3.2339 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{d+e x} \, dx\)

Optimal. Leaf size=544 \[ -\frac{(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac{3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac{(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}-\frac{x \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^7}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac{c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac{B c^3 (d+e x)^7}{7 e^8} \]

[Out]

-(((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)))*x)/e^7) - (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)))*(d + e*x)^2)/(2*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^
3*d^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d
+ e*x)^3)/(3*e^8) - ((B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c
*e*(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^4)/(4*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2
+ b^2*e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^5)/(5*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6)/(6*e^8) +
(B*c^3*(d + e*x)^7)/(7*e^8) - ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/e^8

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Rubi [A]  time = 1.19474, antiderivative size = 541, 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} \[ -\frac{(d+e x)^3 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{3 e^8}-\frac{3 c (d+e x)^5 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{5 e^8}-\frac{(d+e x)^4 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8}-\frac{3 (d+e x)^2 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8}+\frac{x \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{e^7}-\frac{(B d-A e) \log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^8}-\frac{c^2 (d+e x)^6 (-A c e-3 b B e+7 B c d)}{6 e^8}+\frac{B c^3 (d+e x)^7}{7 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((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))*x)/e^7 - (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)))*(d + e*x)^2)/(2*e^8) - ((A*e*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 - 2*c*e*(5*b*d - 3*a*e)) - B*(35*c^3*d
^4 - b^2*e^3*(4*b*d - 3*a*e) - 30*c^2*d^2*e*(2*b*d - a*e) + 3*c*e^2*(10*b^2*d^2 - 8*a*b*d*e + a^2*e^2)))*(d +
e*x)^3)/(3*e^8) - ((B*(35*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(5*b*d - 2*a*e) - 15*c^2*d*e*(3*b*d - a*e)) - 3*A*c*e*
(5*c^2*d^2 + b^2*e^2 - c*e*(5*b*d - a*e)))*(d + e*x)^4)/(4*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b
^2*e^2 - c*e*(6*b*d - a*e)))*(d + e*x)^5)/(5*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6)/(6*e^8) + (B
*c^3*(d + e*x)^7)/(7*e^8) - ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*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} \, dx &=\int \left (\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}+\frac{(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (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 ) (d+e x)}{e^7}+\frac{\left (-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^2}{e^7}+\frac{\left (-B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^3}{e^7}+\frac{3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^4}{e^7}+\frac{c^2 (-7 B c d+3 b B e+A c e) (d+e x)^5}{e^7}+\frac{B c^3 (d+e x)^6}{e^7}\right ) \, dx\\ &=\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 ) x}{e^7}-\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 ) (d+e x)^2}{2 e^8}-\frac{\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) (d+e x)^3}{3 e^8}-\frac{\left (B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right ) (d+e x)^4}{4 e^8}-\frac{3 c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right ) (d+e x)^5}{5 e^8}-\frac{c^2 (7 B c d-3 b B e-A c e) (d+e x)^6}{6 e^8}+\frac{B c^3 (d+e x)^7}{7 e^8}-\frac{(B d-A e) \left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^8}\\ \end{align*}

Mathematica [A]  time = 0.512166, size = 700, normalized size = 1.29 \[ \frac{e x \left (7 A e \left (15 c e^2 \left (6 a^2 e^2 (e x-2 d)+4 a b e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^2 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+10 b e^3 \left (18 a^2 e^2+9 a b e (e x-2 d)+b^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+3 c^2 e \left (5 a e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+b \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+c^3 \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+B \left (21 c e^2 \left (10 a^2 e^2 \left (6 d^2-3 d e x+2 e^2 x^2\right )+10 a b e \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )+b^2 \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )\right )+35 e^3 \left (18 a^2 b e^2 (e x-2 d)+12 a^3 e^3+6 a b^2 e \left (6 d^2-3 d e x+2 e^2 x^2\right )+b^3 \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+21 c^2 e \left (a e \left (20 d^2 e^2 x^2-30 d^3 e x+60 d^4-15 d e^3 x^3+12 e^4 x^4\right )+b \left (-20 d^3 e^2 x^2+15 d^2 e^3 x^3+30 d^4 e x-60 d^5-12 d e^4 x^4+10 e^5 x^5\right )\right )+c^3 \left (140 d^4 e^2 x^2-105 d^3 e^3 x^3+84 d^2 e^4 x^4-210 d^5 e x+420 d^6-70 d e^5 x^5+60 e^6 x^6\right )\right )\right )-420 (B d-A e) \log (d+e x) \left (e (a e-b d)+c d^2\right )^3}{420 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(7*A*e*(c^3*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + 10*b*e
^3*(18*a^2*e^2 + 9*a*b*e*(-2*d + e*x) + b^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2)) + 15*c*e^2*(6*a^2*e^2*(-2*d + e*x)
+ 4*a*b*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + b^2*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 3*c^2*e*(5*a*
e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 1
2*e^4*x^4))) + B*(c^3*(420*d^6 - 210*d^5*e*x + 140*d^4*e^2*x^2 - 105*d^3*e^3*x^3 + 84*d^2*e^4*x^4 - 70*d*e^5*x
^5 + 60*e^6*x^6) + 35*e^3*(12*a^3*e^3 + 18*a^2*b*e^2*(-2*d + e*x) + 6*a*b^2*e*(6*d^2 - 3*d*e*x + 2*e^2*x^2) +
b^3*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3)) + 21*c*e^2*(10*a^2*e^2*(6*d^2 - 3*d*e*x + 2*e^2*x^2) + 10
*a*b*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + b^2*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*
x^3 + 12*e^4*x^4)) + 21*c^2*e*(a*e*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) + b*(-60
*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5)))) - 420*(B*d - A*e)*(c*d^2 +
e*(-(b*d) + a*e))^3*Log[d + e*x])/(420*e^8)

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Maple [B]  time = 0.009, size = 1319, normalized size = 2.4 \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),x)

[Out]

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

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Maxima [A]  time = 0.993927, size = 1137, normalized size = 2.09 \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),x, algorithm="maxima")

[Out]

1/420*(60*B*c^3*e^6*x^7 - 70*(B*c^3*d*e^5 - (3*B*b*c^2 + A*c^3)*e^6)*x^6 + 84*(B*c^3*d^2*e^4 - (3*B*b*c^2 + A*
c^3)*d*e^5 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^6)*x^5 - 105*(B*c^3*d^3*e^3 - (3*B*b*c^2 + A*c^3)*d^2*e^4 + 3*(B*
b^2*c + (B*a + A*b)*c^2)*d*e^5 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^6)*x^4 + 140*(B*c^3*d^4*e^2 - (
3*B*b*c^2 + A*c^3)*d^3*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^4 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*
c)*d*e^5 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^6)*x^3 - 210*(B*c^3*d^5*e - (3*B*b*c^2 + A*c^3)*d^4*e
^2 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^4 + (3*B*a*b^2
+ A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^5 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^6)*x^2 + 420*(B*c^3*d^6 - (3*B*b*c^
2 + A*c^3)*d^5*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^3
+ (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*e^4 - 3*(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)*x)/e^7 - (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)*log(e*x + d)/e^8

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Fricas [A]  time = 1.10058, size = 1762, normalized size = 3.24 \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),x, algorithm="fricas")

[Out]

1/420*(60*B*c^3*e^7*x^7 - 70*(B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 84*(B*c^3*d^2*e^5 - (3*B*b*c^2 + A*
c^3)*d*e^6 + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^7)*x^5 - 105*(B*c^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^2*e^5 + 3*(B*
b^2*c + (B*a + A*b)*c^2)*d*e^6 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 140*(B*c^3*d^4*e^3 - (
3*B*b*c^2 + A*c^3)*d^3*e^4 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - (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 - 210*(B*c^3*d^5*e^2 - (3*B*b*c^2 + A*c^3)*d^4
*e^3 + 3*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - (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 - 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 420*(B*c^3*d^6*e - (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 + (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 + (B*a^3
+ 3*A*a^2*b)*e^7)*x - 420*(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)*log(e*x + d))/e^8

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Sympy [A]  time = 3.96452, size = 949, normalized size = 1.74 \begin{align*} \frac{B c^{3} x^{7}}{7 e} + \frac{x^{6} \left (A c^{3} e + 3 B b c^{2} e - B c^{3} d\right )}{6 e^{2}} + \frac{x^{5} \left (3 A b c^{2} e^{2} - A c^{3} d e + 3 B a c^{2} e^{2} + 3 B b^{2} c e^{2} - 3 B b c^{2} d e + B c^{3} d^{2}\right )}{5 e^{3}} + \frac{x^{4} \left (3 A a c^{2} e^{3} + 3 A b^{2} c e^{3} - 3 A b c^{2} d e^{2} + A c^{3} d^{2} e + 6 B a b c e^{3} - 3 B a c^{2} d e^{2} + B b^{3} e^{3} - 3 B b^{2} c d e^{2} + 3 B b c^{2} d^{2} e - B c^{3} d^{3}\right )}{4 e^{4}} + \frac{x^{3} \left (6 A a b c e^{4} - 3 A a c^{2} d e^{3} + A b^{3} e^{4} - 3 A b^{2} c d e^{3} + 3 A b c^{2} d^{2} e^{2} - A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 3 B a b^{2} e^{4} - 6 B a b c d e^{3} + 3 B a c^{2} d^{2} e^{2} - B b^{3} d e^{3} + 3 B b^{2} c d^{2} e^{2} - 3 B b c^{2} d^{3} e + B c^{3} d^{4}\right )}{3 e^{5}} + \frac{x^{2} \left (3 A a^{2} c e^{5} + 3 A a b^{2} e^{5} - 6 A a b c d e^{4} + 3 A a c^{2} d^{2} e^{3} - A b^{3} d e^{4} + 3 A b^{2} c d^{2} e^{3} - 3 A b c^{2} d^{3} e^{2} + A c^{3} d^{4} e + 3 B a^{2} b e^{5} - 3 B a^{2} c d e^{4} - 3 B a b^{2} d e^{4} + 6 B a b c d^{2} e^{3} - 3 B a c^{2} d^{3} e^{2} + B b^{3} d^{2} e^{3} - 3 B b^{2} c d^{3} e^{2} + 3 B b c^{2} d^{4} e - B c^{3} d^{5}\right )}{2 e^{6}} + \frac{x \left (3 A a^{2} b e^{6} - 3 A a^{2} c d e^{5} - 3 A a b^{2} d e^{5} + 6 A a b c d^{2} e^{4} - 3 A a c^{2} d^{3} e^{3} + A b^{3} d^{2} e^{4} - 3 A b^{2} c d^{3} e^{3} + 3 A b c^{2} d^{4} e^{2} - A c^{3} d^{5} e + B a^{3} e^{6} - 3 B a^{2} b d e^{5} + 3 B a^{2} c d^{2} e^{4} + 3 B a b^{2} d^{2} e^{4} - 6 B a b c d^{3} e^{3} + 3 B a c^{2} d^{4} e^{2} - B b^{3} d^{3} e^{3} + 3 B b^{2} c d^{4} e^{2} - 3 B b c^{2} d^{5} e + B c^{3} d^{6}\right )}{e^{7}} - \frac{\left (- A e + B d\right ) \left (a e^{2} - b d e + c d^{2}\right )^{3} \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),x)

[Out]

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

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Giac [B]  time = 1.11278, size = 1526, normalized size = 2.81 \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),x, algorithm="giac")

[Out]

-(B*c^3*d^7 - 3*B*b*c^2*d^6*e - A*c^3*d^6*e + 3*B*b^2*c*d^5*e^2 + 3*B*a*c^2*d^5*e^2 + 3*A*b*c^2*d^5*e^2 - B*b^
3*d^4*e^3 - 6*B*a*b*c*d^4*e^3 - 3*A*b^2*c*d^4*e^3 - 3*A*a*c^2*d^4*e^3 + 3*B*a*b^2*d^3*e^4 + A*b^3*d^3*e^4 + 3*
B*a^2*c*d^3*e^4 + 6*A*a*b*c*d^3*e^4 - 3*B*a^2*b*d^2*e^5 - 3*A*a*b^2*d^2*e^5 - 3*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)*e^(-8)*log(abs(x*e + d)) + 1/420*(60*B*c^3*x^7*e^6 - 70*B*c^3*d*x^6*e^5 + 84*B*
c^3*d^2*x^5*e^4 - 105*B*c^3*d^3*x^4*e^3 + 140*B*c^3*d^4*x^3*e^2 - 210*B*c^3*d^5*x^2*e + 420*B*c^3*d^6*x + 210*
B*b*c^2*x^6*e^6 + 70*A*c^3*x^6*e^6 - 252*B*b*c^2*d*x^5*e^5 - 84*A*c^3*d*x^5*e^5 + 315*B*b*c^2*d^2*x^4*e^4 + 10
5*A*c^3*d^2*x^4*e^4 - 420*B*b*c^2*d^3*x^3*e^3 - 140*A*c^3*d^3*x^3*e^3 + 630*B*b*c^2*d^4*x^2*e^2 + 210*A*c^3*d^
4*x^2*e^2 - 1260*B*b*c^2*d^5*x*e - 420*A*c^3*d^5*x*e + 252*B*b^2*c*x^5*e^6 + 252*B*a*c^2*x^5*e^6 + 252*A*b*c^2
*x^5*e^6 - 315*B*b^2*c*d*x^4*e^5 - 315*B*a*c^2*d*x^4*e^5 - 315*A*b*c^2*d*x^4*e^5 + 420*B*b^2*c*d^2*x^3*e^4 + 4
20*B*a*c^2*d^2*x^3*e^4 + 420*A*b*c^2*d^2*x^3*e^4 - 630*B*b^2*c*d^3*x^2*e^3 - 630*B*a*c^2*d^3*x^2*e^3 - 630*A*b
*c^2*d^3*x^2*e^3 + 1260*B*b^2*c*d^4*x*e^2 + 1260*B*a*c^2*d^4*x*e^2 + 1260*A*b*c^2*d^4*x*e^2 + 105*B*b^3*x^4*e^
6 + 630*B*a*b*c*x^4*e^6 + 315*A*b^2*c*x^4*e^6 + 315*A*a*c^2*x^4*e^6 - 140*B*b^3*d*x^3*e^5 - 840*B*a*b*c*d*x^3*
e^5 - 420*A*b^2*c*d*x^3*e^5 - 420*A*a*c^2*d*x^3*e^5 + 210*B*b^3*d^2*x^2*e^4 + 1260*B*a*b*c*d^2*x^2*e^4 + 630*A
*b^2*c*d^2*x^2*e^4 + 630*A*a*c^2*d^2*x^2*e^4 - 420*B*b^3*d^3*x*e^3 - 2520*B*a*b*c*d^3*x*e^3 - 1260*A*b^2*c*d^3
*x*e^3 - 1260*A*a*c^2*d^3*x*e^3 + 420*B*a*b^2*x^3*e^6 + 140*A*b^3*x^3*e^6 + 420*B*a^2*c*x^3*e^6 + 840*A*a*b*c*
x^3*e^6 - 630*B*a*b^2*d*x^2*e^5 - 210*A*b^3*d*x^2*e^5 - 630*B*a^2*c*d*x^2*e^5 - 1260*A*a*b*c*d*x^2*e^5 + 1260*
B*a*b^2*d^2*x*e^4 + 420*A*b^3*d^2*x*e^4 + 1260*B*a^2*c*d^2*x*e^4 + 2520*A*a*b*c*d^2*x*e^4 + 630*B*a^2*b*x^2*e^
6 + 630*A*a*b^2*x^2*e^6 + 630*A*a^2*c*x^2*e^6 - 1260*B*a^2*b*d*x*e^5 - 1260*A*a*b^2*d*x*e^5 - 1260*A*a^2*c*d*x
*e^5 + 420*B*a^3*x*e^6 + 1260*A*a^2*b*x*e^6)*e^(-7)