3.2345 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.81, antiderivative size = 539, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {771} \[ \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{3 e^8 (d+e x)^3}+\frac {3 c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{4 e^8 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{5 e^8 (d+e x)^5}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{6 e^8 (d+e x)^6}-\frac {c^2 \log (d+e x) (-A c e-3 b B e+7 B c d)}{e^8}+\frac {B c^3 x}{e^7} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x]

[Out]

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

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Mathematica [A]  time = 0.65, size = 868, normalized size = 1.60 \[ -\frac {60 c^2 (7 B c d-3 b B e-A c e) \log (d+e x) (d+e x)^6+A e \left (-d \left (147 d^5+822 e x d^4+1875 e^2 x^2 d^3+2200 e^3 x^3 d^2+1350 e^4 x^4 d+360 e^5 x^5\right ) c^3+6 e \left (a e \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right )+5 b \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right )\right ) c^2+3 e^2 \left (2 \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b^2+2 a e \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b+a^2 e^2 \left (d^2+6 e x d+15 e^2 x^2\right )\right ) c+e^3 \left (\left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b^3+3 a e \left (d^2+6 e x d+15 e^2 x^2\right ) b^2+6 a^2 e^2 (d+6 e x) b+10 a^3 e^3\right )\right )+B \left (\left (669 d^7+3594 e x d^6+7725 e^2 x^2 d^5+8200 e^3 x^3 d^4+4050 e^4 x^4 d^3+360 e^5 x^5 d^2-360 e^6 x^6 d-60 e^7 x^7\right ) c^3+3 e \left (10 a e \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right )-b d \left (147 d^5+822 e x d^4+1875 e^2 x^2 d^3+2200 e^3 x^3 d^2+1350 e^4 x^4 d+360 e^5 x^5\right )\right ) c^2+3 e^2 \left (10 \left (d^5+6 e x d^4+15 e^2 x^2 d^3+20 e^3 x^3 d^2+15 e^4 x^4 d+6 e^5 x^5\right ) b^2+4 a e \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b+a^2 e^2 \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right )\right ) c+e^3 \left (2 \left (d^4+6 e x d^3+15 e^2 x^2 d^2+20 e^3 x^3 d+15 e^4 x^4\right ) b^3+3 a e \left (d^3+6 e x d^2+15 e^2 x^2 d+20 e^3 x^3\right ) b^2+3 a^2 e^2 \left (d^2+6 e x d+15 e^2 x^2\right ) b+2 a^3 e^3 (d+6 e x)\right )\right )}{60 e^8 (d+e x)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x]

[Out]

-1/60*(A*e*(-(c^3*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*x^3 + 1350*d*e^4*x^4 + 360*e^5*x^
5)) + e^3*(10*a^3*e^3 + 6*a^2*b*e^2*(d + 6*e*x) + 3*a*b^2*e*(d^2 + 6*d*e*x + 15*e^2*x^2) + b^3*(d^3 + 6*d^2*e*
x + 15*d*e^2*x^2 + 20*e^3*x^3)) + 3*c*e^2*(a^2*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 2*a*b*e*(d^3 + 6*d^2*e*x + 1
5*d*e^2*x^2 + 20*e^3*x^3) + 2*b^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 6*c^2*e*(a
*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 5*b*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^2 + 20
*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5))) + B*(c^3*(669*d^7 + 3594*d^6*e*x + 7725*d^5*e^2*x^2 + 8200*d^4*e^3*
x^3 + 4050*d^3*e^4*x^4 + 360*d^2*e^5*x^5 - 360*d*e^6*x^6 - 60*e^7*x^7) + e^3*(2*a^3*e^3*(d + 6*e*x) + 3*a^2*b*
e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + 3*a*b^2*e*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e^3*x^3) + 2*b^3*(d^4 + 6*d^
3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4)) + 3*c*e^2*(a^2*e^2*(d^3 + 6*d^2*e*x + 15*d*e^2*x^2 + 20*e
^3*x^3) + 4*a*b*e*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3 + 15*e^4*x^4) + 10*b^2*(d^5 + 6*d^4*e*x + 1
5*d^3*e^2*x^2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5)) + 3*c^2*e*(10*a*e*(d^5 + 6*d^4*e*x + 15*d^3*e^2*x^
2 + 20*d^2*e^3*x^3 + 15*d*e^4*x^4 + 6*e^5*x^5) - b*d*(147*d^5 + 822*d^4*e*x + 1875*d^3*e^2*x^2 + 2200*d^2*e^3*
x^3 + 1350*d*e^4*x^4 + 360*e^5*x^5))) + 60*c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^6*Log[d + e*x])/(e^8*(d +
e*x)^6)

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fricas [B]  time = 0.94, size = 1145, normalized size = 2.12 \[ \text {result too large to display} \]

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)^7,x, algorithm="fricas")

[Out]

1/60*(60*B*c^3*e^7*x^7 + 360*B*c^3*d*e^6*x^6 - 669*B*c^3*d^7 - 10*A*a^3*e^7 + 147*(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 - 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 - 2*(B*a^3 + 3*A*a^2*b)*d*e^6
- 180*(2*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 - 30*(135*B*c^3*d
^3*e^4 - 45*(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 - 20*(410*B*c^3*d^4*e^3 - 110*(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 + 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 - 15*(515*B*c^3*d^5*e^2 - 125*(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
+ 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 +
3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^2 - 6*(599*B*c^3*d^6*e - 137*(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 + 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 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6 + 2*(B*a^3 + 3*A*a^2*b)*e^7)*x - 60*(7*B*c
^3*d^7 - (3*B*b*c^2 + A*c^3)*d^6*e + (7*B*c^3*d*e^6 - (3*B*b*c^2 + A*c^3)*e^7)*x^6 + 6*(7*B*c^3*d^2*e^5 - (3*B
*b*c^2 + A*c^3)*d*e^6)*x^5 + 15*(7*B*c^3*d^3*e^4 - (3*B*b*c^2 + A*c^3)*d^2*e^5)*x^4 + 20*(7*B*c^3*d^4*e^3 - (3
*B*b*c^2 + A*c^3)*d^3*e^4)*x^3 + 15*(7*B*c^3*d^5*e^2 - (3*B*b*c^2 + A*c^3)*d^4*e^3)*x^2 + 6*(7*B*c^3*d^6*e - (
3*B*b*c^2 + A*c^3)*d^5*e^2)*x)*log(e*x + d))/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^3*e^11*x^3 + 15
*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8)

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giac [A]  time = 0.23, size = 989, normalized size = 1.83 \[ B c^{3} x e^{\left (-7\right )} - {\left (7 \, B c^{3} d - 3 \, B b c^{2} e - A c^{3} e\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) - \frac {{\left (669 \, B c^{3} d^{7} - 441 \, B b c^{2} d^{6} e - 147 \, A c^{3} d^{6} e + 30 \, B b^{2} c d^{5} e^{2} + 30 \, B a c^{2} d^{5} e^{2} + 30 \, A b c^{2} d^{5} e^{2} + 2 \, B b^{3} d^{4} e^{3} + 12 \, B a b c d^{4} e^{3} + 6 \, A b^{2} c d^{4} e^{3} + 6 \, 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} + 180 \, {\left (7 \, B c^{3} d^{2} e^{5} - 6 \, B b c^{2} d e^{6} - 2 \, A c^{3} d e^{6} + B b^{2} c e^{7} + B a c^{2} e^{7} + A b c^{2} e^{7}\right )} x^{5} + 2 \, B a^{3} d e^{6} + 6 \, A a^{2} b d e^{6} + 30 \, {\left (175 \, B c^{3} d^{3} e^{4} - 135 \, B b c^{2} d^{2} e^{5} - 45 \, 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}\right )} x^{4} + 10 \, A a^{3} e^{7} + 20 \, {\left (455 \, B c^{3} d^{4} e^{3} - 330 \, B b c^{2} d^{3} e^{4} - 110 \, A c^{3} d^{3} e^{4} + 30 \, B b^{2} c d^{2} e^{5} + 30 \, B a c^{2} d^{2} e^{5} + 30 \, A b c^{2} d^{2} e^{5} + 2 \, B b^{3} d e^{6} + 12 \, B a b c d e^{6} + 6 \, A b^{2} c d e^{6} + 6 \, 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}\right )} x^{3} + 15 \, {\left (539 \, B c^{3} d^{5} e^{2} - 375 \, B b c^{2} d^{4} e^{3} - 125 \, A c^{3} d^{4} e^{3} + 30 \, B b^{2} c d^{3} e^{4} + 30 \, B a c^{2} d^{3} e^{4} + 30 \, 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} + 3 \, B a^{2} b e^{7} + 3 \, A a b^{2} e^{7} + 3 \, A a^{2} c e^{7}\right )} x^{2} + 6 \, {\left (609 \, B c^{3} d^{6} e - 411 \, B b c^{2} d^{5} e^{2} - 137 \, A c^{3} d^{5} e^{2} + 30 \, B b^{2} c d^{4} e^{3} + 30 \, B a c^{2} d^{4} e^{3} + 30 \, A b c^{2} d^{4} e^{3} + 2 \, B b^{3} d^{3} e^{4} + 12 \, B a b c d^{3} e^{4} + 6 \, A b^{2} c d^{3} e^{4} + 6 \, 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} + 3 \, B a^{2} b d e^{6} + 3 \, A a b^{2} d e^{6} + 3 \, A a^{2} c d e^{6} + 2 \, B a^{3} e^{7} + 6 \, A a^{2} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{60 \, {\left (x e + d\right )}^{6}} \]

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)^7,x, algorithm="giac")

[Out]

B*c^3*x*e^(-7) - (7*B*c^3*d - 3*B*b*c^2*e - A*c^3*e)*e^(-8)*log(abs(x*e + d)) - 1/60*(669*B*c^3*d^7 - 441*B*b*
c^2*d^6*e - 147*A*c^3*d^6*e + 30*B*b^2*c*d^5*e^2 + 30*B*a*c^2*d^5*e^2 + 30*A*b*c^2*d^5*e^2 + 2*B*b^3*d^4*e^3 +
12*B*a*b*c*d^4*e^3 + 6*A*b^2*c*d^4*e^3 + 6*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 + 180*(7*B*c^3*d^2*e^5 -
6*B*b*c^2*d*e^6 - 2*A*c^3*d*e^6 + B*b^2*c*e^7 + B*a*c^2*e^7 + A*b*c^2*e^7)*x^5 + 2*B*a^3*d*e^6 + 6*A*a^2*b*d*
e^6 + 30*(175*B*c^3*d^3*e^4 - 135*B*b*c^2*d^2*e^5 - 45*A*c^3*d^2*e^5 + 15*B*b^2*c*d*e^6 + 15*B*a*c^2*d*e^6 + 1
5*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 + 10*A*a^3*e^7 + 20*(455*B*c^
3*d^4*e^3 - 330*B*b*c^2*d^3*e^4 - 110*A*c^3*d^3*e^4 + 30*B*b^2*c*d^2*e^5 + 30*B*a*c^2*d^2*e^5 + 30*A*b*c^2*d^2
*e^5 + 2*B*b^3*d*e^6 + 12*B*a*b*c*d*e^6 + 6*A*b^2*c*d*e^6 + 6*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 + 15*(539*B*c^3*d^5*e^2 - 375*B*b*c^2*d^4*e^3 - 125*A*c^3*d^4*e^3 + 30*B*b^2*c*
d^3*e^4 + 30*B*a*c^2*d^3*e^4 + 30*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 + 3*B*a^2*b*e^7 + 3*A*a
*b^2*e^7 + 3*A*a^2*c*e^7)*x^2 + 6*(609*B*c^3*d^6*e - 411*B*b*c^2*d^5*e^2 - 137*A*c^3*d^5*e^2 + 30*B*b^2*c*d^4*
e^3 + 30*B*a*c^2*d^4*e^3 + 30*A*b*c^2*d^4*e^3 + 2*B*b^3*d^3*e^4 + 12*B*a*b*c*d^3*e^4 + 6*A*b^2*c*d^3*e^4 + 6*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 + 3*B*a^2*b*d*e^6 +
3*A*a*b^2*d*e^6 + 3*A*a^2*c*d*e^6 + 2*B*a^3*e^7 + 6*A*a^2*b*e^7)*x)*e^(-8)/(x*e + d)^6

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maple [B]  time = 0.06, size = 1656, normalized size = 3.06 \[ \text {result too large to display} \]

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)^7,x)

[Out]

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

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maxima [A]  time = 0.71, size = 905, normalized size = 1.67 \[ -\frac {669 \, B c^{3} d^{7} + 10 \, A a^{3} e^{7} - 147 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3} e^{4} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} e^{5} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 180 \, {\left (7 \, B c^{3} d^{2} e^{5} - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{7}\right )} x^{5} + 30 \, {\left (175 \, B c^{3} d^{3} e^{4} - 45 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{6} + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{7}\right )} x^{4} + 20 \, {\left (455 \, B c^{3} d^{4} e^{3} - 110 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{5} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{6} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{7}\right )} x^{3} + 15 \, {\left (539 \, B c^{3} d^{5} e^{2} - 125 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} e^{5} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{6} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 6 \, {\left (609 \, B c^{3} d^{6} e - 137 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 30 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} + 2 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} e^{4} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} e^{5} + 3 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e^{6} + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{60 \, {\left (e^{14} x^{6} + 6 \, d e^{13} x^{5} + 15 \, d^{2} e^{12} x^{4} + 20 \, d^{3} e^{11} x^{3} + 15 \, d^{4} e^{10} x^{2} + 6 \, d^{5} e^{9} x + d^{6} e^{8}\right )}} + \frac {B c^{3} x}{e^{7}} - \frac {{\left (7 \, B c^{3} d - {\left (3 \, B b c^{2} + A c^{3}\right )} e\right )} \log \left (e x + d\right )}{e^{8}} \]

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)^7,x, algorithm="maxima")

[Out]

-1/60*(669*B*c^3*d^7 + 10*A*a^3*e^7 - 147*(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 +
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 + 2*(B*a^3 + 3*A*a^2*b)*d*e^6 + 180*(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 + 30*(175*B*c^3*d^3*e^4 - 45*(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 + 20*(455*B*c^3*
d^4*e^3 - 110*(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 + 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 + 15*(539*B*c^3*d^5*e^2 - 12
5*(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 + 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 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^7)*x^
2 + 6*(609*B*c^3*d^6*e - 137*(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 + 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 + 3*(B*a^2*b
+ A*a*b^2 + A*a^2*c)*d*e^6 + 2*(B*a^3 + 3*A*a^2*b)*e^7)*x)/(e^14*x^6 + 6*d*e^13*x^5 + 15*d^2*e^12*x^4 + 20*d^
3*e^11*x^3 + 15*d^4*e^10*x^2 + 6*d^5*e^9*x + d^6*e^8) + B*c^3*x/e^7 - (7*B*c^3*d - (3*B*b*c^2 + A*c^3)*e)*log(
e*x + d)/e^8

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mupad [B]  time = 2.54, size = 1598, normalized size = 2.95 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^7,x)

[Out]

-(10*A*a^3*e^7 + 669*B*c^3*d^7 + 2*B*a^3*d*e^6 - 147*A*c^3*d^6*e + 420*B*c^3*d^7*log(d + e*x) + 12*B*a^3*e^7*x
+ A*b^3*d^3*e^4 + 2*B*b^3*d^4*e^3 + 20*A*b^3*e^7*x^3 + 30*B*b^3*e^7*x^4 - 60*B*c^3*e^7*x^7 + 3594*B*c^3*d^6*e
*x + 3*A*a*b^2*d^2*e^5 + 6*A*a*c^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 + 3*B*a*b^2*d^3*e^4 + 3*B*a^2*b*d^2*e^5 + 30*A*
b*c^2*d^5*e^2 + 6*A*b^2*c*d^4*e^3 + 30*B*a*c^2*d^5*e^2 + 3*B*a^2*c*d^3*e^4 + 30*B*b^2*c*d^5*e^2 + 45*A*a*b^2*e
^7*x^2 + 45*A*a^2*c*e^7*x^2 + 45*B*a^2*b*e^7*x^2 + 60*B*a*b^2*e^7*x^3 + 90*A*a*c^2*e^7*x^4 + 60*B*a^2*c*e^7*x^
3 + 90*A*b^2*c*e^7*x^4 + 180*A*b*c^2*e^7*x^5 + 180*B*a*c^2*e^7*x^5 + 6*A*b^3*d^2*e^5*x + 15*A*b^3*d*e^6*x^2 +
180*B*b^2*c*e^7*x^5 - 822*A*c^3*d^5*e^2*x + 12*B*b^3*d^3*e^4*x + 40*B*b^3*d*e^6*x^3 - 360*A*c^3*d*e^6*x^5 - 36
0*B*c^3*d*e^6*x^6 - 60*A*c^3*e^7*x^6*log(d + e*x) - 1875*A*c^3*d^4*e^3*x^2 + 30*B*b^3*d^2*e^5*x^2 - 2200*A*c^3
*d^3*e^4*x^3 - 1350*A*c^3*d^2*e^5*x^4 + 7725*B*c^3*d^5*e^2*x^2 + 8200*B*c^3*d^4*e^3*x^3 + 4050*B*c^3*d^3*e^4*x
^4 + 360*B*c^3*d^2*e^5*x^5 + 6*A*a^2*b*d*e^6 - 441*B*b*c^2*d^6*e - 60*A*c^3*d^6*e*log(d + e*x) + 36*A*a^2*b*e^
7*x + 90*A*a*c^2*d^2*e^5*x^2 + 450*A*b*c^2*d^3*e^4*x^2 + 90*A*b^2*c*d^2*e^5*x^2 + 450*B*a*c^2*d^3*e^4*x^2 + 60
0*A*b*c^2*d^2*e^5*x^3 + 600*B*a*c^2*d^2*e^5*x^3 - 5625*B*b*c^2*d^4*e^3*x^2 + 450*B*b^2*c*d^3*e^4*x^2 - 6600*B*
b*c^2*d^3*e^4*x^3 + 600*B*b^2*c*d^2*e^5*x^3 - 4050*B*b*c^2*d^2*e^5*x^4 - 900*A*c^3*d^4*e^3*x^2*log(d + e*x) -
1200*A*c^3*d^3*e^4*x^3*log(d + e*x) - 900*A*c^3*d^2*e^5*x^4*log(d + e*x) + 6300*B*c^3*d^5*e^2*x^2*log(d + e*x)
+ 8400*B*c^3*d^4*e^3*x^3*log(d + e*x) + 6300*B*c^3*d^3*e^4*x^4*log(d + e*x) + 2520*B*c^3*d^2*e^5*x^5*log(d +
e*x) + 6*A*a*b*c*d^3*e^4 + 12*B*a*b*c*d^4*e^3 - 180*B*b*c^2*d^6*e*log(d + e*x) + 120*A*a*b*c*e^7*x^3 + 18*A*a*
b^2*d*e^6*x + 180*B*a*b*c*e^7*x^4 + 18*A*a^2*c*d*e^6*x + 18*B*a^2*b*d*e^6*x + 2520*B*c^3*d^6*e*x*log(d + e*x)
+ 36*A*a*c^2*d^3*e^4*x + 18*B*a*b^2*d^2*e^5*x + 45*B*a*b^2*d*e^6*x^2 + 120*A*a*c^2*d*e^6*x^3 + 180*A*b*c^2*d^4
*e^3*x + 36*A*b^2*c*d^3*e^4*x + 180*B*a*c^2*d^4*e^3*x + 18*B*a^2*c*d^2*e^5*x + 45*B*a^2*c*d*e^6*x^2 + 120*A*b^
2*c*d*e^6*x^3 + 450*A*b*c^2*d*e^6*x^4 + 450*B*a*c^2*d*e^6*x^4 - 2466*B*b*c^2*d^5*e^2*x + 180*B*b^2*c*d^4*e^3*x
+ 450*B*b^2*c*d*e^6*x^4 - 1080*B*b*c^2*d*e^6*x^5 - 180*B*b*c^2*e^7*x^6*log(d + e*x) - 360*A*c^3*d^5*e^2*x*log
(d + e*x) - 360*A*c^3*d*e^6*x^5*log(d + e*x) + 420*B*c^3*d*e^6*x^6*log(d + e*x) + 180*B*a*b*c*d^2*e^5*x^2 - 10
80*B*b*c^2*d^5*e^2*x*log(d + e*x) - 1080*B*b*c^2*d*e^6*x^5*log(d + e*x) - 2700*B*b*c^2*d^4*e^3*x^2*log(d + e*x
) - 3600*B*b*c^2*d^3*e^4*x^3*log(d + e*x) - 2700*B*b*c^2*d^2*e^5*x^4*log(d + e*x) + 36*A*a*b*c*d^2*e^5*x + 90*
A*a*b*c*d*e^6*x^2 + 72*B*a*b*c*d^3*e^4*x + 240*B*a*b*c*d*e^6*x^3)/(60*e^8*(d + e*x)^6)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**7,x)

[Out]

Timed out

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