3.22.2 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^5} \, dx\)

Optimal. Leaf size=533 \[ \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{e^8 (d+e x)}-\frac {c x \left (A c e (5 c d-3 b e)-3 B \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^7}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8 (d+e x)^2}-\frac {\log (d+e x) \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{4 e^8 (d+e x)^4}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{3 e^8 (d+e x)^3}-\frac {c^2 x^2 (-A c e-3 b B e+5 B c d)}{2 e^6}+\frac {B c^3 x^3}{3 e^5} \]

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Rubi [A]  time = 0.94, antiderivative size = 531, 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} \begin {gather*} \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 )}{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 )}{2 e^8 (d+e x)^2}-\frac {c x \left (A c e (5 c d-3 b e)-3 B \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^7}-\frac {\log (d+e x) \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 )}{e^8}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{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 )}{3 e^8 (d+e x)^3}-\frac {c^2 x^2 (-A c e-3 b B e+5 B c d)}{2 e^6}+\frac {B c^3 x^3}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^5,x]

[Out]

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

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Mathematica [A]  time = 0.25, size = 496, normalized size = 0.93 \begin {gather*} \frac {-\frac {12 \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 )}{d+e x}+12 c e x \left (3 B \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+A c e (3 b e-5 c d)\right )+\frac {18 \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)^2}+12 \log (d+e x) \left (3 A c e \left (c e (a e-5 b d)+b^2 e^2+5 c^2 d^2\right )+B \left (15 c^2 d e (3 b d-a e)+3 b c e^2 (2 a e-5 b d)+b^3 e^3-35 c^3 d^3\right )\right )+\frac {3 (B d-A e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^4}-\frac {4 \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)^3}+6 c^2 e^2 x^2 (A c e+3 b B e-5 B c d)+4 B c^3 e^3 x^3}{12 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^5,x]

[Out]

(12*c*e*(A*c*e*(-5*c*d + 3*b*e) + 3*B*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e)))*x + 6*c^2*e^2*(-5*B*c*d + 3*
b*B*e + A*c*e)*x^2 + 4*B*c^3*e^3*x^3 + (3*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)/(d + e*x)^4 - (4*(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)^3 + (18*(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)^2 - (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*(3
5*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
))))/(d + e*x) + 12*(3*A*c*e*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e)) + B*(-35*c^3*d^3 + b^3*e^3 + 15*c^2*d*
e*(3*b*d - a*e) + 3*b*c*e^2*(-5*b*d + 2*a*e)))*Log[d + e*x])/(12*e^8)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^5} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^5,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(a + b*x + c*x^2)^3)/(d + e*x)^5, x]

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fricas [B]  time = 0.40, size = 1343, normalized size = 2.52

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)^5,x, algorithm="fricas")

[Out]

1/12*(4*B*c^3*e^7*x^7 - 319*B*c^3*d^7 - 3*A*a^3*e^7 + 171*(3*B*b*c^2 + A*c^3)*d^6*e - 231*(B*b^2*c + (B*a + A*
b)*c^2)*d^5*e^2 + 25*(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 - 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 - 2*(7*B*c^3*d*e^6 -
3*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 12*(7*B*c^3*d^2*e^5 - 3*(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 + 4*(139*B*c^3*d^3*e^4 - 51*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 36*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6)*
x^4 + 4*(136*B*c^3*d^4*e^3 - 24*(3*B*b*c^2 + A*c^3)*d^3*e^4 - 36*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^5 + 12*(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 - 6*(74
*B*c^3*d^5*e^2 - 66*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 126*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^4 - 18*(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 + 3*(B*a^2*b + A*a*
b^2 + A*a^2*c)*e^7)*x^2 - 4*(214*B*c^3*d^6*e - 126*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 186*(B*b^2*c + (B*a + A*b)*c^
2)*d^4*e^3 - 22*(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 + 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 - 12*(35*B*c^3*d^7 - 15*(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 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*
d^4*e^3 + (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 + 4*(35*B*c^3*d^4*e^3 - 15*(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 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e^6)*x^3 + 6*(35*B*c^3*d^5*e^2
- 15*(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 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b +
A*b^2)*c)*d^2*e^5)*x^2 + 4*(35*B*c^3*d^6*e - 15*(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 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*e^4)*x)*log(e*x + d))/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^
2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8)

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giac [B]  time = 0.42, size = 1567, normalized size = 2.94

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)^5,x, algorithm="giac")

[Out]

1/6*(2*B*c^3 - 3*(7*B*c^3*d*e - 3*B*b*c^2*e^2 - A*c^3*e^2)*e^(-1)/(x*e + d) + 18*(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)*(x*e + d)^3*e^(-8) + (35*
B*c^3*d^3 - 45*B*b*c^2*d^2*e - 15*A*c^3*d^2*e + 15*B*b^2*c*d*e^2 + 15*B*a*c^2*d*e^2 + 15*A*b*c^2*d*e^2 - B*b^3
*e^3 - 6*B*a*b*c*e^3 - 3*A*b^2*c*e^3 - 3*A*a*c^2*e^3)*e^(-8)*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - 1/12*(420*
B*c^3*d^4*e^36/(x*e + d) - 126*B*c^3*d^5*e^36/(x*e + d)^2 + 28*B*c^3*d^6*e^36/(x*e + d)^3 - 3*B*c^3*d^7*e^36/(
x*e + d)^4 - 720*B*b*c^2*d^3*e^37/(x*e + d) - 240*A*c^3*d^3*e^37/(x*e + d) + 270*B*b*c^2*d^4*e^37/(x*e + d)^2
+ 90*A*c^3*d^4*e^37/(x*e + d)^2 - 72*B*b*c^2*d^5*e^37/(x*e + d)^3 - 24*A*c^3*d^5*e^37/(x*e + d)^3 + 9*B*b*c^2*
d^6*e^37/(x*e + d)^4 + 3*A*c^3*d^6*e^37/(x*e + d)^4 + 360*B*b^2*c*d^2*e^38/(x*e + d) + 360*B*a*c^2*d^2*e^38/(x
*e + d) + 360*A*b*c^2*d^2*e^38/(x*e + d) - 180*B*b^2*c*d^3*e^38/(x*e + d)^2 - 180*B*a*c^2*d^3*e^38/(x*e + d)^2
- 180*A*b*c^2*d^3*e^38/(x*e + d)^2 + 60*B*b^2*c*d^4*e^38/(x*e + d)^3 + 60*B*a*c^2*d^4*e^38/(x*e + d)^3 + 60*A
*b*c^2*d^4*e^38/(x*e + d)^3 - 9*B*b^2*c*d^5*e^38/(x*e + d)^4 - 9*B*a*c^2*d^5*e^38/(x*e + d)^4 - 9*A*b*c^2*d^5*
e^38/(x*e + d)^4 - 48*B*b^3*d*e^39/(x*e + d) - 288*B*a*b*c*d*e^39/(x*e + d) - 144*A*b^2*c*d*e^39/(x*e + d) - 1
44*A*a*c^2*d*e^39/(x*e + d) + 36*B*b^3*d^2*e^39/(x*e + d)^2 + 216*B*a*b*c*d^2*e^39/(x*e + d)^2 + 108*A*b^2*c*d
^2*e^39/(x*e + d)^2 + 108*A*a*c^2*d^2*e^39/(x*e + d)^2 - 16*B*b^3*d^3*e^39/(x*e + d)^3 - 96*B*a*b*c*d^3*e^39/(
x*e + d)^3 - 48*A*b^2*c*d^3*e^39/(x*e + d)^3 - 48*A*a*c^2*d^3*e^39/(x*e + d)^3 + 3*B*b^3*d^4*e^39/(x*e + d)^4
+ 18*B*a*b*c*d^4*e^39/(x*e + d)^4 + 9*A*b^2*c*d^4*e^39/(x*e + d)^4 + 9*A*a*c^2*d^4*e^39/(x*e + d)^4 + 36*B*a*b
^2*e^40/(x*e + d) + 12*A*b^3*e^40/(x*e + d) + 36*B*a^2*c*e^40/(x*e + d) + 72*A*a*b*c*e^40/(x*e + d) - 54*B*a*b
^2*d*e^40/(x*e + d)^2 - 18*A*b^3*d*e^40/(x*e + d)^2 - 54*B*a^2*c*d*e^40/(x*e + d)^2 - 108*A*a*b*c*d*e^40/(x*e
+ d)^2 + 36*B*a*b^2*d^2*e^40/(x*e + d)^3 + 12*A*b^3*d^2*e^40/(x*e + d)^3 + 36*B*a^2*c*d^2*e^40/(x*e + d)^3 + 7
2*A*a*b*c*d^2*e^40/(x*e + d)^3 - 9*B*a*b^2*d^3*e^40/(x*e + d)^4 - 3*A*b^3*d^3*e^40/(x*e + d)^4 - 9*B*a^2*c*d^3
*e^40/(x*e + d)^4 - 18*A*a*b*c*d^3*e^40/(x*e + d)^4 + 18*B*a^2*b*e^41/(x*e + d)^2 + 18*A*a*b^2*e^41/(x*e + d)^
2 + 18*A*a^2*c*e^41/(x*e + d)^2 - 24*B*a^2*b*d*e^41/(x*e + d)^3 - 24*A*a*b^2*d*e^41/(x*e + d)^3 - 24*A*a^2*c*d
*e^41/(x*e + d)^3 + 9*B*a^2*b*d^2*e^41/(x*e + d)^4 + 9*A*a*b^2*d^2*e^41/(x*e + d)^4 + 9*A*a^2*c*d^2*e^41/(x*e
+ d)^4 + 4*B*a^3*e^42/(x*e + d)^3 + 12*A*a^2*b*e^42/(x*e + d)^3 - 3*B*a^3*d*e^42/(x*e + d)^4 - 9*A*a^2*b*d*e^4
2/(x*e + d)^4 + 3*A*a^3*e^43/(x*e + d)^4)*e^(-44)

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maple [B]  time = 0.07, size = 1605, normalized size = 3.01

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)^5,x)

[Out]

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

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maxima [A]  time = 0.72, size = 884, normalized size = 1.66 \begin {gather*} -\frac {319 \, B c^{3} d^{7} + 3 \, A a^{3} e^{7} - 171 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 231 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 25 \, {\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} + 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} + {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{6} + 12 \, {\left (35 \, B c^{3} d^{4} e^{3} - 20 \, {\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} - 4 \, {\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} + 18 \, {\left (63 \, B c^{3} d^{5} e^{2} - 35 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 50 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} e^{4} - 6 \, {\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} + {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{7}\right )} x^{2} + 4 \, {\left (259 \, B c^{3} d^{6} e - 141 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 195 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 22 \, {\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} + 3 \, {\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} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{7}\right )} x}{12 \, {\left (e^{12} x^{4} + 4 \, d e^{11} x^{3} + 6 \, d^{2} e^{10} x^{2} + 4 \, d^{3} e^{9} x + d^{4} e^{8}\right )}} + \frac {2 \, B c^{3} e^{2} x^{3} - 3 \, {\left (5 \, B c^{3} d e - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{2} + 6 \, {\left (15 \, B c^{3} d^{2} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e + 3 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} e^{2}\right )} x}{6 \, e^{7}} - \frac {{\left (35 \, B c^{3} d^{3} - 15 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{2} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \end {gather*}

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)^5,x, algorithm="maxima")

[Out]

-1/12*(319*B*c^3*d^7 + 3*A*a^3*e^7 - 171*(3*B*b*c^2 + A*c^3)*d^6*e + 231*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 -
25*(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 + 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 + 12*(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 + 18*(63*B*c^3*d^5*e^2 - 35*(3*B*b*c^2 + A*c^3)*d^4*
e^3 + 50*(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 + (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 + 4*(259*B*c^3*d^6*e - 141
*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 195*(B*b^2*c + (B*a + A*b)*c^2)*d^4*e^3 - 22*(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 + 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)/(e^12*x^4 + 4*d*e^11*x^3 + 6*d^2*e^10*x^2 + 4*d^3*e^9*x + d^4*e^8) + 1/6*(2*
B*c^3*e^2*x^3 - 3*(5*B*c^3*d*e - (3*B*b*c^2 + A*c^3)*e^2)*x^2 + 6*(15*B*c^3*d^2 - 5*(3*B*b*c^2 + A*c^3)*d*e +
3*(B*b^2*c + (B*a + A*b)*c^2)*e^2)*x)/e^7 - (35*B*c^3*d^3 - 15*(3*B*b*c^2 + A*c^3)*d^2*e + 15*(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)*log(e*x + d)/e^8

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mupad [B]  time = 2.54, size = 1106, normalized size = 2.08 \begin {gather*} x^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{2\,e^5}-\frac {5\,B\,c^3\,d}{2\,e^6}\right )-\frac {\frac {B\,a^3\,d\,e^6+3\,A\,a^3\,e^7+3\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6+9\,B\,a^2\,c\,d^3\,e^4+3\,A\,a^2\,c\,d^2\,e^5+9\,B\,a\,b^2\,d^3\,e^4+3\,A\,a\,b^2\,d^2\,e^5-150\,B\,a\,b\,c\,d^4\,e^3+18\,A\,a\,b\,c\,d^3\,e^4+231\,B\,a\,c^2\,d^5\,e^2-75\,A\,a\,c^2\,d^4\,e^3-25\,B\,b^3\,d^4\,e^3+3\,A\,b^3\,d^3\,e^4+231\,B\,b^2\,c\,d^5\,e^2-75\,A\,b^2\,c\,d^4\,e^3-513\,B\,b\,c^2\,d^6\,e+231\,A\,b\,c^2\,d^5\,e^2+319\,B\,c^3\,d^7-171\,A\,c^3\,d^6\,e}{12\,e}+x^3\,\left (3\,B\,a^2\,c\,e^6+3\,B\,a\,b^2\,e^6-24\,B\,a\,b\,c\,d\,e^5+6\,A\,a\,b\,c\,e^6+30\,B\,a\,c^2\,d^2\,e^4-12\,A\,a\,c^2\,d\,e^5-4\,B\,b^3\,d\,e^5+A\,b^3\,e^6+30\,B\,b^2\,c\,d^2\,e^4-12\,A\,b^2\,c\,d\,e^5-60\,B\,b\,c^2\,d^3\,e^3+30\,A\,b\,c^2\,d^2\,e^4+35\,B\,c^3\,d^4\,e^2-20\,A\,c^3\,d^3\,e^3\right )+x\,\left (\frac {B\,a^3\,e^6}{3}+B\,a^2\,b\,d\,e^5+A\,a^2\,b\,e^6+3\,B\,a^2\,c\,d^2\,e^4+A\,a^2\,c\,d\,e^5+3\,B\,a\,b^2\,d^2\,e^4+A\,a\,b^2\,d\,e^5-44\,B\,a\,b\,c\,d^3\,e^3+6\,A\,a\,b\,c\,d^2\,e^4+65\,B\,a\,c^2\,d^4\,e^2-22\,A\,a\,c^2\,d^3\,e^3-\frac {22\,B\,b^3\,d^3\,e^3}{3}+A\,b^3\,d^2\,e^4+65\,B\,b^2\,c\,d^4\,e^2-22\,A\,b^2\,c\,d^3\,e^3-141\,B\,b\,c^2\,d^5\,e+65\,A\,b\,c^2\,d^4\,e^2+\frac {259\,B\,c^3\,d^6}{3}-47\,A\,c^3\,d^5\,e\right )+x^2\,\left (\frac {3\,B\,a^2\,b\,e^6}{2}+\frac {9\,B\,a^2\,c\,d\,e^5}{2}+\frac {3\,A\,a^2\,c\,e^6}{2}+\frac {9\,B\,a\,b^2\,d\,e^5}{2}+\frac {3\,A\,a\,b^2\,e^6}{2}-54\,B\,a\,b\,c\,d^2\,e^4+9\,A\,a\,b\,c\,d\,e^5+75\,B\,a\,c^2\,d^3\,e^3-27\,A\,a\,c^2\,d^2\,e^4-9\,B\,b^3\,d^2\,e^4+\frac {3\,A\,b^3\,d\,e^5}{2}+75\,B\,b^2\,c\,d^3\,e^3-27\,A\,b^2\,c\,d^2\,e^4-\frac {315\,B\,b\,c^2\,d^4\,e^2}{2}+75\,A\,b\,c^2\,d^3\,e^3+\frac {189\,B\,c^3\,d^5\,e}{2}-\frac {105\,A\,c^3\,d^4\,e^2}{2}\right )}{d^4\,e^7+4\,d^3\,e^8\,x+6\,d^2\,e^9\,x^2+4\,d\,e^{10}\,x^3+e^{11}\,x^4}-x\,\left (\frac {5\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^5}-\frac {5\,B\,c^3\,d}{e^6}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^5}+\frac {10\,B\,c^3\,d^2}{e^7}\right )+\frac {\ln \left (d+e\,x\right )\,\left (B\,b^3\,e^3-15\,B\,b^2\,c\,d\,e^2+3\,A\,b^2\,c\,e^3+45\,B\,b\,c^2\,d^2\,e-15\,A\,b\,c^2\,d\,e^2+6\,B\,a\,b\,c\,e^3-35\,B\,c^3\,d^3+15\,A\,c^3\,d^2\,e-15\,B\,a\,c^2\,d\,e^2+3\,A\,a\,c^2\,e^3\right )}{e^8}+\frac {B\,c^3\,x^3}{3\,e^5} \end {gather*}

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)^5,x)

[Out]

x^2*((A*c^3 + 3*B*b*c^2)/(2*e^5) - (5*B*c^3*d)/(2*e^6)) - ((3*A*a^3*e^7 + 319*B*c^3*d^7 + B*a^3*d*e^6 - 171*A*
c^3*d^6*e + 3*A*b^3*d^3*e^4 - 25*B*b^3*d^4*e^3 + 3*A*a*b^2*d^2*e^5 - 75*A*a*c^2*d^4*e^3 + 3*A*a^2*c*d^2*e^5 +
9*B*a*b^2*d^3*e^4 + 3*B*a^2*b*d^2*e^5 + 231*A*b*c^2*d^5*e^2 - 75*A*b^2*c*d^4*e^3 + 231*B*a*c^2*d^5*e^2 + 9*B*a
^2*c*d^3*e^4 + 231*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 - 513*B*b*c^2*d^6*e + 18*A*a*b*c*d^3*e^4 - 150*B*a*b*c*d^
4*e^3)/(12*e) + x^3*(A*b^3*e^6 + 3*B*a*b^2*e^6 + 3*B*a^2*c*e^6 - 4*B*b^3*d*e^5 - 20*A*c^3*d^3*e^3 + 35*B*c^3*d
^4*e^2 + 30*A*b*c^2*d^2*e^4 + 30*B*a*c^2*d^2*e^4 - 60*B*b*c^2*d^3*e^3 + 30*B*b^2*c*d^2*e^4 + 6*A*a*b*c*e^6 - 1
2*A*a*c^2*d*e^5 - 12*A*b^2*c*d*e^5 - 24*B*a*b*c*d*e^5) + x*((B*a^3*e^6)/3 + (259*B*c^3*d^6)/3 + A*a^2*b*e^6 -
47*A*c^3*d^5*e + A*b^3*d^2*e^4 - (22*B*b^3*d^3*e^3)/3 - 22*A*a*c^2*d^3*e^3 + 3*B*a*b^2*d^2*e^4 + 65*A*b*c^2*d^
4*e^2 - 22*A*b^2*c*d^3*e^3 + 65*B*a*c^2*d^4*e^2 + 3*B*a^2*c*d^2*e^4 + 65*B*b^2*c*d^4*e^2 + A*a*b^2*d*e^5 + A*a
^2*c*d*e^5 + B*a^2*b*d*e^5 - 141*B*b*c^2*d^5*e + 6*A*a*b*c*d^2*e^4 - 44*B*a*b*c*d^3*e^3) + x^2*((3*A*a*b^2*e^6
)/2 + (3*A*a^2*c*e^6)/2 + (3*B*a^2*b*e^6)/2 + (3*A*b^3*d*e^5)/2 + (189*B*c^3*d^5*e)/2 - (105*A*c^3*d^4*e^2)/2
- 9*B*b^3*d^2*e^4 - 27*A*a*c^2*d^2*e^4 + 75*A*b*c^2*d^3*e^3 - 27*A*b^2*c*d^2*e^4 + 75*B*a*c^2*d^3*e^3 - (315*B
*b*c^2*d^4*e^2)/2 + 75*B*b^2*c*d^3*e^3 + (9*B*a*b^2*d*e^5)/2 + (9*B*a^2*c*d*e^5)/2 - 54*B*a*b*c*d^2*e^4 + 9*A*
a*b*c*d*e^5))/(d^4*e^7 + e^11*x^4 + 4*d^3*e^8*x + 4*d*e^10*x^3 + 6*d^2*e^9*x^2) - x*((5*d*((A*c^3 + 3*B*b*c^2)
/e^5 - (5*B*c^3*d)/e^6))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^5 + (10*B*c^3*d^2)/e^7) + (log(d + e*x)*(B*
b^3*e^3 - 35*B*c^3*d^3 + 3*A*a*c^2*e^3 + 3*A*b^2*c*e^3 + 15*A*c^3*d^2*e + 6*B*a*b*c*e^3 - 15*A*b*c^2*d*e^2 - 1
5*B*a*c^2*d*e^2 + 45*B*b*c^2*d^2*e - 15*B*b^2*c*d*e^2))/e^8 + (B*c^3*x^3)/(3*e^5)

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

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)**5,x)

[Out]

Timed out

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