3.2344 \(\int \frac{(A+B x) (a+b x+c x^2)^3}{(d+e x)^6} \, dx\)

Optimal. Leaf size=534 \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{2 e^8 (d+e x)^2}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^8 (d+e x)}+\frac{\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)^3}-\frac{3 c \log (d+e x) \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8}+\frac{\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{4 e^8 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8 (d+e x)^5}-\frac{c^2 x (-A c e-3 b B e+6 B c d)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.901447, antiderivative size = 532, 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} \[ \frac{A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{2 e^8 (d+e x)^2}+\frac{B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{e^8 (d+e x)}+\frac{\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)^3}-\frac{3 c \log (d+e x) \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8}-\frac{\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{4 e^8 (d+e x)^4}+\frac{(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{5 e^8 (d+e x)^5}-\frac{c^2 x (-A c e-3 b B e+6 B c d)}{e^7}+\frac{B c^3 x^2}{2 e^6} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x]

[Out]

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

Mathematica [A]  time = 0.6466, size = 885, normalized size = 1.66 \[ \frac{60 c \left (A c e (b e-2 c d)+B \left (7 c^2 d^2+b^2 e^2+c e (a e-6 b d)\right )\right ) \log (d+e x) (d+e x)^5+A e \left (-2 \left (87 d^6+375 e x d^5+600 e^2 x^2 d^4+400 e^3 x^3 d^3+50 e^4 x^4 d^2-50 e^5 x^5 d-10 e^6 x^6\right ) c^3+e \left (b d \left (137 d^4+625 e x d^3+1100 e^2 x^2 d^2+900 e^3 x^3 d+300 e^4 x^4\right )-12 a e \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right )\right ) c^2-2 e^2 \left (6 \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right ) b^2+3 a e \left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right ) b+a^2 e^2 \left (d^2+5 e x d+10 e^2 x^2\right )\right ) c-e^3 \left (\left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right ) b^3+2 a e \left (d^2+5 e x d+10 e^2 x^2\right ) b^2+3 a^2 e^2 (d+5 e x) b+4 a^3 e^3\right )\right )+B \left (\left (459 d^7+1875 e x d^6+2700 e^2 x^2 d^5+1300 e^3 x^3 d^4-400 e^4 x^4 d^3-500 e^5 x^5 d^2-70 e^6 x^6 d+10 e^7 x^7\right ) c^3+e \left (a d e \left (137 d^4+625 e x d^3+1100 e^2 x^2 d^2+900 e^3 x^3 d+300 e^4 x^4\right )-6 b \left (87 d^6+375 e x d^5+600 e^2 x^2 d^4+400 e^3 x^3 d^3+50 e^4 x^4 d^2-50 e^5 x^5 d-10 e^6 x^6\right )\right ) c^2+e^2 \left (d \left (137 d^4+625 e x d^3+1100 e^2 x^2 d^2+900 e^3 x^3 d+300 e^4 x^4\right ) b^2-24 a e \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right ) b-3 a^2 e^2 \left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right )\right ) c-e^3 \left (4 \left (d^4+5 e x d^3+10 e^2 x^2 d^2+10 e^3 x^3 d+5 e^4 x^4\right ) b^3+3 a e \left (d^3+5 e x d^2+10 e^2 x^2 d+10 e^3 x^3\right ) b^2+2 a^2 e^2 \left (d^2+5 e x d+10 e^2 x^2\right ) b+a^3 e^3 (d+5 e x)\right )\right )}{20 e^8 (d+e x)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^6,x]

[Out]

(A*e*(-2*c^3*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d*e^5*x^5 - 10*e^
6*x^6) - e^3*(4*a^3*e^3 + 3*a^2*b*e^2*(d + 5*e*x) + 2*a*b^2*e*(d^2 + 5*d*e*x + 10*e^2*x^2) + b^3*(d^3 + 5*d^2*
e*x + 10*d*e^2*x^2 + 10*e^3*x^3)) - 2*c*e^2*(a^2*e^2*(d^2 + 5*d*e*x + 10*e^2*x^2) + 3*a*b*e*(d^3 + 5*d^2*e*x +
10*d*e^2*x^2 + 10*e^3*x^3) + 6*b^2*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c^2*e*(-1
2*a*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + b*d*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^
2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4))) + B*(c^3*(459*d^7 + 1875*d^6*e*x + 2700*d^5*e^2*x^2 + 1300*d^4*e^3*x^3
- 400*d^3*e^4*x^4 - 500*d^2*e^5*x^5 - 70*d*e^6*x^6 + 10*e^7*x^7) - e^3*(a^3*e^3*(d + 5*e*x) + 2*a^2*b*e^2*(d^2
+ 5*d*e*x + 10*e^2*x^2) + 3*a*b^2*e*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3) + 4*b^3*(d^4 + 5*d^3*e*x +
10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4)) + c*e^2*(-3*a^2*e^2*(d^3 + 5*d^2*e*x + 10*d*e^2*x^2 + 10*e^3*x^3)
- 24*a*b*e*(d^4 + 5*d^3*e*x + 10*d^2*e^2*x^2 + 10*d*e^3*x^3 + 5*e^4*x^4) + b^2*d*(137*d^4 + 625*d^3*e*x + 1100
*d^2*e^2*x^2 + 900*d*e^3*x^3 + 300*e^4*x^4)) + c^2*e*(a*d*e*(137*d^4 + 625*d^3*e*x + 1100*d^2*e^2*x^2 + 900*d*
e^3*x^3 + 300*e^4*x^4) - 6*b*(87*d^6 + 375*d^5*e*x + 600*d^4*e^2*x^2 + 400*d^3*e^3*x^3 + 50*d^2*e^4*x^4 - 50*d
*e^5*x^5 - 10*e^6*x^6))) + 60*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*Log[d + e*x])/(20*e^8*(d + e*x)^5)

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Maple [B]  time = 0.018, size = 1637, normalized size = 3.1 \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)^6,x)

[Out]

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

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Maxima [A]  time = 1.1421, size = 1212, 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)^6,x, algorithm="maxima")

[Out]

1/20*(459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A*c^3)*d^6*e + 137*(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 -
2*(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 + 20*(35*B*c^3*d^3*e^4 - 15*(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 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 +
10*(245*B*c^3*d^4*e^3 - 100*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(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 + 10*(329*B*c^
3*d^5*e^2 - 130*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(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 - 2*(B*a^2*b + A*a*b^2 + A
*a^2*c)*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 154*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(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 - 2*(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^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^
11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*e^8) + 1/2*(B*c^3*e*x^2 - 2*(6*B*c^3*d - (3*B*b*c^2 + A*c^3)*e)*x
)/e^7 + 3*(7*B*c^3*d^2 - 2*(3*B*b*c^2 + A*c^3)*d*e + (B*b^2*c + (B*a + A*b)*c^2)*e^2)*log(e*x + d)/e^8

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Fricas [B]  time = 1.07514, size = 2639, normalized size = 4.94 \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)^6,x, algorithm="fricas")

[Out]

1/20*(10*B*c^3*e^7*x^7 + 459*B*c^3*d^7 - 4*A*a^3*e^7 - 174*(3*B*b*c^2 + A*c^3)*d^6*e + 137*(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 - 2*(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 - 10*(7*B*c^3*d*e^6 - 2
*(3*B*b*c^2 + A*c^3)*e^7)*x^6 - 100*(5*B*c^3*d^2*e^5 - (3*B*b*c^2 + A*c^3)*d*e^6)*x^5 - 20*(20*B*c^3*d^3*e^4 +
5*(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 + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^
2)*c)*e^7)*x^4 + 10*(130*B*c^3*d^4*e^3 - 80*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 90*(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 + 10*(270*B*c^3*d^5*e^2 - 120*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 110*(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 - 2*(B*a^2
*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 + 5*(375*B*c^3*d^6*e - 150*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 125*(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 - 2*(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 - 2
*(3*B*b*c^2 + A*c^3)*d^6*e + (B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 + (7*B*c^3*d^2*e^5 - 2*(3*B*b*c^2 + A*c^3)*d*
e^6 + (B*b^2*c + (B*a + A*b)*c^2)*e^7)*x^5 + 5*(7*B*c^3*d^3*e^4 - 2*(3*B*b*c^2 + A*c^3)*d^2*e^5 + (B*b^2*c + (
B*a + A*b)*c^2)*d*e^6)*x^4 + 10*(7*B*c^3*d^4*e^3 - 2*(3*B*b*c^2 + A*c^3)*d^3*e^4 + (B*b^2*c + (B*a + A*b)*c^2)
*d^2*e^5)*x^3 + 10*(7*B*c^3*d^5*e^2 - 2*(3*B*b*c^2 + A*c^3)*d^4*e^3 + (B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4)*x^2
+ 5*(7*B*c^3*d^6*e - 2*(3*B*b*c^2 + A*c^3)*d^5*e^2 + (B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3)*x)*log(e*x + d))/(e
^13*x^5 + 5*d*e^12*x^4 + 10*d^2*e^11*x^3 + 10*d^3*e^10*x^2 + 5*d^4*e^9*x + d^5*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)**6,x)

[Out]

Timed out

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Giac [A]  time = 1.10744, size = 1347, normalized size = 2.52 \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)^6,x, algorithm="giac")

[Out]

3*(7*B*c^3*d^2 - 6*B*b*c^2*d*e - 2*A*c^3*d*e + B*b^2*c*e^2 + B*a*c^2*e^2 + A*b*c^2*e^2)*e^(-8)*log(abs(x*e + d
)) + 1/2*(B*c^3*x^2*e^6 - 12*B*c^3*d*x*e^5 + 6*B*b*c^2*x*e^6 + 2*A*c^3*x*e^6)*e^(-12) + 1/20*(459*B*c^3*d^7 -
522*B*b*c^2*d^6*e - 174*A*c^3*d^6*e + 137*B*b^2*c*d^5*e^2 + 137*B*a*c^2*d^5*e^2 + 137*A*b*c^2*d^5*e^2 - 4*B*b^
3*d^4*e^3 - 24*B*a*b*c*d^4*e^3 - 12*A*b^2*c*d^4*e^3 - 12*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 - 2*B*a^2*b*d^2*e^5 - 2*A*a*b^2*d^2*e^5 - 2*A*a^2*c*d^2*e^5 - B*a^3*d*e
^6 - 3*A*a^2*b*d*e^6 + 20*(35*B*c^3*d^3*e^4 - 45*B*b*c^2*d^2*e^5 - 15*A*c^3*d^2*e^5 + 15*B*b^2*c*d*e^6 + 15*B*
a*c^2*d*e^6 + 15*A*b*c^2*d*e^6 - B*b^3*e^7 - 6*B*a*b*c*e^7 - 3*A*b^2*c*e^7 - 3*A*a*c^2*e^7)*x^4 - 4*A*a^3*e^7
+ 10*(245*B*c^3*d^4*e^3 - 300*B*b*c^2*d^3*e^4 - 100*A*c^3*d^3*e^4 + 90*B*b^2*c*d^2*e^5 + 90*B*a*c^2*d^2*e^5 +
90*A*b*c^2*d^2*e^5 - 4*B*b^3*d*e^6 - 24*B*a*b*c*d*e^6 - 12*A*b^2*c*d*e^6 - 12*A*a*c^2*d*e^6 - 3*B*a*b^2*e^7 -
A*b^3*e^7 - 3*B*a^2*c*e^7 - 6*A*a*b*c*e^7)*x^3 + 10*(329*B*c^3*d^5*e^2 - 390*B*b*c^2*d^4*e^3 - 130*A*c^3*d^4*e
^3 + 110*B*b^2*c*d^3*e^4 + 110*B*a*c^2*d^3*e^4 + 110*A*b*c^2*d^3*e^4 - 4*B*b^3*d^2*e^5 - 24*B*a*b*c*d^2*e^5 -
12*A*b^2*c*d^2*e^5 - 12*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 -
2*B*a^2*b*e^7 - 2*A*a*b^2*e^7 - 2*A*a^2*c*e^7)*x^2 + 5*(399*B*c^3*d^6*e - 462*B*b*c^2*d^5*e^2 - 154*A*c^3*d^5*
e^2 + 125*B*b^2*c*d^4*e^3 + 125*B*a*c^2*d^4*e^3 + 125*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 - 3*B*a*b^2*d^2*e^5 - A*b^3*d^2*e^5 - 3*B*a^2*c*d^2*e^5 - 6*A*a*b*c*d
^2*e^5 - 2*B*a^2*b*d*e^6 - 2*A*a*b^2*d*e^6 - 2*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)^
5