3.2342 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{(d+e x)^4} \, dx\)

Optimal. Leaf size=521 \[ -\frac{\log (d+e x) \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{e^8}-\frac{c x^2 \left (A c e (4 c d-3 b e)-B \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{2 e^6}-\frac{x \left (B (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-A c e \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{e^7}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac{\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{2 e^8 (d+e x)^2}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac{c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac{B c^3 x^4}{4 e^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.990182, antiderivative size = 519, 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} \[ -\frac{\log (d+e x) \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{e^8}-\frac{c x^2 \left (A c e (4 c d-3 b e)-B \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{2 e^6}-\frac{x \left (B (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-A c e \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{e^7}+\frac{3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac{c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac{B c^3 x^4}{4 e^4} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

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

Mathematica [A]  time = 0.321938, size = 488, normalized size = 0.94 \[ \frac{12 \log (d+e x) \left (B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )+b^2 e^3 (3 a e-4 b d)+30 c^2 d^2 e (a e-2 b d)+35 c^3 d^4\right )+A e (b e-2 c d) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )\right )+6 c e^2 x^2 \left (B \left (3 c e (a e-4 b d)+3 b^2 e^2+10 c^2 d^2\right )+A c e (3 b e-4 c d)\right )+12 e x \left (A c e \left (3 c e (a e-4 b d)+3 b^2 e^2+10 c^2 d^2\right )-B (2 c d-b e) \left (2 c e (3 a e-5 b d)+b^2 e^2+10 c^2 d^2\right )\right )+\frac{36 \left (e (a e-b d)+c d^2\right ) \left (B \left (c d e (3 a e-8 b d)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )\right )}{d+e x}-\frac{6 \left (e (a e-b d)+c d^2\right )^2 \left (B e (a e-4 b d)+3 A e (b e-2 c d)+7 B c d^2\right )}{(d+e x)^2}+\frac{4 (B d-A e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^3}+4 c^2 e^3 x^3 (A c e+3 b B e-4 B c d)+3 B c^3 e^4 x^4}{12 e^8} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^4,x]

[Out]

(12*e*(A*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)) - B*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(
-5*b*d + 3*a*e)))*x + 6*c*e^2*(A*c*e*(-4*c*d + 3*b*e) + B*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)))*x^2
+ 4*c^2*e^3*(-4*B*c*d + 3*b*B*e + A*c*e)*x^3 + 3*B*c^3*e^4*x^4 + (4*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)
/(d + e*x)^3 - (6*(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)^2 + (36*(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))))/(d + e*x) + 12*(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 + 30*c^2*d^2*e*(-2*b*d + a*e) + b^2*e^3*(-4*b*d + 3*a*e) + 3*c*e^2*(10*b^2*d
^2 - 8*a*b*d*e + a^2*e^2)))*Log[d + e*x])/(12*e^8)

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Maple [B]  time = 0.02, size = 1545, normalized size = 3. \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)^4,x)

[Out]

1/4*B*c^3*x^4/e^4+12/e^5/(e*x+d)^2*B*a*b*c*d^3+18/e^4/(e*x+d)*A*a*b*c*d-36/e^5/(e*x+d)*B*a*b*c*d^2-24/e^5*ln(e
*x+d)*B*a*b*c*d+2/e^4/(e*x+d)^3*A*d^3*a*b*c-2/e^5/(e*x+d)^3*B*a*b*c*d^4-9/e^4/(e*x+d)^2*A*a*b*c*d^2-12/e^5*B*a
*c^2*d*x-12/e^5*B*b^2*c*d*x+30/e^6*B*b*c^2*d^2*x-12/e^5*A*b*c^2*d*x+6/e^4*B*a*b*c*x+30/e^6*ln(e*x+d)*A*b*c^2*d
^2+30/e^6*ln(e*x+d)*B*a*c^2*d^2+30/e^6*ln(e*x+d)*B*b^2*c*d^2-60/e^7*ln(e*x+d)*B*b*c^2*d^3+1/e^2/(e*x+d)^3*A*d*
a^2*b-1/e^3/(e*x+d)^3*A*a^2*c*d^2-1/e^3/(e*x+d)^3*A*d^2*a*b^2-1/e^5/(e*x+d)^3*A*a*c^2*d^4-1/e^5/(e*x+d)^3*A*b^
2*c*d^4+1/e^6/(e*x+d)^3*A*b*c^2*d^5-1/e^3/(e*x+d)^3*B*d^2*a^2*b+1/e^4/(e*x+d)^3*B*a^2*c*d^3+1/e^4/(e*x+d)^3*B*
d^3*a*b^2+1/e^6/(e*x+d)^3*B*a*c^2*d^5+1/e^6/(e*x+d)^3*B*b^2*c*d^5-1/e^7/(e*x+d)^3*B*b*c^2*d^6-6/e^5*B*x^2*b*c^
2*d-12/e^5*ln(e*x+d)*A*b^2*c*d-12/e^5*ln(e*x+d)*A*a*c^2*d-18/e^5/(e*x+d)*A*a*c^2*d^2-18/e^5/(e*x+d)*A*b^2*c*d^
2+30/e^6/(e*x+d)*A*b*c^2*d^3+6/e^4*ln(e*x+d)*A*a*b*c+3/e^3/(e*x+d)^2*B*a^2*b*d-9/2/e^4/(e*x+d)^2*B*a^2*c*d^2-9
/2/e^4/(e*x+d)^2*B*a*b^2*d^2-15/2/e^6/(e*x+d)^2*B*a*c^2*d^4-15/2/e^6/(e*x+d)^2*B*b^2*c*d^4+9/e^7/(e*x+d)^2*B*b
*c^2*d^5+6/e^5/(e*x+d)^2*A*b^2*c*d^3-15/2/e^6/(e*x+d)^2*A*b*c^2*d^4+3/e^3/(e*x+d)^2*A*a^2*c*d+3/e^3/(e*x+d)^2*
A*a*b^2*d+6/e^5/(e*x+d)^2*A*a*c^2*d^3+9/e^4/(e*x+d)*B*a^2*c*d+9/e^4/(e*x+d)*B*a*b^2*d+30/e^6/(e*x+d)*B*a*c^2*d
^3+30/e^6/(e*x+d)*B*b^2*c*d^3-45/e^7/(e*x+d)*B*b*c^2*d^4-1/2/e^2/(e*x+d)^2*B*a^3+1/3/e^4*A*x^3*c^3+1/e^4*B*b^3
*x+1/e^4*ln(e*x+d)*A*b^3-1/3/e/(e*x+d)^3*A*a^3-1/3/e^7/(e*x+d)^3*A*c^3*d^6+1/3/e^2/(e*x+d)^3*B*a^3*d-1/3/e^5/(
e*x+d)^3*B*b^3*d^4+1/3/e^8/(e*x+d)^3*B*c^3*d^7-3/2/e^2/(e*x+d)^2*A*a^2*b-3/2/e^4/(e*x+d)^2*A*b^3*d^2+3/e^7/(e*
x+d)^2*A*c^3*d^5+2/e^5/(e*x+d)^2*B*b^3*d^3-7/2/e^8/(e*x+d)^2*B*c^3*d^6-3/e^3/(e*x+d)*A*a^2*c-3/e^3/(e*x+d)*A*a
*b^2+3/e^4/(e*x+d)*A*b^3*d-20/e^7*ln(e*x+d)*A*c^3*d^3+3/2/e^4*A*x^2*b*c^2-2/e^5*A*x^2*c^3*d+3/2/e^4*B*x^2*a*c^
2+3/2/e^4*B*x^2*b^2*c+5/e^6*B*x^2*c^3*d^2+3/e^4*A*a*c^2*x+3/e^4*A*b^2*c*x+10/e^6*A*c^3*d^2*x-20/e^7*B*c^3*d^3*
x+1/e^4*B*x^3*b*c^2-4/3/e^5*B*x^3*c^3*d-6/e^5/(e*x+d)*B*b^3*d^2+21/e^8/(e*x+d)*B*c^3*d^5+3/e^4*ln(e*x+d)*B*a^2
*c+3/e^4*ln(e*x+d)*B*a*b^2-4/e^5*ln(e*x+d)*B*b^3*d+35/e^8*ln(e*x+d)*B*c^3*d^4+1/3/e^4/(e*x+d)^3*A*d^3*b^3-15/e
^7/(e*x+d)*A*c^3*d^4-3/e^3/(e*x+d)*B*a^2*b

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Maxima [A]  time = 1.09927, size = 1181, normalized size = 2.27 \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)^4,x, algorithm="maxima")

[Out]

1/6*(107*B*c^3*d^7 - 2*A*a^3*e^7 - 74*(3*B*b*c^2 + A*c^3)*d^6*e + 141*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 26
*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 11*(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 + 18*(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 + 3*(77*B*c^3*d^
6*e - 54*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 105*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 20*(B*b^3 + 3*A*a*c^2 + 3*(2*
B*a*b + A*b^2)*c)*d^3*e^4 + 9*(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^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8) + 1/12*(3*B*c^3*e^
3*x^4 - 4*(4*B*c^3*d*e^2 - (3*B*b*c^2 + A*c^3)*e^3)*x^3 + 6*(10*B*c^3*d^2*e - 4*(3*B*b*c^2 + A*c^3)*d*e^2 + 3*
(B*b^2*c + (B*a + A*b)*c^2)*e^3)*x^2 - 12*(20*B*c^3*d^3 - 10*(3*B*b*c^2 + A*c^3)*d^2*e + 12*(B*b^2*c + (B*a +
A*b)*c^2)*d*e^2 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^3)*x)/e^7 + (35*B*c^3*d^4 - 20*(3*B*b*c^2 + A*
c^3)*d^3*e + 30*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^2 - 4*(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)*log(e*x + d)/e^8

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Fricas [B]  time = 1.12616, size = 2936, normalized size = 5.64 \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)^4,x, algorithm="fricas")

[Out]

1/12*(3*B*c^3*e^7*x^7 + 214*B*c^3*d^7 - 4*A*a^3*e^7 - 148*(3*B*b*c^2 + A*c^3)*d^6*e + 282*(B*b^2*c + (B*a + A*
b)*c^2)*d^5*e^2 - 52*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 22*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 +
2*A*a*b)*c)*d^3*e^4 - 12*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - 2*(B*a^3 + 3*A*a^2*b)*d*e^6 - (7*B*c^3*d*e^6
- 4*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 3*(7*B*c^3*d^2*e^5 - 4*(3*B*b*c^2 + A*c^3)*d*e^6 + 6*(B*b^2*c + (B*a + A*b)
*c^2)*e^7)*x^5 - 3*(35*B*c^3*d^3*e^4 - 20*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 30*(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 - 2*(278*B*c^3*d^4*e^3 - 146*(3*B*b*c^2 + A*c^3)*d^3*e
^4 + 189*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 - 18*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6)*x^3 - 6*(
68*B*c^3*d^5*e^2 - 26*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 9*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 + 6*(B*b^3 + 3*A*a*c
^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 - 6*(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 + 6*(37*B*c^3*d^6*e - 34*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 81*(B*b^2*c + (B*a + A*b)*c^2)*d
^4*e^3 - 18*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4 + 9*(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 + 12*(35*B*c^3*d^7 - 20*(3*B*b*
c^2 + A*c^3)*d^6*e + 30*(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*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 + (35*B*c^3*d^4*e^3 - 20*(3*B*b*c^2 + A*c^3)*d^3*e
^4 + 30*(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*B*a*b^2
+ A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^7)*x^3 + 3*(35*B*c^3*d^5*e^2 - 20*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 30*(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*B*a*b^2 + A*b^3 + 3*
(B*a^2 + 2*A*a*b)*c)*d*e^6)*x^2 + 3*(35*B*c^3*d^6*e - 20*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 30*(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*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*
a*b)*c)*d^2*e^5)*x)*log(e*x + d))/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9*x + d^3*e^8)

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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)**3/(e*x+d)**4,x)

[Out]

Timed out

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Giac [A]  time = 1.1449, size = 1378, normalized size = 2.64 \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)^4,x, algorithm="giac")

[Out]

(35*B*c^3*d^4 - 60*B*b*c^2*d^3*e - 20*A*c^3*d^3*e + 30*B*b^2*c*d^2*e^2 + 30*B*a*c^2*d^2*e^2 + 30*A*b*c^2*d^2*e
^2 - 4*B*b^3*d*e^3 - 24*B*a*b*c*d*e^3 - 12*A*b^2*c*d*e^3 - 12*A*a*c^2*d*e^3 + 3*B*a*b^2*e^4 + A*b^3*e^4 + 3*B*
a^2*c*e^4 + 6*A*a*b*c*e^4)*e^(-8)*log(abs(x*e + d)) + 1/12*(3*B*c^3*x^4*e^12 - 16*B*c^3*d*x^3*e^11 + 60*B*c^3*
d^2*x^2*e^10 - 240*B*c^3*d^3*x*e^9 + 12*B*b*c^2*x^3*e^12 + 4*A*c^3*x^3*e^12 - 72*B*b*c^2*d*x^2*e^11 - 24*A*c^3
*d*x^2*e^11 + 360*B*b*c^2*d^2*x*e^10 + 120*A*c^3*d^2*x*e^10 + 18*B*b^2*c*x^2*e^12 + 18*B*a*c^2*x^2*e^12 + 18*A
*b*c^2*x^2*e^12 - 144*B*b^2*c*d*x*e^11 - 144*B*a*c^2*d*x*e^11 - 144*A*b*c^2*d*x*e^11 + 12*B*b^3*x*e^12 + 72*B*
a*b*c*x*e^12 + 36*A*b^2*c*x*e^12 + 36*A*a*c^2*x*e^12)*e^(-16) + 1/6*(107*B*c^3*d^7 - 222*B*b*c^2*d^6*e - 74*A*
c^3*d^6*e + 141*B*b^2*c*d^5*e^2 + 141*B*a*c^2*d^5*e^2 + 141*A*b*c^2*d^5*e^2 - 26*B*b^3*d^4*e^3 - 156*B*a*b*c*d
^4*e^3 - 78*A*b^2*c*d^4*e^3 - 78*A*a*c^2*d^4*e^3 + 33*B*a*b^2*d^3*e^4 + 11*A*b^3*d^3*e^4 + 33*B*a^2*c*d^3*e^4
+ 66*A*a*b*c*d^3*e^4 - 6*B*a^2*b*d^2*e^5 - 6*A*a*b^2*d^2*e^5 - 6*A*a^2*c*d^2*e^5 - B*a^3*d*e^6 - 3*A*a^2*b*d*e
^6 - 2*A*a^3*e^7 + 18*(7*B*c^3*d^5*e^2 - 15*B*b*c^2*d^4*e^3 - 5*A*c^3*d^4*e^3 + 10*B*b^2*c*d^3*e^4 + 10*B*a*c^
2*d^3*e^4 + 10*A*b*c^2*d^3*e^4 - 2*B*b^3*d^2*e^5 - 12*B*a*b*c*d^2*e^5 - 6*A*b^2*c*d^2*e^5 - 6*A*a*c^2*d^2*e^5
+ 3*B*a*b^2*d*e^6 + A*b^3*d*e^6 + 3*B*a^2*c*d*e^6 + 6*A*a*b*c*d*e^6 - B*a^2*b*e^7 - A*a*b^2*e^7 - A*a^2*c*e^7)
*x^2 + 3*(77*B*c^3*d^6*e - 162*B*b*c^2*d^5*e^2 - 54*A*c^3*d^5*e^2 + 105*B*b^2*c*d^4*e^3 + 105*B*a*c^2*d^4*e^3
+ 105*A*b*c^2*d^4*e^3 - 20*B*b^3*d^3*e^4 - 120*B*a*b*c*d^3*e^4 - 60*A*b^2*c*d^3*e^4 - 60*A*a*c^2*d^3*e^4 + 27*
B*a*b^2*d^2*e^5 + 9*A*b^3*d^2*e^5 + 27*B*a^2*c*d^2*e^5 + 54*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)^3