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

Optimal. Leaf size=531 \[ -\frac {x \left (A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac {c x^3 \left (A c e (c d-b e)-B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{e^5}-\frac {3 \log (d+e x) \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8}-\frac {x^2 \left (B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{2 e^6}+\frac {\left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{e^8 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac {B c^3 x^5}{5 e^3} \]

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Rubi [A]  time = 1.08, antiderivative size = 530, 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 {x \left (A e \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )-b^2 e^3 (b d-a e)-2 c^2 d^2 e (5 b d-3 a e)+5 c^3 d^4\right )\right )}{e^7}-\frac {c x^3 \left (A c e (c d-b e)-B \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{e^5}-\frac {x^2 \left (B \left (-9 c^2 d e (2 b d-a e)+3 b c e^2 (3 b d-2 a e)-b^3 e^3+10 c^3 d^3\right )-3 A c e \left (-c e (3 b d-a e)+b^2 e^2+2 c^2 d^2\right )\right )}{2 e^6}-\frac {3 \log (d+e x) \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}-\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 )}{e^8 (d+e x)}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{2 e^8 (d+e x)^2}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac {B c^3 x^5}{5 e^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(((A*e*(10*c^3*d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*B*(5*c^3*d^4 - 2*c^2*
d^2*e*(5*b*d - 3*a*e) - b^2*e^3*(b*d - a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x)/e^7) - ((B*(10*c^3*
d^3 - b^3*e^3 + 3*b*c*e^2*(3*b*d - 2*a*e) - 9*c^2*d*e*(2*b*d - a*e)) - 3*A*c*e*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b
*d - a*e)))*x^2)/(2*e^6) - (c*(A*c*e*(c*d - b*e) - B*(2*c^2*d^2 + b^2*e^2 - c*e*(3*b*d - a*e)))*x^3)/e^5 - (c^
2*(3*B*c*d - 3*b*B*e - A*c*e)*x^4)/(4*e^4) + (B*c^3*x^5)/(5*e^3) + ((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3)/(2*
e^8*(d + e*x)^2) - ((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)))/(e^8*(d +
e*x)) - (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)))*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)^3} \, dx &=\int \left (\frac {-A e \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )+3 B \left (5 c^3 d^4-2 c^2 d^2 e (5 b d-3 a e)-b^2 e^3 (b d-a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )}{e^7}+\frac {\left (-B \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )+3 A c e \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x}{e^6}+\frac {3 c \left (-A c e (c d-b e)+B \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^2}{e^5}+\frac {c^2 (-3 B c d+3 b B e+A c e) x^3}{e^4}+\frac {B c^3 x^4}{e^3}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (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 )}{e^7 (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^7 (d+e x)}\right ) \, dx\\ &=-\frac {\left (A e \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 B \left (5 c^3 d^4-2 c^2 d^2 e (5 b d-3 a e)-b^2 e^3 (b d-a e)+c e^2 \left (6 b^2 d^2-6 a b d e+a^2 e^2\right )\right )\right ) x}{e^7}-\frac {\left (B \left (10 c^3 d^3-b^3 e^3+3 b c e^2 (3 b d-2 a e)-9 c^2 d e (2 b d-a e)\right )-3 A c e \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c \left (A c e (c d-b e)-B \left (2 c^2 d^2+b^2 e^2-c e (3 b d-a e)\right )\right ) x^3}{e^5}-\frac {c^2 (3 B c d-3 b B e-A c e) x^4}{4 e^4}+\frac {B c^3 x^5}{5 e^3}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{2 e^8 (d+e x)^2}-\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^8 (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 ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]  time = 0.28, size = 503, normalized size = 0.95 \begin {gather*} \frac {20 e x \left (3 B \left (c e^2 \left (a^2 e^2-6 a b d e+6 b^2 d^2\right )+b^2 e^3 (a e-b d)+2 c^2 d^2 e (3 a e-5 b d)+5 c^3 d^4\right )+A e \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )\right )+20 c e^3 x^3 \left (B \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+A c e (b e-c d)\right )-60 \log (d+e x) \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 )+10 e^2 x^2 \left (3 A c e \left (c e (a e-3 b d)+b^2 e^2+2 c^2 d^2\right )+B \left (9 c^2 d e (2 b d-a e)+3 b c e^2 (2 a e-3 b d)+b^3 e^3-10 c^3 d^3\right )\right )-\frac {20 \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}+\frac {10 (B d-A e) \left (e (a e-b d)+c d^2\right )^3}{(d+e x)^2}+5 c^2 e^4 x^4 (A c e+3 b B e-3 B c d)+4 B c^3 e^5 x^5}{20 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e*(A*e*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c*e^2*(-3*b*d + 2*a*e)) + 3*B*(5*c^3*d^4 + b
^2*e^3*(-(b*d) + a*e) + 2*c^2*d^2*e*(-5*b*d + 3*a*e) + c*e^2*(6*b^2*d^2 - 6*a*b*d*e + a^2*e^2)))*x + 10*e^2*(3
*A*c*e*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)) + B*(-10*c^3*d^3 + b^3*e^3 + 9*c^2*d*e*(2*b*d - a*e) + 3*b*c
*e^2*(-3*b*d + 2*a*e)))*x^2 + 20*c*e^3*(A*c*e*(-(c*d) + b*e) + B*(2*c^2*d^2 + b^2*e^2 + c*e*(-3*b*d + a*e)))*x
^3 + 5*c^2*e^4*(-3*B*c*d + 3*b*B*e + A*c*e)*x^4 + 4*B*c^3*e^5*x^5 + (10*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))
^3)/(d + e*x)^2 - (20*(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) - 60*(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)))*Log[d + e*x])/(20*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)^3} \, 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)^3,x]

[Out]

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

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fricas [B]  time = 0.40, size = 1311, normalized size = 2.47

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)^3,x, algorithm="fricas")

[Out]

1/20*(4*B*c^3*e^7*x^7 - 130*B*c^3*d^7 - 10*A*a^3*e^7 + 110*(3*B*b*c^2 + A*c^3)*d^6*e - 270*(B*b^2*c + (B*a + A
*b)*c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 - 50*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 +
2*A*a*b)*c)*d^3*e^4 + 90*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 - 10*(B*a^3 + 3*A*a^2*b)*d*e^6 - (7*B*c^3*d*e^
6 - 5*(3*B*b*c^2 + A*c^3)*e^7)*x^6 + 2*(7*B*c^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d*e^6 + 10*(B*b^2*c + (B*a + A
*b)*c^2)*e^7)*x^5 - 5*(7*B*c^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^2*e^5 + 10*(B*b^2*c + (B*a + A*b)*c^2)*d*e^6
- 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 - 5*(3*B*b*c^2 + A*c^3)*d^3*e^4
+ 10*(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 + 10*(50*B*c^3*d^5*e^2 - 34*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c
+ (B*a + A*b)*c^2)*d^3*e^4 - 11*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^5 + 4*(3*B*a*b^2 + A*b^3 +
3*(B*a^2 + 2*A*a*b)*c)*d*e^6)*x^2 + 20*(8*B*c^3*d^6*e - 4*(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 - 2*(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 - 60*(7*B*c^3*d^7 - 5*(3*
B*b*c^2 + A*c^3)*d^6*e + 10*(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 - (B*a^2*b + A*a*b^2 + A*a^2*c)*d^2*e^5 + (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 + 2*(7*B*c^3*d^6*e - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 10*(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 - (
B*a^2*b + A*a*b^2 + A*a^2*c)*d*e^6)*x)*log(e*x + d))/(e^10*x^2 + 2*d*e^9*x + d^2*e^8)

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giac [A]  time = 0.20, size = 1048, normalized size = 1.97 \begin {gather*} -3 \, {\left (7 \, B c^{3} d^{5} - 15 \, B b c^{2} d^{4} e - 5 \, A c^{3} d^{4} e + 10 \, B b^{2} c d^{3} e^{2} + 10 \, B a c^{2} d^{3} e^{2} + 10 \, A b c^{2} d^{3} e^{2} - 2 \, B b^{3} d^{2} e^{3} - 12 \, B a b c d^{2} e^{3} - 6 \, A b^{2} c d^{2} e^{3} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a b^{2} d e^{4} + A b^{3} d e^{4} + 3 \, B a^{2} c d e^{4} + 6 \, A a b c d e^{4} - B a^{2} b e^{5} - A a b^{2} e^{5} - A a^{2} c e^{5}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{20} \, {\left (4 \, B c^{3} x^{5} e^{12} - 15 \, B c^{3} d x^{4} e^{11} + 40 \, B c^{3} d^{2} x^{3} e^{10} - 100 \, B c^{3} d^{3} x^{2} e^{9} + 300 \, B c^{3} d^{4} x e^{8} + 15 \, B b c^{2} x^{4} e^{12} + 5 \, A c^{3} x^{4} e^{12} - 60 \, B b c^{2} d x^{3} e^{11} - 20 \, A c^{3} d x^{3} e^{11} + 180 \, B b c^{2} d^{2} x^{2} e^{10} + 60 \, A c^{3} d^{2} x^{2} e^{10} - 600 \, B b c^{2} d^{3} x e^{9} - 200 \, A c^{3} d^{3} x e^{9} + 20 \, B b^{2} c x^{3} e^{12} + 20 \, B a c^{2} x^{3} e^{12} + 20 \, A b c^{2} x^{3} e^{12} - 90 \, B b^{2} c d x^{2} e^{11} - 90 \, B a c^{2} d x^{2} e^{11} - 90 \, A b c^{2} d x^{2} e^{11} + 360 \, B b^{2} c d^{2} x e^{10} + 360 \, B a c^{2} d^{2} x e^{10} + 360 \, A b c^{2} d^{2} x e^{10} + 10 \, B b^{3} x^{2} e^{12} + 60 \, B a b c x^{2} e^{12} + 30 \, A b^{2} c x^{2} e^{12} + 30 \, A a c^{2} x^{2} e^{12} - 60 \, B b^{3} d x e^{11} - 360 \, B a b c d x e^{11} - 180 \, A b^{2} c d x e^{11} - 180 \, A a c^{2} d x e^{11} + 60 \, B a b^{2} x e^{12} + 20 \, A b^{3} x e^{12} + 60 \, B a^{2} c x e^{12} + 120 \, A a b c x e^{12}\right )} e^{\left (-15\right )} - \frac {{\left (13 \, B c^{3} d^{7} - 33 \, B b c^{2} d^{6} e - 11 \, A c^{3} d^{6} e + 27 \, B b^{2} c d^{5} e^{2} + 27 \, B a c^{2} d^{5} e^{2} + 27 \, A b c^{2} d^{5} e^{2} - 7 \, B b^{3} d^{4} e^{3} - 42 \, B a b c d^{4} e^{3} - 21 \, A b^{2} c d^{4} e^{3} - 21 \, A a c^{2} d^{4} e^{3} + 15 \, B a b^{2} d^{3} e^{4} + 5 \, A b^{3} d^{3} e^{4} + 15 \, B a^{2} c d^{3} e^{4} + 30 \, A a b c d^{3} e^{4} - 9 \, B a^{2} b d^{2} e^{5} - 9 \, A a b^{2} d^{2} e^{5} - 9 \, 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} + 2 \, {\left (7 \, B c^{3} d^{6} e - 18 \, B b c^{2} d^{5} e^{2} - 6 \, A c^{3} d^{5} e^{2} + 15 \, B b^{2} c d^{4} e^{3} + 15 \, B a c^{2} d^{4} e^{3} + 15 \, 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} + 9 \, B a b^{2} d^{2} e^{5} + 3 \, A b^{3} d^{2} e^{5} + 9 \, B a^{2} c d^{2} e^{5} + 18 \, 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}\right )} x\right )} e^{\left (-8\right )}}{2 \, {\left (x e + d\right )}^{2}} \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)^3,x, algorithm="giac")

[Out]

-3*(7*B*c^3*d^5 - 15*B*b*c^2*d^4*e - 5*A*c^3*d^4*e + 10*B*b^2*c*d^3*e^2 + 10*B*a*c^2*d^3*e^2 + 10*A*b*c^2*d^3*
e^2 - 2*B*b^3*d^2*e^3 - 12*B*a*b*c*d^2*e^3 - 6*A*b^2*c*d^2*e^3 - 6*A*a*c^2*d^2*e^3 + 3*B*a*b^2*d*e^4 + A*b^3*d
*e^4 + 3*B*a^2*c*d*e^4 + 6*A*a*b*c*d*e^4 - B*a^2*b*e^5 - A*a*b^2*e^5 - A*a^2*c*e^5)*e^(-8)*log(abs(x*e + d)) +
1/20*(4*B*c^3*x^5*e^12 - 15*B*c^3*d*x^4*e^11 + 40*B*c^3*d^2*x^3*e^10 - 100*B*c^3*d^3*x^2*e^9 + 300*B*c^3*d^4*
x*e^8 + 15*B*b*c^2*x^4*e^12 + 5*A*c^3*x^4*e^12 - 60*B*b*c^2*d*x^3*e^11 - 20*A*c^3*d*x^3*e^11 + 180*B*b*c^2*d^2
*x^2*e^10 + 60*A*c^3*d^2*x^2*e^10 - 600*B*b*c^2*d^3*x*e^9 - 200*A*c^3*d^3*x*e^9 + 20*B*b^2*c*x^3*e^12 + 20*B*a
*c^2*x^3*e^12 + 20*A*b*c^2*x^3*e^12 - 90*B*b^2*c*d*x^2*e^11 - 90*B*a*c^2*d*x^2*e^11 - 90*A*b*c^2*d*x^2*e^11 +
360*B*b^2*c*d^2*x*e^10 + 360*B*a*c^2*d^2*x*e^10 + 360*A*b*c^2*d^2*x*e^10 + 10*B*b^3*x^2*e^12 + 60*B*a*b*c*x^2*
e^12 + 30*A*b^2*c*x^2*e^12 + 30*A*a*c^2*x^2*e^12 - 60*B*b^3*d*x*e^11 - 360*B*a*b*c*d*x*e^11 - 180*A*b^2*c*d*x*
e^11 - 180*A*a*c^2*d*x*e^11 + 60*B*a*b^2*x*e^12 + 20*A*b^3*x*e^12 + 60*B*a^2*c*x*e^12 + 120*A*a*b*c*x*e^12)*e^
(-15) - 1/2*(13*B*c^3*d^7 - 33*B*b*c^2*d^6*e - 11*A*c^3*d^6*e + 27*B*b^2*c*d^5*e^2 + 27*B*a*c^2*d^5*e^2 + 27*A
*b*c^2*d^5*e^2 - 7*B*b^3*d^4*e^3 - 42*B*a*b*c*d^4*e^3 - 21*A*b^2*c*d^4*e^3 - 21*A*a*c^2*d^4*e^3 + 15*B*a*b^2*d
^3*e^4 + 5*A*b^3*d^3*e^4 + 15*B*a^2*c*d^3*e^4 + 30*A*a*b*c*d^3*e^4 - 9*B*a^2*b*d^2*e^5 - 9*A*a*b^2*d^2*e^5 - 9
*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 + 2*(7*B*c^3*d^6*e - 18*B*b*c^2*d^5*e^2 - 6*A*c^3
*d^5*e^2 + 15*B*b^2*c*d^4*e^3 + 15*B*a*c^2*d^4*e^3 + 15*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 + 9*B*a*b^2*d^2*e^5 + 3*A*b^3*d^2*e^5 + 9*B*a^2*c*d^2*e^5 + 18*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)^2

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maple [B]  time = 0.06, size = 1483, normalized size = 2.79

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)^3,x)

[Out]

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

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maxima [A]  time = 0.79, size = 861, normalized size = 1.62 \begin {gather*} -\frac {13 \, B c^{3} d^{7} + A a^{3} e^{7} - 11 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 27 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5} e^{2} - 7 \, {\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} + 5 \, {\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} - 9 \, {\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} + 2 \, {\left (7 \, B c^{3} d^{6} e - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} e^{3} - 4 \, {\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} - 6 \, {\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}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B c^{3} e^{4} x^{5} - 5 \, {\left (3 \, B c^{3} d e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B c^{3} d^{2} e^{2} - {\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^{4}\right )} x^{3} - 10 \, {\left (10 \, B c^{3} d^{3} e - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 9 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d e^{3} - {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B c^{3} d^{4} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} e^{2} - 3 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{4}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B c^{3} d^{5} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 10 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} 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^{2} e^{3} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d e^{4} - {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{5}\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)^3,x, algorithm="maxima")

[Out]

-1/2*(13*B*c^3*d^7 + A*a^3*e^7 - 11*(3*B*b*c^2 + A*c^3)*d^6*e + 27*(B*b^2*c + (B*a + A*b)*c^2)*d^5*e^2 - 7*(B*
b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4*e^3 + 5*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3*e^4 - 9*(
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^6*e - 6*(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 - 4*(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 - 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^10*x^2 + 2*d*e^9*x + d^2*e^8) + 1/20*(4*B*c^3*e^4*x^5 - 5*(3*B*c^3*d*e^3 - (3*B*b*c^2 + A*c^3)*e^4
)*x^4 + 20*(2*B*c^3*d^2*e^2 - (3*B*b*c^2 + A*c^3)*d*e^3 + (B*b^2*c + (B*a + A*b)*c^2)*e^4)*x^3 - 10*(10*B*c^3*
d^3*e - 6*(3*B*b*c^2 + A*c^3)*d^2*e^2 + 9*(B*b^2*c + (B*a + A*b)*c^2)*d*e^3 - (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b
+ A*b^2)*c)*e^4)*x^2 + 20*(15*B*c^3*d^4 - 10*(3*B*b*c^2 + A*c^3)*d^3*e + 18*(B*b^2*c + (B*a + A*b)*c^2)*d^2*e^
2 - 3*(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)*x)/
e^7 - 3*(7*B*c^3*d^5 - 5*(3*B*b*c^2 + A*c^3)*d^4*e + 10*(B*b^2*c + (B*a + A*b)*c^2)*d^3*e^2 - 2*(B*b^3 + 3*A*a
*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*e^3 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e^4 - (B*a^2*b + A*a*b^2
+ A*a^2*c)*e^5)*log(e*x + d)/e^8

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mupad [B]  time = 2.51, size = 1297, normalized size = 2.44 \begin {gather*} x\,\left (\frac {3\,B\,c\,a^2+3\,B\,a\,b^2+6\,A\,c\,a\,b+A\,b^3}{e^3}+\frac {3\,d^2\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e^2}-\frac {3\,d\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e^3}-\frac {3\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^2}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e}-\frac {B\,c^3\,d^3}{e^6}\right )}{e}-\frac {d^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e^3}\right )-\frac {\frac {B\,a^3\,d\,e^6+A\,a^3\,e^7-9\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6+15\,B\,a^2\,c\,d^3\,e^4-9\,A\,a^2\,c\,d^2\,e^5+15\,B\,a\,b^2\,d^3\,e^4-9\,A\,a\,b^2\,d^2\,e^5-42\,B\,a\,b\,c\,d^4\,e^3+30\,A\,a\,b\,c\,d^3\,e^4+27\,B\,a\,c^2\,d^5\,e^2-21\,A\,a\,c^2\,d^4\,e^3-7\,B\,b^3\,d^4\,e^3+5\,A\,b^3\,d^3\,e^4+27\,B\,b^2\,c\,d^5\,e^2-21\,A\,b^2\,c\,d^4\,e^3-33\,B\,b\,c^2\,d^6\,e+27\,A\,b\,c^2\,d^5\,e^2+13\,B\,c^3\,d^7-11\,A\,c^3\,d^6\,e}{2\,e}+x\,\left (B\,a^3\,e^6-6\,B\,a^2\,b\,d\,e^5+3\,A\,a^2\,b\,e^6+9\,B\,a^2\,c\,d^2\,e^4-6\,A\,a^2\,c\,d\,e^5+9\,B\,a\,b^2\,d^2\,e^4-6\,A\,a\,b^2\,d\,e^5-24\,B\,a\,b\,c\,d^3\,e^3+18\,A\,a\,b\,c\,d^2\,e^4+15\,B\,a\,c^2\,d^4\,e^2-12\,A\,a\,c^2\,d^3\,e^3-4\,B\,b^3\,d^3\,e^3+3\,A\,b^3\,d^2\,e^4+15\,B\,b^2\,c\,d^4\,e^2-12\,A\,b^2\,c\,d^3\,e^3-18\,B\,b\,c^2\,d^5\,e+15\,A\,b\,c^2\,d^4\,e^2+7\,B\,c^3\,d^6-6\,A\,c^3\,d^5\,e\right )}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}-x^3\,\left (\frac {d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{3\,e^3}+\frac {B\,c^3\,d^2}{e^5}\right )+x^4\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{4\,e^3}-\frac {3\,B\,c^3\,d}{4\,e^4}\right )+x^2\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{2\,e^3}-\frac {3\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{2\,e^2}+\frac {3\,d\,\left (\frac {3\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^3}-\frac {3\,B\,c^3\,d}{e^4}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{2\,e}-\frac {B\,c^3\,d^3}{2\,e^6}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,b\,e^5-9\,B\,a^2\,c\,d\,e^4+3\,A\,a^2\,c\,e^5-9\,B\,a\,b^2\,d\,e^4+3\,A\,a\,b^2\,e^5+36\,B\,a\,b\,c\,d^2\,e^3-18\,A\,a\,b\,c\,d\,e^4-30\,B\,a\,c^2\,d^3\,e^2+18\,A\,a\,c^2\,d^2\,e^3+6\,B\,b^3\,d^2\,e^3-3\,A\,b^3\,d\,e^4-30\,B\,b^2\,c\,d^3\,e^2+18\,A\,b^2\,c\,d^2\,e^3+45\,B\,b\,c^2\,d^4\,e-30\,A\,b\,c^2\,d^3\,e^2-21\,B\,c^3\,d^5+15\,A\,c^3\,d^4\,e\right )}{e^8}+\frac {B\,c^3\,x^5}{5\,e^3} \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)^3,x)

[Out]

x*((A*b^3 + 3*B*a*b^2 + 3*B*a^2*c + 6*A*a*b*c)/e^3 + (3*d^2*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))
/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^3 + (3*B*c^3*d^2)/e^5))/e^2 - (3*d*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c
+ 6*B*a*b*c)/e^3 - (3*d^2*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^2 + (3*d*((3*d*((A*c^3 + 3*B*b*c^2)/e
^3 - (3*B*c^3*d)/e^4))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^3 + (3*B*c^3*d^2)/e^5))/e - (B*c^3*d^3)/e^6))
/e - (d^3*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e^3) - ((A*a^3*e^7 + 13*B*c^3*d^7 + B*a^3*d*e^6 - 11*A*
c^3*d^6*e + 5*A*b^3*d^3*e^4 - 7*B*b^3*d^4*e^3 - 9*A*a*b^2*d^2*e^5 - 21*A*a*c^2*d^4*e^3 - 9*A*a^2*c*d^2*e^5 + 1
5*B*a*b^2*d^3*e^4 - 9*B*a^2*b*d^2*e^5 + 27*A*b*c^2*d^5*e^2 - 21*A*b^2*c*d^4*e^3 + 27*B*a*c^2*d^5*e^2 + 15*B*a^
2*c*d^3*e^4 + 27*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 - 33*B*b*c^2*d^6*e + 30*A*a*b*c*d^3*e^4 - 42*B*a*b*c*d^4*e^
3)/(2*e) + x*(B*a^3*e^6 + 7*B*c^3*d^6 + 3*A*a^2*b*e^6 - 6*A*c^3*d^5*e + 3*A*b^3*d^2*e^4 - 4*B*b^3*d^3*e^3 - 12
*A*a*c^2*d^3*e^3 + 9*B*a*b^2*d^2*e^4 + 15*A*b*c^2*d^4*e^2 - 12*A*b^2*c*d^3*e^3 + 15*B*a*c^2*d^4*e^2 + 9*B*a^2*
c*d^2*e^4 + 15*B*b^2*c*d^4*e^2 - 6*A*a*b^2*d*e^5 - 6*A*a^2*c*d*e^5 - 6*B*a^2*b*d*e^5 - 18*B*b*c^2*d^5*e + 18*A
*a*b*c*d^2*e^4 - 24*B*a*b*c*d^3*e^3))/(d^2*e^7 + e^9*x^2 + 2*d*e^8*x) - x^3*((d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*
B*c^3*d)/e^4))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/(3*e^3) + (B*c^3*d^2)/e^5) + x^4*((A*c^3 + 3*B*b*c^2)/(
4*e^3) - (3*B*c^3*d)/(4*e^4)) + x^2*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*B*a*b*c)/(2*e^3) - (3*d^2*((A*c^3 + 3*
B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/(2*e^2) + (3*d*((3*d*((A*c^3 + 3*B*b*c^2)/e^3 - (3*B*c^3*d)/e^4))/e - (3*A*b*
c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^3 + (3*B*c^3*d^2)/e^5))/(2*e) - (B*c^3*d^3)/(2*e^6)) + (log(d + e*x)*(3*A*a*b^2
*e^5 - 21*B*c^3*d^5 + 3*A*a^2*c*e^5 + 3*B*a^2*b*e^5 - 3*A*b^3*d*e^4 + 15*A*c^3*d^4*e + 6*B*b^3*d^2*e^3 + 18*A*
a*c^2*d^2*e^3 - 30*A*b*c^2*d^3*e^2 + 18*A*b^2*c*d^2*e^3 - 30*B*a*c^2*d^3*e^2 - 30*B*b^2*c*d^3*e^2 - 9*B*a*b^2*
d*e^4 - 9*B*a^2*c*d*e^4 + 45*B*b*c^2*d^4*e + 36*B*a*b*c*d^2*e^3 - 18*A*a*b*c*d*e^4))/e^8 + (B*c^3*x^5)/(5*e^3)

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sympy [B]  time = 86.43, size = 1149, normalized size = 2.16 \begin {gather*} \frac {B c^{3} x^{5}}{5 e^{3}} + x^{4} \left (\frac {A c^{3}}{4 e^{3}} + \frac {3 B b c^{2}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}}\right ) + x^{3} \left (\frac {A b c^{2}}{e^{3}} - \frac {A c^{3} d}{e^{4}} + \frac {B a c^{2}}{e^{3}} + \frac {B b^{2} c}{e^{3}} - \frac {3 B b c^{2} d}{e^{4}} + \frac {2 B c^{3} d^{2}}{e^{5}}\right ) + x^{2} \left (\frac {3 A a c^{2}}{2 e^{3}} + \frac {3 A b^{2} c}{2 e^{3}} - \frac {9 A b c^{2} d}{2 e^{4}} + \frac {3 A c^{3} d^{2}}{e^{5}} + \frac {3 B a b c}{e^{3}} - \frac {9 B a c^{2} d}{2 e^{4}} + \frac {B b^{3}}{2 e^{3}} - \frac {9 B b^{2} c d}{2 e^{4}} + \frac {9 B b c^{2} d^{2}}{e^{5}} - \frac {5 B c^{3} d^{3}}{e^{6}}\right ) + x \left (\frac {6 A a b c}{e^{3}} - \frac {9 A a c^{2} d}{e^{4}} + \frac {A b^{3}}{e^{3}} - \frac {9 A b^{2} c d}{e^{4}} + \frac {18 A b c^{2} d^{2}}{e^{5}} - \frac {10 A c^{3} d^{3}}{e^{6}} + \frac {3 B a^{2} c}{e^{3}} + \frac {3 B a b^{2}}{e^{3}} - \frac {18 B a b c d}{e^{4}} + \frac {18 B a c^{2} d^{2}}{e^{5}} - \frac {3 B b^{3} d}{e^{4}} + \frac {18 B b^{2} c d^{2}}{e^{5}} - \frac {30 B b c^{2} d^{3}}{e^{6}} + \frac {15 B c^{3} d^{4}}{e^{7}}\right ) + \frac {- A a^{3} e^{7} - 3 A a^{2} b d e^{6} + 9 A a^{2} c d^{2} e^{5} + 9 A a b^{2} d^{2} e^{5} - 30 A a b c d^{3} e^{4} + 21 A a c^{2} d^{4} e^{3} - 5 A b^{3} d^{3} e^{4} + 21 A b^{2} c d^{4} e^{3} - 27 A b c^{2} d^{5} e^{2} + 11 A c^{3} d^{6} e - B a^{3} d e^{6} + 9 B a^{2} b d^{2} e^{5} - 15 B a^{2} c d^{3} e^{4} - 15 B a b^{2} d^{3} e^{4} + 42 B a b c d^{4} e^{3} - 27 B a c^{2} d^{5} e^{2} + 7 B b^{3} d^{4} e^{3} - 27 B b^{2} c d^{5} e^{2} + 33 B b c^{2} d^{6} e - 13 B c^{3} d^{7} + x \left (- 6 A a^{2} b e^{7} + 12 A a^{2} c d e^{6} + 12 A a b^{2} d e^{6} - 36 A a b c d^{2} e^{5} + 24 A a c^{2} d^{3} e^{4} - 6 A b^{3} d^{2} e^{5} + 24 A b^{2} c d^{3} e^{4} - 30 A b c^{2} d^{4} e^{3} + 12 A c^{3} d^{5} e^{2} - 2 B a^{3} e^{7} + 12 B a^{2} b d e^{6} - 18 B a^{2} c d^{2} e^{5} - 18 B a b^{2} d^{2} e^{5} + 48 B a b c d^{3} e^{4} - 30 B a c^{2} d^{4} e^{3} + 8 B b^{3} d^{3} e^{4} - 30 B b^{2} c d^{4} e^{3} + 36 B b c^{2} d^{5} e^{2} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} + \frac {3 \left (a e^{2} - b d e + c d^{2}\right ) \left (A a c e^{3} + A b^{2} e^{3} - 5 A b c d e^{2} + 5 A c^{2} d^{2} e + B a b e^{3} - 3 B a c d e^{2} - 2 B b^{2} d e^{2} + 8 B b c d^{2} e - 7 B c^{2} d^{3}\right ) \log {\left (d + e x \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)**3,x)

[Out]

B*c**3*x**5/(5*e**3) + x**4*(A*c**3/(4*e**3) + 3*B*b*c**2/(4*e**3) - 3*B*c**3*d/(4*e**4)) + x**3*(A*b*c**2/e**
3 - A*c**3*d/e**4 + B*a*c**2/e**3 + B*b**2*c/e**3 - 3*B*b*c**2*d/e**4 + 2*B*c**3*d**2/e**5) + x**2*(3*A*a*c**2
/(2*e**3) + 3*A*b**2*c/(2*e**3) - 9*A*b*c**2*d/(2*e**4) + 3*A*c**3*d**2/e**5 + 3*B*a*b*c/e**3 - 9*B*a*c**2*d/(
2*e**4) + B*b**3/(2*e**3) - 9*B*b**2*c*d/(2*e**4) + 9*B*b*c**2*d**2/e**5 - 5*B*c**3*d**3/e**6) + x*(6*A*a*b*c/
e**3 - 9*A*a*c**2*d/e**4 + A*b**3/e**3 - 9*A*b**2*c*d/e**4 + 18*A*b*c**2*d**2/e**5 - 10*A*c**3*d**3/e**6 + 3*B
*a**2*c/e**3 + 3*B*a*b**2/e**3 - 18*B*a*b*c*d/e**4 + 18*B*a*c**2*d**2/e**5 - 3*B*b**3*d/e**4 + 18*B*b**2*c*d**
2/e**5 - 30*B*b*c**2*d**3/e**6 + 15*B*c**3*d**4/e**7) + (-A*a**3*e**7 - 3*A*a**2*b*d*e**6 + 9*A*a**2*c*d**2*e*
*5 + 9*A*a*b**2*d**2*e**5 - 30*A*a*b*c*d**3*e**4 + 21*A*a*c**2*d**4*e**3 - 5*A*b**3*d**3*e**4 + 21*A*b**2*c*d*
*4*e**3 - 27*A*b*c**2*d**5*e**2 + 11*A*c**3*d**6*e - B*a**3*d*e**6 + 9*B*a**2*b*d**2*e**5 - 15*B*a**2*c*d**3*e
**4 - 15*B*a*b**2*d**3*e**4 + 42*B*a*b*c*d**4*e**3 - 27*B*a*c**2*d**5*e**2 + 7*B*b**3*d**4*e**3 - 27*B*b**2*c*
d**5*e**2 + 33*B*b*c**2*d**6*e - 13*B*c**3*d**7 + x*(-6*A*a**2*b*e**7 + 12*A*a**2*c*d*e**6 + 12*A*a*b**2*d*e**
6 - 36*A*a*b*c*d**2*e**5 + 24*A*a*c**2*d**3*e**4 - 6*A*b**3*d**2*e**5 + 24*A*b**2*c*d**3*e**4 - 30*A*b*c**2*d*
*4*e**3 + 12*A*c**3*d**5*e**2 - 2*B*a**3*e**7 + 12*B*a**2*b*d*e**6 - 18*B*a**2*c*d**2*e**5 - 18*B*a*b**2*d**2*
e**5 + 48*B*a*b*c*d**3*e**4 - 30*B*a*c**2*d**4*e**3 + 8*B*b**3*d**3*e**4 - 30*B*b**2*c*d**4*e**3 + 36*B*b*c**2
*d**5*e**2 - 14*B*c**3*d**6*e))/(2*d**2*e**8 + 4*d*e**9*x + 2*e**10*x**2) + 3*(a*e**2 - b*d*e + c*d**2)*(A*a*c
*e**3 + A*b**2*e**3 - 5*A*b*c*d*e**2 + 5*A*c**2*d**2*e + B*a*b*e**3 - 3*B*a*c*d*e**2 - 2*B*b**2*d*e**2 + 8*B*b
*c*d**2*e - 7*B*c**2*d**3)*log(d + e*x)/e**8

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