2 b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{4 e^4}-\frac{x^3 \left (B (c d-b e) \left (-c e (5 b d-6 a e)+b^2 e^2+4 c^2 d^2\right )-3 A c e \left (-c e (2 b d-a e)+b^2 e^2+c^2 d^2\right )\right )}{3 e^5}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{e^8 (d+e x)}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^8}-\frac{c^2 x^5 (-A c e-3 b B e+2 B c d)}{5 e^3}+\frac{B c^3 x^6}{6 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.687577, size = 922, normalized size = 1.76 \[ \frac{60 \left (7 B c d^2+B e (a e-4 b d)+3 A e (b e-2 c d)\right ) (d+e x) \log (d+e x) \left (c d^2+e (a e-b d)\right )^2+3 A e \left (\left (-20 d^6+100 e x d^5+60 e^2 x^2 d^4-20 e^3 x^3 d^3+10 e^4 x^4 d^2-6 e^5 x^5 d+4 e^6 x^6\right ) c^3+5 e \left (4 a e \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right )+b \left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right )\right ) c^2+20 e^2 \left (\left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right ) b^2+3 a e \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right ) b+3 a^2 e^2 \left (-d^2+e x d+e^2 x^2\right )\right ) c+10 e^3 \left (\left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right ) b^3+6 a e \left (-d^2+e x d+e^2 x^2\right ) b^2+6 a^2 d e^2 b-2 a^3 e^3\right )\right )+B \left (\left (60 d^7-360 e x d^6-210 e^2 x^2 d^5+70 e^3 x^3 d^4-35 e^4 x^4 d^3+21 e^5 x^5 d^2-14 e^6 x^6 d+10 e^7 x^7\right ) c^3+3 e \left (5 a e \left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right )-6 b \left (10 d^6-50 e x d^5-30 e^2 x^2 d^4+10 e^3 x^3 d^3-5 e^4 x^4 d^2+3 e^5 x^5 d-2 e^6 x^6\right )\right ) c^2+15 e^2 \left (\left (12 d^5-48 e x d^4-30 e^2 x^2 d^3+10 e^3 x^3 d^2-5 e^4 x^4 d+3 e^5 x^5\right ) b^2+8 a e \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right ) b+6 a^2 e^2 \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right )\right ) c+10 e^3 \left (2 \left (-3 d^4+9 e x d^3+6 e^2 x^2 d^2-2 e^3 x^3 d+e^4 x^4\right ) b^3+9 a e \left (2 d^3-4 e x d^2-3 e^2 x^2 d+e^3 x^3\right ) b^2+18 a^2 e^2 \left (-d^2+e x d+e^2 x^2\right ) b+6 a^3 d e^3\right )\right )}{60 e^8 (d+e x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*A*e*(c^3*(-20*d^6 + 100*d^5*e*x + 60*d^4*e^2*x^2 - 20*d^3*e^3*x^3 + 10*d^2*e^4*x^4 - 6*d*e^5*x^5 + 4*e^6*x^
6) + 10*e^3*(6*a^2*b*d*e^2 - 2*a^3*e^3 + 6*a*b^2*e*(-d^2 + d*e*x + e^2*x^2) + b^3*(2*d^3 - 4*d^2*e*x - 3*d*e^2
*x^2 + e^3*x^3)) + 20*c*e^2*(3*a^2*e^2*(-d^2 + d*e*x + e^2*x^2) + 3*a*b*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e
^3*x^3) + b^2*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4)) + 5*c^2*e*(4*a*e*(-3*d^4 + 9*d^3*e
*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e
^4*x^4 + 3*e^5*x^5))) + B*(c^3*(60*d^7 - 360*d^6*e*x - 210*d^5*e^2*x^2 + 70*d^4*e^3*x^3 - 35*d^3*e^4*x^4 + 21*
d^2*e^5*x^5 - 14*d*e^6*x^6 + 10*e^7*x^7) + 10*e^3*(6*a^3*d*e^3 + 18*a^2*b*e^2*(-d^2 + d*e*x + e^2*x^2) + 9*a*b
^2*e*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 2*b^3*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 - 2*d*e^3*x^3 + e
^4*x^4)) + 15*c*e^2*(6*a^2*e^2*(2*d^3 - 4*d^2*e*x - 3*d*e^2*x^2 + e^3*x^3) + 8*a*b*e*(-3*d^4 + 9*d^3*e*x + 6*d
^2*e^2*x^2 - 2*d*e^3*x^3 + e^4*x^4) + b^2*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4
+ 3*e^5*x^5)) + 3*c^2*e*(5*a*e*(12*d^5 - 48*d^4*e*x - 30*d^3*e^2*x^2 + 10*d^2*e^3*x^3 - 5*d*e^4*x^4 + 3*e^5*x
^5) - 6*b*(10*d^6 - 50*d^5*e*x - 30*d^4*e^2*x^2 + 10*d^3*e^3*x^3 - 5*d^2*e^4*x^4 + 3*d*e^5*x^5 - 2*e^6*x^6)))
+ 60*(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)*Log[d + e*
x])/(60*e^8*(d + e*x))

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Maple [B]  time = 0.017, size = 1404, normalized size = 2.7 \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)^2,x)

[Out]

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

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Maxima [A]  time = 1.04648, size = 1152, 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)^2,x, algorithm="maxima")

[Out]

(B*c^3*d^7 - A*a^3*e^7 - (3*B*b*c^2 + A*c^3)*d^6*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^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 - 3*(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)/(e^9*x + d*e^8) + 1/60*(10*B*c^3*e^5*x^6 - 12*(2*B*c^3*d*e^4
- (3*B*b*c^2 + A*c^3)*e^5)*x^5 + 15*(3*B*c^3*d^2*e^3 - 2*(3*B*b*c^2 + A*c^3)*d*e^4 + 3*(B*b^2*c + (B*a + A*b)
*c^2)*e^5)*x^4 - 20*(4*B*c^3*d^3*e^2 - 3*(3*B*b*c^2 + A*c^3)*d^2*e^3 + 6*(B*b^2*c + (B*a + A*b)*c^2)*d*e^4 - (
B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^5)*x^3 + 30*(5*B*c^3*d^4*e - 4*(3*B*b*c^2 + A*c^3)*d^3*e^2 + 9*(B
*b^2*c + (B*a + A*b)*c^2)*d^2*e^3 - 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^4 + (3*B*a*b^2 + A*b^3 +
3*(B*a^2 + 2*A*a*b)*c)*e^5)*x^2 - 60*(6*B*c^3*d^5 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 12*(B*b^2*c + (B*a + A*b)*c
^2)*d^3*e^2 - 3*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^3 + 2*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*
b)*c)*d*e^4 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^5)*x)/e^7 + (7*B*c^3*d^6 - 6*(3*B*b*c^2 + A*c^3)*d^5*e + 15*(B
*b^2*c + (B*a + A*b)*c^2)*d^4*e^2 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^3 + 3*(3*B*a*b^2 + A*b
^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^4 - 6*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^5 + (B*a^3 + 3*A*a^2*b)*e^6)*log(e*x
+ d)/e^8

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Fricas [B]  time = 1.17571, size = 2545, normalized size = 4.85 \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)^2,x, algorithm="fricas")

[Out]

1/60*(10*B*c^3*e^7*x^7 + 60*B*c^3*d^7 - 60*A*a^3*e^7 - 60*(3*B*b*c^2 + A*c^3)*d^6*e + 180*(B*b^2*c + (B*a + A*
b)*c^2)*d^5*e^2 - 60*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 60*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 +
2*A*a*b)*c)*d^3*e^4 - 180*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + 60*(B*a^3 + 3*A*a^2*b)*d*e^6 - 2*(7*B*c^3*d*
e^6 - 6*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7*B*c^3*d^2*e^5 - 6*(3*B*b*c^2 + A*c^3)*d*e^6 + 15*(B*b^2*c + (B*a +
A*b)*c^2)*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 6*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d*e^
6 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 10*(7*B*c^3*d^4*e^3 - 6*(3*B*b*c^2 + A*c^3)*d^3*e
^4 + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6 + 3*(3*B*a*b
^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 - 30*(7*B*c^3*d^5*e^2 - 6*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 15*(B*b^2
*c + (B*a + A*b)*c^2)*d^3*e^4 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + 3*(3*B*a*b^2 + A*b^3 +
3*(B*a^2 + 2*A*a*b)*c)*d*e^6 - 6*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 - 60*(6*B*c^3*d^6*e - 5*(3*B*b*c^2 +
A*c^3)*d^5*e^2 + 12*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 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 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6)*x + 60*(7*B
*c^3*d^7 - 6*(3*B*b*c^2 + A*c^3)*d^6*e + 15*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*
B*a*b + A*b^2)*c)*d^4*e^3 + 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 6*(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 + (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)*log(e*x + d))/(e
^9*x + d*e^8)

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Sympy [B]  time = 12.9959, size = 1023, normalized size = 1.95 \begin{align*} \frac{B c^{3} x^{6}}{6 e^{2}} + \frac{- A a^{3} e^{7} + 3 A a^{2} b d e^{6} - 3 A a^{2} c d^{2} e^{5} - 3 A a b^{2} d^{2} e^{5} + 6 A a b c d^{3} e^{4} - 3 A a c^{2} d^{4} e^{3} + A b^{3} d^{3} e^{4} - 3 A b^{2} c d^{4} e^{3} + 3 A b c^{2} d^{5} e^{2} - A c^{3} d^{6} e + B a^{3} d e^{6} - 3 B a^{2} b d^{2} e^{5} + 3 B a^{2} c d^{3} e^{4} + 3 B a b^{2} d^{3} e^{4} - 6 B a b c d^{4} e^{3} + 3 B a c^{2} d^{5} e^{2} - B b^{3} d^{4} e^{3} + 3 B b^{2} c d^{5} e^{2} - 3 B b c^{2} d^{6} e + B c^{3} d^{7}}{d e^{8} + e^{9} x} + \frac{x^{5} \left (A c^{3} e + 3 B b c^{2} e - 2 B c^{3} d\right )}{5 e^{3}} + \frac{x^{4} \left (3 A b c^{2} e^{2} - 2 A c^{3} d e + 3 B a c^{2} e^{2} + 3 B b^{2} c e^{2} - 6 B b c^{2} d e + 3 B c^{3} d^{2}\right )}{4 e^{4}} + \frac{x^{3} \left (3 A a c^{2} e^{3} + 3 A b^{2} c e^{3} - 6 A b c^{2} d e^{2} + 3 A c^{3} d^{2} e + 6 B a b c e^{3} - 6 B a c^{2} d e^{2} + B b^{3} e^{3} - 6 B b^{2} c d e^{2} + 9 B b c^{2} d^{2} e - 4 B c^{3} d^{3}\right )}{3 e^{5}} + \frac{x^{2} \left (6 A a b c e^{4} - 6 A a c^{2} d e^{3} + A b^{3} e^{4} - 6 A b^{2} c d e^{3} + 9 A b c^{2} d^{2} e^{2} - 4 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 3 B a b^{2} e^{4} - 12 B a b c d e^{3} + 9 B a c^{2} d^{2} e^{2} - 2 B b^{3} d e^{3} + 9 B b^{2} c d^{2} e^{2} - 12 B b c^{2} d^{3} e + 5 B c^{3} d^{4}\right )}{2 e^{6}} + \frac{x \left (3 A a^{2} c e^{5} + 3 A a b^{2} e^{5} - 12 A a b c d e^{4} + 9 A a c^{2} d^{2} e^{3} - 2 A b^{3} d e^{4} + 9 A b^{2} c d^{2} e^{3} - 12 A b c^{2} d^{3} e^{2} + 5 A c^{3} d^{4} e + 3 B a^{2} b e^{5} - 6 B a^{2} c d e^{4} - 6 B a b^{2} d e^{4} + 18 B a b c d^{2} e^{3} - 12 B a c^{2} d^{3} e^{2} + 3 B b^{3} d^{2} e^{3} - 12 B b^{2} c d^{3} e^{2} + 15 B b c^{2} d^{4} e - 6 B c^{3} d^{5}\right )}{e^{7}} + \frac{\left (a e^{2} - b d e + c d^{2}\right )^{2} \left (3 A b e^{2} - 6 A c d e + B a e^{2} - 4 B b d e + 7 B c d^{2}\right ) \log{\left (d + e x \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+a)**3/(e*x+d)**2,x)

[Out]

B*c**3*x**6/(6*e**2) + (-A*a**3*e**7 + 3*A*a**2*b*d*e**6 - 3*A*a**2*c*d**2*e**5 - 3*A*a*b**2*d**2*e**5 + 6*A*a
*b*c*d**3*e**4 - 3*A*a*c**2*d**4*e**3 + A*b**3*d**3*e**4 - 3*A*b**2*c*d**4*e**3 + 3*A*b*c**2*d**5*e**2 - A*c**
3*d**6*e + B*a**3*d*e**6 - 3*B*a**2*b*d**2*e**5 + 3*B*a**2*c*d**3*e**4 + 3*B*a*b**2*d**3*e**4 - 6*B*a*b*c*d**4
*e**3 + 3*B*a*c**2*d**5*e**2 - B*b**3*d**4*e**3 + 3*B*b**2*c*d**5*e**2 - 3*B*b*c**2*d**6*e + B*c**3*d**7)/(d*e
**8 + e**9*x) + x**5*(A*c**3*e + 3*B*b*c**2*e - 2*B*c**3*d)/(5*e**3) + x**4*(3*A*b*c**2*e**2 - 2*A*c**3*d*e +
3*B*a*c**2*e**2 + 3*B*b**2*c*e**2 - 6*B*b*c**2*d*e + 3*B*c**3*d**2)/(4*e**4) + x**3*(3*A*a*c**2*e**3 + 3*A*b**
2*c*e**3 - 6*A*b*c**2*d*e**2 + 3*A*c**3*d**2*e + 6*B*a*b*c*e**3 - 6*B*a*c**2*d*e**2 + B*b**3*e**3 - 6*B*b**2*c
*d*e**2 + 9*B*b*c**2*d**2*e - 4*B*c**3*d**3)/(3*e**5) + x**2*(6*A*a*b*c*e**4 - 6*A*a*c**2*d*e**3 + A*b**3*e**4
- 6*A*b**2*c*d*e**3 + 9*A*b*c**2*d**2*e**2 - 4*A*c**3*d**3*e + 3*B*a**2*c*e**4 + 3*B*a*b**2*e**4 - 12*B*a*b*c
*d*e**3 + 9*B*a*c**2*d**2*e**2 - 2*B*b**3*d*e**3 + 9*B*b**2*c*d**2*e**2 - 12*B*b*c**2*d**3*e + 5*B*c**3*d**4)/
(2*e**6) + x*(3*A*a**2*c*e**5 + 3*A*a*b**2*e**5 - 12*A*a*b*c*d*e**4 + 9*A*a*c**2*d**2*e**3 - 2*A*b**3*d*e**4 +
9*A*b**2*c*d**2*e**3 - 12*A*b*c**2*d**3*e**2 + 5*A*c**3*d**4*e + 3*B*a**2*b*e**5 - 6*B*a**2*c*d*e**4 - 6*B*a*
b**2*d*e**4 + 18*B*a*b*c*d**2*e**3 - 12*B*a*c**2*d**3*e**2 + 3*B*b**3*d**2*e**3 - 12*B*b**2*c*d**3*e**2 + 15*B
*b*c**2*d**4*e - 6*B*c**3*d**5)/e**7 + (a*e**2 - b*d*e + c*d**2)**2*(3*A*b*e**2 - 6*A*c*d*e + B*a*e**2 - 4*B*b
*d*e + 7*B*c*d**2)*log(d + e*x)/e**8

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Giac [B]  time = 1.18401, size = 1620, normalized size = 3.09 \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)^2,x, algorithm="giac")

[Out]

1/60*(10*B*c^3 - 12*(7*B*c^3*d*e - 3*B*b*c^2*e^2 - A*c^3*e^2)*e^(-1)/(x*e + d) + 45*(7*B*c^3*d^2*e^2 - 6*B*b*c
^2*d*e^3 - 2*A*c^3*d*e^3 + B*b^2*c*e^4 + B*a*c^2*e^4 + A*b*c^2*e^4)*e^(-2)/(x*e + d)^2 - 20*(35*B*c^3*d^3*e^3
- 45*B*b*c^2*d^2*e^4 - 15*A*c^3*d^2*e^4 + 15*B*b^2*c*d*e^5 + 15*B*a*c^2*d*e^5 + 15*A*b*c^2*d*e^5 - B*b^3*e^6 -
6*B*a*b*c*e^6 - 3*A*b^2*c*e^6 - 3*A*a*c^2*e^6)*e^(-3)/(x*e + d)^3 + 30*(35*B*c^3*d^4*e^4 - 60*B*b*c^2*d^3*e^5
- 20*A*c^3*d^3*e^5 + 30*B*b^2*c*d^2*e^6 + 30*B*a*c^2*d^2*e^6 + 30*A*b*c^2*d^2*e^6 - 4*B*b^3*d*e^7 - 24*B*a*b*
c*d*e^7 - 12*A*b^2*c*d*e^7 - 12*A*a*c^2*d*e^7 + 3*B*a*b^2*e^8 + A*b^3*e^8 + 3*B*a^2*c*e^8 + 6*A*a*b*c*e^8)*e^(
-4)/(x*e + d)^4 - 180*(7*B*c^3*d^5*e^5 - 15*B*b*c^2*d^4*e^6 - 5*A*c^3*d^4*e^6 + 10*B*b^2*c*d^3*e^7 + 10*B*a*c^
2*d^3*e^7 + 10*A*b*c^2*d^3*e^7 - 2*B*b^3*d^2*e^8 - 12*B*a*b*c*d^2*e^8 - 6*A*b^2*c*d^2*e^8 - 6*A*a*c^2*d^2*e^8
+ 3*B*a*b^2*d*e^9 + A*b^3*d*e^9 + 3*B*a^2*c*d*e^9 + 6*A*a*b*c*d*e^9 - B*a^2*b*e^10 - A*a*b^2*e^10 - A*a^2*c*e^
10)*e^(-5)/(x*e + d)^5)*(x*e + d)^6*e^(-8) - (7*B*c^3*d^6 - 18*B*b*c^2*d^5*e - 6*A*c^3*d^5*e + 15*B*b^2*c*d^4*
e^2 + 15*B*a*c^2*d^4*e^2 + 15*A*b*c^2*d^4*e^2 - 4*B*b^3*d^3*e^3 - 24*B*a*b*c*d^3*e^3 - 12*A*b^2*c*d^3*e^3 - 12
*A*a*c^2*d^3*e^3 + 9*B*a*b^2*d^2*e^4 + 3*A*b^3*d^2*e^4 + 9*B*a^2*c*d^2*e^4 + 18*A*a*b*c*d^2*e^4 - 6*B*a^2*b*d*
e^5 - 6*A*a*b^2*d*e^5 - 6*A*a^2*c*d*e^5 + B*a^3*e^6 + 3*A*a^2*b*e^6)*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^
2) + (B*c^3*d^7*e^6/(x*e + d) - 3*B*b*c^2*d^6*e^7/(x*e + d) - A*c^3*d^6*e^7/(x*e + d) + 3*B*b^2*c*d^5*e^8/(x*e
+ d) + 3*B*a*c^2*d^5*e^8/(x*e + d) + 3*A*b*c^2*d^5*e^8/(x*e + d) - B*b^3*d^4*e^9/(x*e + d) - 6*B*a*b*c*d^4*e^
9/(x*e + d) - 3*A*b^2*c*d^4*e^9/(x*e + d) - 3*A*a*c^2*d^4*e^9/(x*e + d) + 3*B*a*b^2*d^3*e^10/(x*e + d) + A*b^3
*d^3*e^10/(x*e + d) + 3*B*a^2*c*d^3*e^10/(x*e + d) + 6*A*a*b*c*d^3*e^10/(x*e + d) - 3*B*a^2*b*d^2*e^11/(x*e +
d) - 3*A*a*b^2*d^2*e^11/(x*e + d) - 3*A*a^2*c*d^2*e^11/(x*e + d) + B*a^3*d*e^12/(x*e + d) + 3*A*a^2*b*d*e^12/(
x*e + d) - A*a^3*e^13/(x*e + d))*e^(-14)