3.1135 \(\int \frac {(A+B x) (b x+c x^2)^3}{(d+e x)^3} \, dx\)

Optimal. Leaf size=359 \[ \frac {3 d (c d-b e) \log (d+e x) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8}-\frac {x (c d-b e) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{e^7}+\frac {x^2 (c d-b e) \left (3 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{2 e^6}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}+\frac {d^3 (B d-A e) (c d-b e)^3}{2 e^8 (d+e x)^2}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (d+e x)}+\frac {c x^3 (c d-b e) (-A c e-b B e+2 B c d)}{e^5}+\frac {B c^3 x^5}{5 e^3} \]

[Out]

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

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Rubi [A]  time = 0.58, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {771} \[ \frac {x^2 (c d-b e) \left (3 A c e (2 c d-b e)-B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )}{2 e^6}-\frac {x (c d-b e) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )}{e^7}+\frac {3 d (c d-b e) \log (d+e x) \left (A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )-B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )\right )}{e^8}-\frac {c^2 x^4 (-A c e-3 b B e+3 B c d)}{4 e^4}-\frac {d^2 (c d-b e)^2 (B d (7 c d-4 b e)-3 A e (2 c d-b e))}{e^8 (d+e x)}+\frac {d^3 (B d-A e) (c d-b e)^3}{2 e^8 (d+e x)^2}+\frac {c x^3 (c d-b e) (-A c e-b B e+2 B c d)}{e^5}+\frac {B c^3 x^5}{5 e^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.15, size = 342, normalized size = 0.95 \[ \frac {10 e^2 x^2 (b e-c d) \left (3 A c e (b e-2 c d)+B \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )\right )+20 e x (b e-c d) \left (A e \left (b^2 e^2-8 b c d e+10 c^2 d^2\right )-3 B d \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )-60 d (c d-b e) \log (d+e x) \left (B d \left (2 b^2 e^2-8 b c d e+7 c^2 d^2\right )-A e \left (b^2 e^2-5 b c d e+5 c^2 d^2\right )\right )+5 c^2 e^4 x^4 (A c e+3 b B e-3 B c d)+\frac {10 d^3 (B d-A e) (c d-b e)^3}{(d+e x)^2}-\frac {20 d^2 (c d-b e)^2 (3 A e (b e-2 c d)+B d (7 c d-4 b e))}{d+e x}-20 c e^3 x^3 (c d-b e) (A c e+b B e-2 B c d)+4 B c^3 e^5 x^5}{20 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(20*e*(-(c*d) + b*e)*(A*e*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2) - 3*B*d*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2))*x + 10
*e^2*(-(c*d) + b*e)*(3*A*c*e*(-2*c*d + b*e) + B*(10*c^2*d^2 - 8*b*c*d*e + b^2*e^2))*x^2 - 20*c*e^3*(c*d - b*e)
*(-2*B*c*d + b*B*e + A*c*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*d^3*(B*d
- A*e)*(c*d - b*e)^3)/(d + e*x)^2 - (20*d^2*(c*d - b*e)^2*(B*d*(7*c*d - 4*b*e) + 3*A*e*(-2*c*d + b*e)))/(d + e
*x) - 60*d*(c*d - b*e)*(-(A*e*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)) + B*d*(7*c^2*d^2 - 8*b*c*d*e + 2*b^2*e^2))*Lo
g[d + e*x])/(20*e^8)

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fricas [B]  time = 0.89, size = 819, normalized size = 2.28 \[ \frac {4 \, B c^{3} e^{7} x^{7} - 130 \, B c^{3} d^{7} - 50 \, A b^{3} d^{3} e^{4} + 110 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e - 270 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} + 70 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} - {\left (7 \, B c^{3} d e^{6} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} e^{7}\right )} x^{6} + 2 \, {\left (7 \, B c^{3} d^{2} e^{5} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{6} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} e^{7}\right )} x^{5} - 5 \, {\left (7 \, B c^{3} d^{3} e^{4} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{5} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{6} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{7}\right )} x^{4} + 20 \, {\left (7 \, B c^{3} d^{4} e^{3} + A b^{3} e^{7} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e^{4} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{5} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{6}\right )} x^{3} + 10 \, {\left (50 \, B c^{3} d^{5} e^{2} + 4 \, A b^{3} d e^{6} - 34 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 63 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 11 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 20 \, {\left (8 \, B c^{3} d^{6} e - 2 \, A b^{3} d^{2} e^{5} - 4 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 3 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} + {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x - 60 \, {\left (7 \, B c^{3} d^{7} + A b^{3} d^{3} e^{4} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + {\left (7 \, B c^{3} d^{5} e^{2} + A b^{3} d e^{6} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e^{3} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{4} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{5}\right )} x^{2} + 2 \, {\left (7 \, B c^{3} d^{6} e + A b^{3} d^{2} e^{5} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x\right )} \log \left (e x + d\right )}{20 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^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 - 50*A*b^3*d^3*e^4 + 110*(3*B*b*c^2 + A*c^3)*d^6*e - 270*(B*b^2*c + A*b*
c^2)*d^5*e^2 + 70*(B*b^3 + 3*A*b^2*c)*d^4*e^3 - (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 + 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 + A*b*c^2)*d*e^6 - 2*(B*b^3 + 3*A*b^2*c)*e^7)*x^4 + 20*(7*B*c^3*d^4*e^3 + A*b^3*e^
7 - 5*(3*B*b*c^2 + A*c^3)*d^3*e^4 + 10*(B*b^2*c + A*b*c^2)*d^2*e^5 - 2*(B*b^3 + 3*A*b^2*c)*d*e^6)*x^3 + 10*(50
*B*c^3*d^5*e^2 + 4*A*b^3*d*e^6 - 34*(3*B*b*c^2 + A*c^3)*d^4*e^3 + 63*(B*b^2*c + A*b*c^2)*d^3*e^4 - 11*(B*b^3 +
3*A*b^2*c)*d^2*e^5)*x^2 + 20*(8*B*c^3*d^6*e - 2*A*b^3*d^2*e^5 - 4*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 3*(B*b^2*c +
A*b*c^2)*d^4*e^3 + (B*b^3 + 3*A*b^2*c)*d^3*e^4)*x - 60*(7*B*c^3*d^7 + A*b^3*d^3*e^4 - 5*(3*B*b*c^2 + A*c^3)*d^
6*e + 10*(B*b^2*c + A*b*c^2)*d^5*e^2 - 2*(B*b^3 + 3*A*b^2*c)*d^4*e^3 + (7*B*c^3*d^5*e^2 + A*b^3*d*e^6 - 5*(3*B
*b*c^2 + A*c^3)*d^4*e^3 + 10*(B*b^2*c + A*b*c^2)*d^3*e^4 - 2*(B*b^3 + 3*A*b^2*c)*d^2*e^5)*x^2 + 2*(7*B*c^3*d^6
*e + A*b^3*d^2*e^5 - 5*(3*B*b*c^2 + A*c^3)*d^5*e^2 + 10*(B*b^2*c + A*b*c^2)*d^4*e^3 - 2*(B*b^3 + 3*A*b^2*c)*d^
3*e^4)*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 = 0.19, size = 599, normalized size = 1.67 \[ -3 \, {\left (7 \, B c^{3} d^{5} - 15 \, B b c^{2} d^{4} e - 5 \, A c^{3} d^{4} e + 10 \, B b^{2} c d^{3} e^{2} + 10 \, A b c^{2} d^{3} e^{2} - 2 \, B b^{3} d^{2} e^{3} - 6 \, A b^{2} c d^{2} e^{3} + A b^{3} d e^{4}\right )} e^{\left (-8\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{20} \, {\left (4 \, B c^{3} x^{5} e^{12} - 15 \, B c^{3} d x^{4} e^{11} + 40 \, B c^{3} d^{2} x^{3} e^{10} - 100 \, B c^{3} d^{3} x^{2} e^{9} + 300 \, B c^{3} d^{4} x e^{8} + 15 \, B b c^{2} x^{4} e^{12} + 5 \, A c^{3} x^{4} e^{12} - 60 \, B b c^{2} d x^{3} e^{11} - 20 \, A c^{3} d x^{3} e^{11} + 180 \, B b c^{2} d^{2} x^{2} e^{10} + 60 \, A c^{3} d^{2} x^{2} e^{10} - 600 \, B b c^{2} d^{3} x e^{9} - 200 \, A c^{3} d^{3} x e^{9} + 20 \, B b^{2} c x^{3} e^{12} + 20 \, A b c^{2} x^{3} e^{12} - 90 \, B b^{2} c d x^{2} e^{11} - 90 \, A b c^{2} d x^{2} e^{11} + 360 \, B b^{2} c d^{2} x e^{10} + 360 \, A b c^{2} d^{2} x e^{10} + 10 \, B b^{3} x^{2} e^{12} + 30 \, A b^{2} c x^{2} e^{12} - 60 \, B b^{3} d x e^{11} - 180 \, A b^{2} c d x e^{11} + 20 \, A b^{3} x e^{12}\right )} e^{\left (-15\right )} - \frac {{\left (13 \, B c^{3} d^{7} - 33 \, B b c^{2} d^{6} e - 11 \, A c^{3} d^{6} e + 27 \, B b^{2} c d^{5} e^{2} + 27 \, A b c^{2} d^{5} e^{2} - 7 \, B b^{3} d^{4} e^{3} - 21 \, A b^{2} c d^{4} e^{3} + 5 \, A b^{3} d^{3} e^{4} + 2 \, {\left (7 \, B c^{3} d^{6} e - 18 \, B b c^{2} d^{5} e^{2} - 6 \, A c^{3} d^{5} e^{2} + 15 \, B b^{2} c d^{4} e^{3} + 15 \, A b c^{2} d^{4} e^{3} - 4 \, B b^{3} d^{3} e^{4} - 12 \, A b^{2} c d^{3} e^{4} + 3 \, A b^{3} d^{2} e^{5}\right )} x\right )} e^{\left (-8\right )}}{2 \, {\left (x e + d\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^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*A*b*c^2*d^3*e^2 - 2*B*b^3*d^2*e^3
- 6*A*b^2*c*d^2*e^3 + A*b^3*d*e^4)*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*A*b*c^2*x^3*e^12 - 90*B*b^2*c*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*A*b*c^2*d^2*x*e^10 + 10*B*b^3*x^2*e^12 + 30*A*b^2*c*x^2*e^12 - 60
*B*b^3*d*x*e^11 - 180*A*b^2*c*d*x*e^11 + 20*A*b^3*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*A*b*c^2*d^5*e^2 - 7*B*b^3*d^4*e^3 - 21*A*b^2*c*d^4*e^3 + 5*A*b^3*d^3*e^4
+ 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*A*b*c^2*d^4*e^3 - 4*B*b^3
*d^3*e^4 - 12*A*b^2*c*d^3*e^4 + 3*A*b^3*d^2*e^5)*x)*e^(-8)/(x*e + d)^2

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maple [B]  time = 0.10, size = 775, normalized size = 2.16 \[ \frac {B \,c^{3} x^{5}}{5 e^{3}}+\frac {A \,c^{3} x^{4}}{4 e^{3}}+\frac {3 B b \,c^{2} x^{4}}{4 e^{3}}-\frac {3 B \,c^{3} d \,x^{4}}{4 e^{4}}+\frac {A b \,c^{2} x^{3}}{e^{3}}-\frac {A \,c^{3} d \,x^{3}}{e^{4}}+\frac {B \,b^{2} c \,x^{3}}{e^{3}}-\frac {3 B b \,c^{2} d \,x^{3}}{e^{4}}+\frac {2 B \,c^{3} d^{2} x^{3}}{e^{5}}+\frac {A \,b^{3} d^{3}}{2 \left (e x +d \right )^{2} e^{4}}-\frac {3 A \,b^{2} c \,d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {3 A \,b^{2} c \,x^{2}}{2 e^{3}}+\frac {3 A b \,c^{2} d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {9 A b \,c^{2} d \,x^{2}}{2 e^{4}}-\frac {A \,c^{3} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {3 A \,c^{3} d^{2} x^{2}}{e^{5}}-\frac {B \,b^{3} d^{4}}{2 \left (e x +d \right )^{2} e^{5}}+\frac {B \,b^{3} x^{2}}{2 e^{3}}+\frac {3 B \,b^{2} c \,d^{5}}{2 \left (e x +d \right )^{2} e^{6}}-\frac {9 B \,b^{2} c d \,x^{2}}{2 e^{4}}-\frac {3 B b \,c^{2} d^{6}}{2 \left (e x +d \right )^{2} e^{7}}+\frac {9 B b \,c^{2} d^{2} x^{2}}{e^{5}}+\frac {B \,c^{3} d^{7}}{2 \left (e x +d \right )^{2} e^{8}}-\frac {5 B \,c^{3} d^{3} x^{2}}{e^{6}}-\frac {3 A \,b^{3} d^{2}}{\left (e x +d \right ) e^{4}}-\frac {3 A \,b^{3} d \ln \left (e x +d \right )}{e^{4}}+\frac {A \,b^{3} x}{e^{3}}+\frac {12 A \,b^{2} c \,d^{3}}{\left (e x +d \right ) e^{5}}+\frac {18 A \,b^{2} c \,d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {9 A \,b^{2} c d x}{e^{4}}-\frac {15 A b \,c^{2} d^{4}}{\left (e x +d \right ) e^{6}}-\frac {30 A b \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {18 A b \,c^{2} d^{2} x}{e^{5}}+\frac {6 A \,c^{3} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {15 A \,c^{3} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {10 A \,c^{3} d^{3} x}{e^{6}}+\frac {4 B \,b^{3} d^{3}}{\left (e x +d \right ) e^{5}}+\frac {6 B \,b^{3} d^{2} \ln \left (e x +d \right )}{e^{5}}-\frac {3 B \,b^{3} d x}{e^{4}}-\frac {15 B \,b^{2} c \,d^{4}}{\left (e x +d \right ) e^{6}}-\frac {30 B \,b^{2} c \,d^{3} \ln \left (e x +d \right )}{e^{6}}+\frac {18 B \,b^{2} c \,d^{2} x}{e^{5}}+\frac {18 B b \,c^{2} d^{5}}{\left (e x +d \right ) e^{7}}+\frac {45 B b \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{7}}-\frac {30 B b \,c^{2} d^{3} x}{e^{6}}-\frac {7 B \,c^{3} d^{6}}{\left (e x +d \right ) e^{8}}-\frac {21 B \,c^{3} d^{5} \ln \left (e x +d \right )}{e^{8}}+\frac {15 B \,c^{3} d^{4} x}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.67, size = 550, normalized size = 1.53 \[ -\frac {13 \, B c^{3} d^{7} + 5 \, A b^{3} d^{3} e^{4} - 11 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6} e + 27 \, {\left (B b^{2} c + A b c^{2}\right )} d^{5} e^{2} - 7 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{4} e^{3} + 2 \, {\left (7 \, B c^{3} d^{6} e + 3 \, A b^{3} d^{2} e^{5} - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} e^{2} + 15 \, {\left (B b^{2} c + A b c^{2}\right )} d^{4} e^{3} - 4 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{3} e^{4}\right )} x}{2 \, {\left (e^{10} x^{2} + 2 \, d e^{9} x + d^{2} e^{8}\right )}} + \frac {4 \, B c^{3} e^{4} x^{5} - 5 \, {\left (3 \, B c^{3} d e^{3} - {\left (3 \, B b c^{2} + A c^{3}\right )} e^{4}\right )} x^{4} + 20 \, {\left (2 \, B c^{3} d^{2} e^{2} - {\left (3 \, B b c^{2} + A c^{3}\right )} d e^{3} + {\left (B b^{2} c + A b c^{2}\right )} e^{4}\right )} x^{3} - 10 \, {\left (10 \, B c^{3} d^{3} e - 6 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} e^{2} + 9 \, {\left (B b^{2} c + A b c^{2}\right )} d e^{3} - {\left (B b^{3} + 3 \, A b^{2} c\right )} e^{4}\right )} x^{2} + 20 \, {\left (15 \, B c^{3} d^{4} + A b^{3} e^{4} - 10 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} e + 18 \, {\left (B b^{2} c + A b c^{2}\right )} d^{2} e^{2} - 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d e^{3}\right )} x}{20 \, e^{7}} - \frac {3 \, {\left (7 \, B c^{3} d^{5} + A b^{3} d e^{4} - 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} e + 10 \, {\left (B b^{2} c + A b c^{2}\right )} d^{3} e^{2} - 2 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} d^{2} e^{3}\right )} \log \left (e x + d\right )}{e^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x^2+b*x)^3/(e*x+d)^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.19, size = 828, normalized size = 2.31 \[ x\,\left (\frac {A\,b^3}{e^3}-\frac {3\,d\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{e^3}+\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\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e}-\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 {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}+\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\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{e^2}\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 )-\frac {x\,\left (-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 )+\frac {-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}}{d^2\,e^7+2\,d\,e^8\,x+e^9\,x^2}+x^2\,\left (\frac {B\,b^3+3\,A\,c\,b^2}{2\,e^3}+\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\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {3\,B\,c^3\,d^2}{e^5}\right )}{2\,e}-\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 {B\,c^3\,d^3}{2\,e^6}\right )-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 {b\,c\,\left (A\,c+B\,b\right )}{e^3}+\frac {B\,c^3\,d^2}{e^5}\right )-\frac {\ln \left (d+e\,x\right )\,\left (-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} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 7.27, size = 660, normalized size = 1.84 \[ \frac {B c^{3} x^{5}}{5 e^{3}} + \frac {3 d \left (b e - c d\right ) \left (- A b^{2} e^{3} + 5 A b c d e^{2} - 5 A c^{2} d^{2} e + 2 B b^{2} d e^{2} - 8 B b c d^{2} e + 7 B c^{2} d^{3}\right ) \log {\left (d + e x \right )}}{e^{8}} + x^{4} \left (\frac {A c^{3}}{4 e^{3}} + \frac {3 B b c^{2}}{4 e^{3}} - \frac {3 B c^{3} d}{4 e^{4}}\right ) + x^{3} \left (\frac {A b c^{2}}{e^{3}} - \frac {A c^{3} d}{e^{4}} + \frac {B b^{2} c}{e^{3}} - \frac {3 B b c^{2} d}{e^{4}} + \frac {2 B c^{3} d^{2}}{e^{5}}\right ) + x^{2} \left (\frac {3 A b^{2} c}{2 e^{3}} - \frac {9 A b c^{2} d}{2 e^{4}} + \frac {3 A c^{3} d^{2}}{e^{5}} + \frac {B b^{3}}{2 e^{3}} - \frac {9 B b^{2} c d}{2 e^{4}} + \frac {9 B b c^{2} d^{2}}{e^{5}} - \frac {5 B c^{3} d^{3}}{e^{6}}\right ) + x \left (\frac {A b^{3}}{e^{3}} - \frac {9 A b^{2} c d}{e^{4}} + \frac {18 A b c^{2} d^{2}}{e^{5}} - \frac {10 A c^{3} d^{3}}{e^{6}} - \frac {3 B b^{3} d}{e^{4}} + \frac {18 B b^{2} c d^{2}}{e^{5}} - \frac {30 B b c^{2} d^{3}}{e^{6}} + \frac {15 B c^{3} d^{4}}{e^{7}}\right ) + \frac {- 5 A b^{3} d^{3} e^{4} + 21 A b^{2} c d^{4} e^{3} - 27 A b c^{2} d^{5} e^{2} + 11 A c^{3} d^{6} e + 7 B b^{3} d^{4} e^{3} - 27 B b^{2} c d^{5} e^{2} + 33 B b c^{2} d^{6} e - 13 B c^{3} d^{7} + x \left (- 6 A b^{3} d^{2} e^{5} + 24 A b^{2} c d^{3} e^{4} - 30 A b c^{2} d^{4} e^{3} + 12 A c^{3} d^{5} e^{2} + 8 B b^{3} d^{3} e^{4} - 30 B b^{2} c d^{4} e^{3} + 36 B b c^{2} d^{5} e^{2} - 14 B c^{3} d^{6} e\right )}{2 d^{2} e^{8} + 4 d e^{9} x + 2 e^{10} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(c*x**2+b*x)**3/(e*x+d)**3,x)

[Out]

B*c**3*x**5/(5*e**3) + 3*d*(b*e - c*d)*(-A*b**2*e**3 + 5*A*b*c*d*e**2 - 5*A*c**2*d**2*e + 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 + 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*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*b**2*c/(2*e**3) - 9*A*b*c**2*d/(2*e**4) + 3*A*c**3*d**2/e**5 + 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*(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*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) + (-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 + 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*b**3*d**2*e**5 + 24*A*b**2*c*d**3*e**4 - 3
0*A*b*c**2*d**4*e**3 + 12*A*c**3*d**5*e**2 + 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)

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