3.24.42 \(\int \frac {(A+B x) (a+b x+c x^2)^3}{(d+e x)^4} \, dx\) [2342]

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

[Out]

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

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Rubi [A]
time = 0.66, antiderivative size = 519, 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 = {785} \begin {gather*} -\frac {\log (d+e x) \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{e^8}-\frac {x \left (B (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-A c e \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{e^7}-\frac {c x^2 \left (A c e (4 c d-3 b e)-B \left (-3 c e (4 b d-a e)+3 b^2 e^2+10 c^2 d^2\right )\right )}{2 e^6}+\frac {3 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{e^8 (d+e x)}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{2 e^8 (d+e x)^2}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8 (d+e x)^3}-\frac {c^2 x^3 (-A c e-3 b B e+4 B c d)}{3 e^5}+\frac {B c^3 x^4}{4 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 785

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^4} \, dx &=\int \left (\frac {-B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )}{e^7}+\frac {c \left (-A c e (4 c d-3 b e)+B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^6}+\frac {c^2 (-4 B c d+3 b B e+A c e) x^2}{e^5}+\frac {B c^3 x^3}{e^4}+\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^4}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7 (d+e x)^3}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^7 (d+e x)^2}+\frac {-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{e^7 (d+e x)}\right ) \, dx\\ &=-\frac {\left (B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-A c e \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x}{e^7}-\frac {c \left (A c e (4 c d-3 b e)-B \left (10 c^2 d^2+3 b^2 e^2-3 c e (4 b d-a e)\right )\right ) x^2}{2 e^6}-\frac {c^2 (4 B c d-3 b B e-A c e) x^3}{3 e^5}+\frac {B c^3 x^4}{4 e^4}+\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{3 e^8 (d+e x)^3}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{2 e^8 (d+e x)^2}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^8 (d+e x)}-\frac {\left (A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{e^8}\\ \end {align*}

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Mathematica [A]
time = 0.18, size = 488, normalized size = 0.94 \begin {gather*} \frac {12 e \left (A c e \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )-B (2 c d-b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )\right ) x+6 c e^2 \left (A c e (-4 c d+3 b e)+B \left (10 c^2 d^2+3 b^2 e^2+3 c e (-4 b d+a e)\right )\right ) x^2+4 c^2 e^3 (-4 B c d+3 b B e+A c e) x^3+3 B c^3 e^4 x^4+\frac {4 (B d-A e) \left (c d^2+e (-b d+a e)\right )^3}{(d+e x)^3}-\frac {6 \left (c d^2+e (-b d+a e)\right )^2 \left (7 B c d^2+B e (-4 b d+a e)+3 A e (-2 c d+b e)\right )}{(d+e x)^2}+\frac {36 \left (c d^2+e (-b d+a e)\right ) \left (-A e \left (5 c^2 d^2+b^2 e^2+c e (-5 b d+a e)\right )+B \left (7 c^2 d^3+b e^2 (2 b d-a e)+c d e (-8 b d+3 a e)\right )\right )}{d+e x}+12 \left (A e (-2 c d+b e) \left (10 c^2 d^2+b^2 e^2+2 c e (-5 b d+3 a e)\right )+B \left (35 c^3 d^4+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 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )\right ) \log (d+e x)}{12 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*e*(A*c*e*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)) - B*(2*c*d - b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*(
-5*b*d + 3*a*e)))*x + 6*c*e^2*(A*c*e*(-4*c*d + 3*b*e) + B*(10*c^2*d^2 + 3*b^2*e^2 + 3*c*e*(-4*b*d + a*e)))*x^2
+ 4*c^2*e^3*(-4*B*c*d + 3*b*B*e + A*c*e)*x^3 + 3*B*c^3*e^4*x^4 + (4*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e))^3)
/(d + e*x)^3 - (6*(c*d^2 + e*(-(b*d) + a*e))^2*(7*B*c*d^2 + B*e*(-4*b*d + a*e) + 3*A*e*(-2*c*d + b*e)))/(d + e
*x)^2 + (36*(c*d^2 + e*(-(b*d) + a*e))*(-(A*e*(5*c^2*d^2 + b^2*e^2 + c*e*(-5*b*d + a*e))) + B*(7*c^2*d^3 + b*e
^2*(2*b*d - a*e) + c*d*e*(-8*b*d + 3*a*e))))/(d + e*x) + 12*(A*e*(-2*c*d + b*e)*(10*c^2*d^2 + b^2*e^2 + 2*c*e*
(-5*b*d + 3*a*e)) + B*(35*c^3*d^4 + 30*c^2*d^2*e*(-2*b*d + a*e) + b^2*e^3*(-4*b*d + 3*a*e) + 3*c*e^2*(10*b^2*d
^2 - 8*a*b*d*e + a^2*e^2)))*Log[d + e*x])/(12*e^8)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1088\) vs. \(2(511)=1022\).
time = 0.07, size = 1089, normalized size = 2.09 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)^4,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.31, size = 946, normalized size = 1.82 \begin {gather*} {\left (35 \, B c^{3} d^{4} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4} - 20 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{3} + 30 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d^{2} + 3 \, {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} c - 4 \, {\left (B b^{3} e^{3} + 3 \, A a c^{2} e^{3} + 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c\right )} d\right )} e^{\left (-8\right )} \log \left (x e + d\right ) + \frac {1}{12} \, {\left (3 \, B c^{3} x^{4} e^{3} - 4 \, {\left (4 \, B c^{3} d e^{2} - 3 \, B b c^{2} e^{3} - A c^{3} e^{3}\right )} x^{3} + 6 \, {\left (10 \, B c^{3} d^{2} e + 3 \, B b^{2} c e^{3} + 3 \, {\left (B a e^{3} + A b e^{3}\right )} c^{2} - 4 \, {\left (3 \, B b c^{2} e^{2} + A c^{3} e^{2}\right )} d\right )} x^{2} - 12 \, {\left (20 \, B c^{3} d^{3} - B b^{3} e^{3} - 3 \, A a c^{2} e^{3} - 10 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{2} - 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c + 12 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d\right )} x\right )} e^{\left (-7\right )} + \frac {107 \, B c^{3} d^{7} - 74 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{6} + 141 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d^{5} - 26 \, {\left (B b^{3} e^{3} + 3 \, A a c^{2} e^{3} + 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c\right )} d^{4} - 2 \, A a^{3} e^{7} + 11 \, {\left (3 \, B a b^{2} e^{4} + A b^{3} e^{4} + 3 \, {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} c\right )} d^{3} - 6 \, {\left (B a^{2} b e^{5} + A a b^{2} e^{5} + A a^{2} c e^{5}\right )} d^{2} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 5 \, {\left (3 \, B b c^{2} e^{3} + A c^{3} e^{3}\right )} d^{4} - B a^{2} b e^{7} - A a b^{2} e^{7} - A a^{2} c e^{7} + 10 \, {\left (B b^{2} c e^{4} + {\left (B a e^{4} + A b e^{4}\right )} c^{2}\right )} d^{3} - 2 \, {\left (B b^{3} e^{5} + 3 \, A a c^{2} e^{5} + 3 \, {\left (2 \, B a b e^{5} + A b^{2} e^{5}\right )} c\right )} d^{2} + {\left (3 \, B a b^{2} e^{6} + A b^{3} e^{6} + 3 \, {\left (B a^{2} e^{6} + 2 \, A a b e^{6}\right )} c\right )} d\right )} x^{2} - {\left (B a^{3} e^{6} + 3 \, A a^{2} b e^{6}\right )} d + 3 \, {\left (77 \, B c^{3} d^{6} e - 54 \, {\left (3 \, B b c^{2} e^{2} + A c^{3} e^{2}\right )} d^{5} + 105 \, {\left (B b^{2} c e^{3} + {\left (B a e^{3} + A b e^{3}\right )} c^{2}\right )} d^{4} - B a^{3} e^{7} - 3 \, A a^{2} b e^{7} - 20 \, {\left (B b^{3} e^{4} + 3 \, A a c^{2} e^{4} + 3 \, {\left (2 \, B a b e^{4} + A b^{2} e^{4}\right )} c\right )} d^{3} + 9 \, {\left (3 \, B a b^{2} e^{5} + A b^{3} e^{5} + 3 \, {\left (B a^{2} e^{5} + 2 \, A a b e^{5}\right )} c\right )} d^{2} - 6 \, {\left (B a^{2} b e^{6} + A a b^{2} e^{6} + A a^{2} c e^{6}\right )} d\right )} x}{6 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\right )}} \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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1332 vs. \(2 (528) = 1056\).
time = 3.89, size = 1332, normalized size = 2.56 \begin {gather*} \frac {214 \, B c^{3} d^{7} + {\left (3 \, B c^{3} x^{7} + 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 18 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} - 4 \, A a^{3} - 36 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} - 6 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{7} - {\left (7 \, B c^{3} d x^{6} + 12 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d x^{5} + 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d x^{4} - 36 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d x^{3} - 36 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d x^{2} + 36 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d x + 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{6} + 3 \, {\left (7 \, B c^{3} d^{2} x^{5} + 20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} x^{4} - 126 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} x^{3} - 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} x^{2} + 18 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} x - 4 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2}\right )} e^{5} - {\left (105 \, B c^{3} d^{3} x^{4} - 292 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} x^{3} + 54 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} x^{2} + 108 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} x - 22 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3}\right )} e^{4} - 2 \, {\left (278 \, B c^{3} d^{4} x^{3} - 78 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} x^{2} - 243 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} x + 26 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4}\right )} e^{3} - 6 \, {\left (68 \, B c^{3} d^{5} x^{2} + 34 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} x - 47 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5}\right )} e^{2} + 74 \, {\left (3 \, B c^{3} d^{6} x - 2 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6}\right )} e + 12 \, {\left (35 \, B c^{3} d^{7} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} e^{7} - {\left (4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d x^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d x^{2}\right )} e^{6} + 3 \, {\left (10 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} x^{3} - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} x^{2} + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} x\right )} e^{5} - {\left (20 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} x^{3} - 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} x^{2} + 12 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} x - {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3}\right )} e^{4} + {\left (35 \, B c^{3} d^{4} x^{3} - 60 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} x^{2} + 90 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} x - 4 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4}\right )} e^{3} + 15 \, {\left (7 \, B c^{3} d^{5} x^{2} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} x + 2 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5}\right )} e^{2} + 5 \, {\left (21 \, B c^{3} d^{6} x - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6}\right )} e\right )} \log \left (x e + d\right )}{12 \, {\left (x^{3} e^{11} + 3 \, d x^{2} e^{10} + 3 \, d^{2} x e^{9} + d^{3} e^{8}\right )}} \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)^4,x, algorithm="fricas")

[Out]

1/12*(214*B*c^3*d^7 + (3*B*c^3*x^7 + 4*(3*B*b*c^2 + A*c^3)*x^6 + 18*(B*b^2*c + (B*a + A*b)*c^2)*x^5 + 12*(B*b^
3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*x^4 - 4*A*a^3 - 36*(B*a^2*b + A*a*b^2 + A*a^2*c)*x^2 - 6*(B*a^3 + 3*A*a
^2*b)*x)*e^7 - (7*B*c^3*d*x^6 + 12*(3*B*b*c^2 + A*c^3)*d*x^5 + 90*(B*b^2*c + (B*a + A*b)*c^2)*d*x^4 - 36*(B*b^
3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*x^3 - 36*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*x^2 + 36*(B*a^
2*b + A*a*b^2 + A*a^2*c)*d*x + 2*(B*a^3 + 3*A*a^2*b)*d)*e^6 + 3*(7*B*c^3*d^2*x^5 + 20*(3*B*b*c^2 + A*c^3)*d^2*
x^4 - 126*(B*b^2*c + (B*a + A*b)*c^2)*d^2*x^3 - 12*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2*x^2 + 18*(3
*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*x - 4*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2)*e^5 - (105*B*c^3*d^3*x^
4 - 292*(3*B*b*c^2 + A*c^3)*d^3*x^3 + 54*(B*b^2*c + (B*a + A*b)*c^2)*d^3*x^2 + 108*(B*b^3 + 3*A*a*c^2 + 3*(2*B
*a*b + A*b^2)*c)*d^3*x - 22*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^3)*e^4 - 2*(278*B*c^3*d^4*x^3 - 78*(
3*B*b*c^2 + A*c^3)*d^4*x^2 - 243*(B*b^2*c + (B*a + A*b)*c^2)*d^4*x + 26*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^
2)*c)*d^4)*e^3 - 6*(68*B*c^3*d^5*x^2 + 34*(3*B*b*c^2 + A*c^3)*d^5*x - 47*(B*b^2*c + (B*a + A*b)*c^2)*d^5)*e^2
+ 74*(3*B*c^3*d^6*x - 2*(3*B*b*c^2 + A*c^3)*d^6)*e + 12*(35*B*c^3*d^7 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*
b)*c)*x^3*e^7 - (4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*x^3 - 3*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a
*b)*c)*d*x^2)*e^6 + 3*(10*(B*b^2*c + (B*a + A*b)*c^2)*d^2*x^3 - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*
d^2*x^2 + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2*x)*e^5 - (20*(3*B*b*c^2 + A*c^3)*d^3*x^3 - 90*(B*b^2
*c + (B*a + A*b)*c^2)*d^3*x^2 + 12*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^3*x - (3*B*a*b^2 + A*b^3 + 3*
(B*a^2 + 2*A*a*b)*c)*d^3)*e^4 + (35*B*c^3*d^4*x^3 - 60*(3*B*b*c^2 + A*c^3)*d^4*x^2 + 90*(B*b^2*c + (B*a + A*b)
*c^2)*d^4*x - 4*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^4)*e^3 + 15*(7*B*c^3*d^5*x^2 - 4*(3*B*b*c^2 + A*
c^3)*d^5*x + 2*(B*b^2*c + (B*a + A*b)*c^2)*d^5)*e^2 + 5*(21*B*c^3*d^6*x - 4*(3*B*b*c^2 + A*c^3)*d^6)*e)*log(x*
e + d))/(x^3*e^11 + 3*d*x^2*e^10 + 3*d^2*x*e^9 + d^3*e^8)

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Sympy [F(-1)] Timed out
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)**4,x)

[Out]

Timed out

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Giac [A]
time = 1.76, size = 1021, normalized size = 1.96 \begin {gather*} {\left (35 \, B c^{3} d^{4} - 60 \, B b c^{2} d^{3} e - 20 \, A c^{3} d^{3} e + 30 \, B b^{2} c d^{2} e^{2} + 30 \, B a c^{2} d^{2} e^{2} + 30 \, A b c^{2} d^{2} e^{2} - 4 \, B b^{3} d e^{3} - 24 \, B a b c d e^{3} - 12 \, A b^{2} c d e^{3} - 12 \, A a c^{2} d e^{3} + 3 \, B a b^{2} e^{4} + A b^{3} e^{4} + 3 \, B a^{2} c e^{4} + 6 \, A a b c e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{12} \, {\left (3 \, B c^{3} x^{4} e^{12} - 16 \, B c^{3} d x^{3} e^{11} + 60 \, B c^{3} d^{2} x^{2} e^{10} - 240 \, B c^{3} d^{3} x e^{9} + 12 \, B b c^{2} x^{3} e^{12} + 4 \, A c^{3} x^{3} e^{12} - 72 \, B b c^{2} d x^{2} e^{11} - 24 \, A c^{3} d x^{2} e^{11} + 360 \, B b c^{2} d^{2} x e^{10} + 120 \, A c^{3} d^{2} x e^{10} + 18 \, B b^{2} c x^{2} e^{12} + 18 \, B a c^{2} x^{2} e^{12} + 18 \, A b c^{2} x^{2} e^{12} - 144 \, B b^{2} c d x e^{11} - 144 \, B a c^{2} d x e^{11} - 144 \, A b c^{2} d x e^{11} + 12 \, B b^{3} x e^{12} + 72 \, B a b c x e^{12} + 36 \, A b^{2} c x e^{12} + 36 \, A a c^{2} x e^{12}\right )} e^{\left (-16\right )} + \frac {{\left (107 \, B c^{3} d^{7} - 222 \, B b c^{2} d^{6} e - 74 \, A c^{3} d^{6} e + 141 \, B b^{2} c d^{5} e^{2} + 141 \, B a c^{2} d^{5} e^{2} + 141 \, A b c^{2} d^{5} e^{2} - 26 \, B b^{3} d^{4} e^{3} - 156 \, B a b c d^{4} e^{3} - 78 \, A b^{2} c d^{4} e^{3} - 78 \, A a c^{2} d^{4} e^{3} + 33 \, B a b^{2} d^{3} e^{4} + 11 \, A b^{3} d^{3} e^{4} + 33 \, B a^{2} c d^{3} e^{4} + 66 \, A a b c d^{3} e^{4} - 6 \, B a^{2} b d^{2} e^{5} - 6 \, A a b^{2} d^{2} e^{5} - 6 \, A a^{2} c d^{2} e^{5} - B a^{3} d e^{6} - 3 \, A a^{2} b d e^{6} - 2 \, A a^{3} e^{7} + 18 \, {\left (7 \, B c^{3} d^{5} e^{2} - 15 \, B b c^{2} d^{4} e^{3} - 5 \, A c^{3} d^{4} e^{3} + 10 \, B b^{2} c d^{3} e^{4} + 10 \, B a c^{2} d^{3} e^{4} + 10 \, A b c^{2} d^{3} e^{4} - 2 \, B b^{3} d^{2} e^{5} - 12 \, B a b c d^{2} e^{5} - 6 \, A b^{2} c d^{2} e^{5} - 6 \, A a c^{2} d^{2} e^{5} + 3 \, B a b^{2} d e^{6} + A b^{3} d e^{6} + 3 \, B a^{2} c d e^{6} + 6 \, A a b c d e^{6} - B a^{2} b e^{7} - A a b^{2} e^{7} - A a^{2} c e^{7}\right )} x^{2} + 3 \, {\left (77 \, B c^{3} d^{6} e - 162 \, B b c^{2} d^{5} e^{2} - 54 \, A c^{3} d^{5} e^{2} + 105 \, B b^{2} c d^{4} e^{3} + 105 \, B a c^{2} d^{4} e^{3} + 105 \, A b c^{2} d^{4} e^{3} - 20 \, B b^{3} d^{3} e^{4} - 120 \, B a b c d^{3} e^{4} - 60 \, A b^{2} c d^{3} e^{4} - 60 \, A a c^{2} d^{3} e^{4} + 27 \, B a b^{2} d^{2} e^{5} + 9 \, A b^{3} d^{2} e^{5} + 27 \, B a^{2} c d^{2} e^{5} + 54 \, A a b c d^{2} e^{5} - 6 \, B a^{2} b d e^{6} - 6 \, A a b^{2} d e^{6} - 6 \, A a^{2} c d e^{6} - B a^{3} e^{7} - 3 \, A a^{2} b e^{7}\right )} x\right )} e^{\left (-8\right )}}{6 \, {\left (x e + d\right )}^{3}} \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)^4,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 2.53, size = 1151, normalized size = 2.21 \begin {gather*} x\,\left (\frac {B\,b^3+3\,A\,b^2\,c+6\,B\,a\,b\,c+3\,A\,a\,c^2}{e^4}-\frac {6\,d^2\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e^2}+\frac {4\,d\,\left (\frac {4\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{e^4}+\frac {6\,B\,c^3\,d^2}{e^6}\right )}{e}-\frac {4\,B\,c^3\,d^3}{e^7}\right )-x^2\,\left (\frac {2\,d\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{e^4}-\frac {4\,B\,c^3\,d}{e^5}\right )}{e}-\frac {3\,B\,b^2\,c+3\,A\,b\,c^2+3\,B\,a\,c^2}{2\,e^4}+\frac {3\,B\,c^3\,d^2}{e^6}\right )-\frac {\frac {B\,a^3\,d\,e^6+2\,A\,a^3\,e^7+6\,B\,a^2\,b\,d^2\,e^5+3\,A\,a^2\,b\,d\,e^6-33\,B\,a^2\,c\,d^3\,e^4+6\,A\,a^2\,c\,d^2\,e^5-33\,B\,a\,b^2\,d^3\,e^4+6\,A\,a\,b^2\,d^2\,e^5+156\,B\,a\,b\,c\,d^4\,e^3-66\,A\,a\,b\,c\,d^3\,e^4-141\,B\,a\,c^2\,d^5\,e^2+78\,A\,a\,c^2\,d^4\,e^3+26\,B\,b^3\,d^4\,e^3-11\,A\,b^3\,d^3\,e^4-141\,B\,b^2\,c\,d^5\,e^2+78\,A\,b^2\,c\,d^4\,e^3+222\,B\,b\,c^2\,d^6\,e-141\,A\,b\,c^2\,d^5\,e^2-107\,B\,c^3\,d^7+74\,A\,c^3\,d^6\,e}{6\,e}+x\,\left (\frac {B\,a^3\,e^6}{2}+3\,B\,a^2\,b\,d\,e^5+\frac {3\,A\,a^2\,b\,e^6}{2}-\frac {27\,B\,a^2\,c\,d^2\,e^4}{2}+3\,A\,a^2\,c\,d\,e^5-\frac {27\,B\,a\,b^2\,d^2\,e^4}{2}+3\,A\,a\,b^2\,d\,e^5+60\,B\,a\,b\,c\,d^3\,e^3-27\,A\,a\,b\,c\,d^2\,e^4-\frac {105\,B\,a\,c^2\,d^4\,e^2}{2}+30\,A\,a\,c^2\,d^3\,e^3+10\,B\,b^3\,d^3\,e^3-\frac {9\,A\,b^3\,d^2\,e^4}{2}-\frac {105\,B\,b^2\,c\,d^4\,e^2}{2}+30\,A\,b^2\,c\,d^3\,e^3+81\,B\,b\,c^2\,d^5\,e-\frac {105\,A\,b\,c^2\,d^4\,e^2}{2}-\frac {77\,B\,c^3\,d^6}{2}+27\,A\,c^3\,d^5\,e\right )+x^2\,\left (3\,B\,a^2\,b\,e^6-9\,B\,a^2\,c\,d\,e^5+3\,A\,a^2\,c\,e^6-9\,B\,a\,b^2\,d\,e^5+3\,A\,a\,b^2\,e^6+36\,B\,a\,b\,c\,d^2\,e^4-18\,A\,a\,b\,c\,d\,e^5-30\,B\,a\,c^2\,d^3\,e^3+18\,A\,a\,c^2\,d^2\,e^4+6\,B\,b^3\,d^2\,e^4-3\,A\,b^3\,d\,e^5-30\,B\,b^2\,c\,d^3\,e^3+18\,A\,b^2\,c\,d^2\,e^4+45\,B\,b\,c^2\,d^4\,e^2-30\,A\,b\,c^2\,d^3\,e^3-21\,B\,c^3\,d^5\,e+15\,A\,c^3\,d^4\,e^2\right )}{d^3\,e^7+3\,d^2\,e^8\,x+3\,d\,e^9\,x^2+e^{10}\,x^3}+x^3\,\left (\frac {A\,c^3+3\,B\,b\,c^2}{3\,e^4}-\frac {4\,B\,c^3\,d}{3\,e^5}\right )+\frac {\ln \left (d+e\,x\right )\,\left (3\,B\,a^2\,c\,e^4+3\,B\,a\,b^2\,e^4-24\,B\,a\,b\,c\,d\,e^3+6\,A\,a\,b\,c\,e^4+30\,B\,a\,c^2\,d^2\,e^2-12\,A\,a\,c^2\,d\,e^3-4\,B\,b^3\,d\,e^3+A\,b^3\,e^4+30\,B\,b^2\,c\,d^2\,e^2-12\,A\,b^2\,c\,d\,e^3-60\,B\,b\,c^2\,d^3\,e+30\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4-20\,A\,c^3\,d^3\,e\right )}{e^8}+\frac {B\,c^3\,x^4}{4\,e^4} \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)^4,x)

[Out]

x*((B*b^3 + 3*A*a*c^2 + 3*A*b^2*c + 6*B*a*b*c)/e^4 - (6*d^2*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e^2 +
(4*d*((4*d*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e - (3*A*b*c^2 + 3*B*a*c^2 + 3*B*b^2*c)/e^4 + (6*B*c^
3*d^2)/e^6))/e - (4*B*c^3*d^3)/e^7) - x^2*((2*d*((A*c^3 + 3*B*b*c^2)/e^4 - (4*B*c^3*d)/e^5))/e - (3*A*b*c^2 +
3*B*a*c^2 + 3*B*b^2*c)/(2*e^4) + (3*B*c^3*d^2)/e^6) - ((2*A*a^3*e^7 - 107*B*c^3*d^7 + B*a^3*d*e^6 + 74*A*c^3*d
^6*e - 11*A*b^3*d^3*e^4 + 26*B*b^3*d^4*e^3 + 6*A*a*b^2*d^2*e^5 + 78*A*a*c^2*d^4*e^3 + 6*A*a^2*c*d^2*e^5 - 33*B
*a*b^2*d^3*e^4 + 6*B*a^2*b*d^2*e^5 - 141*A*b*c^2*d^5*e^2 + 78*A*b^2*c*d^4*e^3 - 141*B*a*c^2*d^5*e^2 - 33*B*a^2
*c*d^3*e^4 - 141*B*b^2*c*d^5*e^2 + 3*A*a^2*b*d*e^6 + 222*B*b*c^2*d^6*e - 66*A*a*b*c*d^3*e^4 + 156*B*a*b*c*d^4*
e^3)/(6*e) + x*((B*a^3*e^6)/2 - (77*B*c^3*d^6)/2 + (3*A*a^2*b*e^6)/2 + 27*A*c^3*d^5*e - (9*A*b^3*d^2*e^4)/2 +
10*B*b^3*d^3*e^3 + 30*A*a*c^2*d^3*e^3 - (27*B*a*b^2*d^2*e^4)/2 - (105*A*b*c^2*d^4*e^2)/2 + 30*A*b^2*c*d^3*e^3
- (105*B*a*c^2*d^4*e^2)/2 - (27*B*a^2*c*d^2*e^4)/2 - (105*B*b^2*c*d^4*e^2)/2 + 3*A*a*b^2*d*e^5 + 3*A*a^2*c*d*e
^5 + 3*B*a^2*b*d*e^5 + 81*B*b*c^2*d^5*e - 27*A*a*b*c*d^2*e^4 + 60*B*a*b*c*d^3*e^3) + x^2*(3*A*a*b^2*e^6 + 3*A*
a^2*c*e^6 + 3*B*a^2*b*e^6 - 3*A*b^3*d*e^5 - 21*B*c^3*d^5*e + 15*A*c^3*d^4*e^2 + 6*B*b^3*d^2*e^4 + 18*A*a*c^2*d
^2*e^4 - 30*A*b*c^2*d^3*e^3 + 18*A*b^2*c*d^2*e^4 - 30*B*a*c^2*d^3*e^3 + 45*B*b*c^2*d^4*e^2 - 30*B*b^2*c*d^3*e^
3 - 9*B*a*b^2*d*e^5 - 9*B*a^2*c*d*e^5 + 36*B*a*b*c*d^2*e^4 - 18*A*a*b*c*d*e^5))/(d^3*e^7 + e^10*x^3 + 3*d^2*e^
8*x + 3*d*e^9*x^2) + x^3*((A*c^3 + 3*B*b*c^2)/(3*e^4) - (4*B*c^3*d)/(3*e^5)) + (log(d + e*x)*(A*b^3*e^4 + 35*B
*c^3*d^4 + 3*B*a*b^2*e^4 + 3*B*a^2*c*e^4 - 20*A*c^3*d^3*e - 4*B*b^3*d*e^3 + 30*A*b*c^2*d^2*e^2 + 30*B*a*c^2*d^
2*e^2 + 30*B*b^2*c*d^2*e^2 + 6*A*a*b*c*e^4 - 12*A*a*c^2*d*e^3 - 12*A*b^2*c*d*e^3 - 60*B*b*c^2*d^3*e - 24*B*a*b
*c*d*e^3))/e^8 + (B*c^3*x^4)/(4*e^4)

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