3.2336 \(\int (A+B x) (d+e x)^2 (a+b x+c x^2)^3 \, dx\)

Optimal. Leaf size=555 \[ -\frac{(d+e x)^6 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{6 e^8}-\frac{3 c (d+e x)^8 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{8 e^8}-\frac{(d+e x)^7 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}-\frac{3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8}-\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{4 e^8}-\frac{(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac{c^2 (d+e x)^9 (-A c e-3 b B e+7 B c d)}{9 e^8}+\frac{B c^3 (d+e x)^{10}}{10 e^8} \]

[Out]

-((B*d - A*e)*(c*d^2 - b*d*e + a*e^2)^3*(d + e*x)^3)/(3*e^8) - ((c*d^2 - b*d*e + a*e^2)^2*(3*A*e*(2*c*d - b*e)
- B*(7*c*d^2 - e*(4*b*d - a*e)))*(d + e*x)^4)/(4*e^8) - (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)))*(d + e*x)^5)/(5*e^8) - ((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)))*(d + e*x)^6)/(6*e^8) - ((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)))*(d + e*x)^7)/(7*e^8) - (3*c*(A*c*e*(2*c*d - b*e) - B*(7*c^2*d^2 + b^2*e^2 - c*e*(6*b*d - a*e)))*(d +
e*x)^8)/(8*e^8) - (c^2*(7*B*c*d - 3*b*B*e - A*c*e)*(d + e*x)^9)/(9*e^8) + (B*c^3*(d + e*x)^10)/(10*e^8)

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Rubi [A]  time = 0.863153, antiderivative size = 553, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {771} \[ -\frac{(d+e x)^6 \left (A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )\right )}{6 e^8}-\frac{3 c (d+e x)^8 \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{8 e^8}-\frac{(d+e x)^7 \left (B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{7 e^8}-\frac{3 (d+e x)^5 \left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{5 e^8}+\frac{(d+e x)^4 \left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{4 e^8}-\frac{(d+e x)^3 (B d-A e) \left (a e^2-b d e+c d^2\right )^3}{3 e^8}-\frac{c^2 (d+e x)^9 (-A c e-3 b B e+7 B c d)}{9 e^8}+\frac{B c^3 (d+e x)^{10}}{10 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.306944, size = 526, normalized size = 0.95 \[ \frac{1}{4} x^4 \left (A \left (6 a^2 c d e+6 a b^2 d e+3 a b \left (a e^2+2 c d^2\right )+b^3 d^2\right )+a B \left (6 a b d e+a \left (a e^2+3 c d^2\right )+3 b^2 d^2\right )\right )+\frac{1}{2} a^2 d x^2 (2 a A e+a B d+3 A b d)+a^3 A d^2 x+\frac{1}{8} c x^8 \left (B \left (3 c e (a e+2 b d)+3 b^2 e^2+c^2 d^2\right )+A c e (3 b e+2 c d)\right )+\frac{1}{7} x^7 \left (3 b c \left (2 a B e^2+2 A c d e+B c d^2\right )+c^2 \left (3 a A e^2+6 a B d e+A c d^2\right )+3 b^2 c e (A e+2 B d)+b^3 B e^2\right )+\frac{1}{6} x^6 \left (3 b^2 \left (a B e^2+2 A c d e+B c d^2\right )+3 b c \left (2 a A e^2+4 a B d e+A c d^2\right )+3 a c \left (a B e^2+2 A c d e+B c d^2\right )+b^3 e (A e+2 B d)\right )+\frac{1}{5} x^5 \left (3 b^2 \left (a A e^2+2 a B d e+A c d^2\right )+3 a b \left (a B e^2+4 A c d e+2 B c d^2\right )+3 a c \left (a A e^2+2 a B d e+A c d^2\right )+b^3 d (2 A e+B d)\right )+\frac{1}{3} a x^3 \left (A \left (6 a b d e+a \left (a e^2+3 c d^2\right )+3 b^2 d^2\right )+a B d (2 a e+3 b d)\right )+\frac{1}{9} c^2 e x^9 (A c e+3 b B e+2 B c d)+\frac{1}{10} B c^3 e^2 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*A*d^2*x + (a^2*d*(3*A*b*d + a*B*d + 2*a*A*e)*x^2)/2 + (a*(a*B*d*(3*b*d + 2*a*e) + A*(3*b^2*d^2 + 6*a*b*d*e
+ a*(3*c*d^2 + a*e^2)))*x^3)/3 + ((A*(b^3*d^2 + 6*a*b^2*d*e + 6*a^2*c*d*e + 3*a*b*(2*c*d^2 + a*e^2)) + a*B*(3
*b^2*d^2 + 6*a*b*d*e + a*(3*c*d^2 + a*e^2)))*x^4)/4 + ((b^3*d*(B*d + 2*A*e) + 3*b^2*(A*c*d^2 + 2*a*B*d*e + a*A
*e^2) + 3*a*c*(A*c*d^2 + 2*a*B*d*e + a*A*e^2) + 3*a*b*(2*B*c*d^2 + 4*A*c*d*e + a*B*e^2))*x^5)/5 + ((b^3*e*(2*B
*d + A*e) + 3*b*c*(A*c*d^2 + 4*a*B*d*e + 2*a*A*e^2) + 3*b^2*(B*c*d^2 + 2*A*c*d*e + a*B*e^2) + 3*a*c*(B*c*d^2 +
2*A*c*d*e + a*B*e^2))*x^6)/6 + ((b^3*B*e^2 + 3*b^2*c*e*(2*B*d + A*e) + c^2*(A*c*d^2 + 6*a*B*d*e + 3*a*A*e^2)
+ 3*b*c*(B*c*d^2 + 2*A*c*d*e + 2*a*B*e^2))*x^7)/7 + (c*(A*c*e*(2*c*d + 3*b*e) + B*(c^2*d^2 + 3*b^2*e^2 + 3*c*e
*(2*b*d + a*e)))*x^8)/8 + (c^2*e*(2*B*c*d + 3*b*B*e + A*c*e)*x^9)/9 + (B*c^3*e^2*x^10)/10

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Maple [A]  time = 0.002, size = 597, normalized size = 1.1 \begin{align*}{\frac{B{c}^{3}{e}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( A{e}^{2}+2\,Bde \right ){c}^{3}+3\,B{e}^{2}b{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{3}+3\, \left ( A{e}^{2}+2\,Bde \right ) b{c}^{2}+B{e}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( A{c}^{3}{d}^{2}+3\, \left ( 2\,Ade+B{d}^{2} \right ) b{c}^{2}+ \left ( A{e}^{2}+2\,Bde \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) +B{e}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) \right ){x}^{7}}{7}}+{\frac{ \left ( 3\,Ab{c}^{2}{d}^{2}+ \left ( 2\,Ade+B{d}^{2} \right ) \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( A{e}^{2}+2\,Bde \right ) \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) +B{e}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) \right ){x}^{6}}{6}}+{\frac{ \left ( A{d}^{2} \left ( a{c}^{2}+2\,{b}^{2}c+c \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( 2\,Ade+B{d}^{2} \right ) \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( A{e}^{2}+2\,Bde \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\,B{a}^{2}b{e}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( A{d}^{2} \left ( 4\,abc+b \left ( 2\,ac+{b}^{2} \right ) \right ) + \left ( 2\,Ade+B{d}^{2} \right ) \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( A{e}^{2}+2\,Bde \right ){a}^{2}b+B{e}^{2}{a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{2} \left ( a \left ( 2\,ac+{b}^{2} \right ) +2\,{b}^{2}a+c{a}^{2} \right ) +3\, \left ( 2\,Ade+B{d}^{2} \right ){a}^{2}b+ \left ( A{e}^{2}+2\,Bde \right ){a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 3\,A{d}^{2}{a}^{2}b+ \left ( 2\,Ade+B{d}^{2} \right ){a}^{3} \right ){x}^{2}}{2}}+A{d}^{2}{a}^{3}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*B*c^3*e^2*x^10+1/9*((A*e^2+2*B*d*e)*c^3+3*B*e^2*b*c^2)*x^9+1/8*((2*A*d*e+B*d^2)*c^3+3*(A*e^2+2*B*d*e)*b*c
^2+B*e^2*(a*c^2+2*b^2*c+c*(2*a*c+b^2)))*x^8+1/7*(A*c^3*d^2+3*(2*A*d*e+B*d^2)*b*c^2+(A*e^2+2*B*d*e)*(a*c^2+2*b^
2*c+c*(2*a*c+b^2))+B*e^2*(4*a*b*c+b*(2*a*c+b^2)))*x^7+1/6*(3*A*b*c^2*d^2+(2*A*d*e+B*d^2)*(a*c^2+2*b^2*c+c*(2*a
*c+b^2))+(A*e^2+2*B*d*e)*(4*a*b*c+b*(2*a*c+b^2))+B*e^2*(a*(2*a*c+b^2)+2*b^2*a+c*a^2))*x^6+1/5*(A*d^2*(a*c^2+2*
b^2*c+c*(2*a*c+b^2))+(2*A*d*e+B*d^2)*(4*a*b*c+b*(2*a*c+b^2))+(A*e^2+2*B*d*e)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*B
*a^2*b*e^2)*x^5+1/4*(A*d^2*(4*a*b*c+b*(2*a*c+b^2))+(2*A*d*e+B*d^2)*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(A*e^2+2*B*
d*e)*a^2*b+B*e^2*a^3)*x^4+1/3*(A*d^2*(a*(2*a*c+b^2)+2*b^2*a+c*a^2)+3*(2*A*d*e+B*d^2)*a^2*b+(A*e^2+2*B*d*e)*a^3
)*x^3+1/2*(3*A*d^2*a^2*b+(2*A*d*e+B*d^2)*a^3)*x^2+A*d^2*a^3*x

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Maxima [A]  time = 0.988254, size = 714, normalized size = 1.29 \begin{align*} \frac{1}{10} \, B c^{3} e^{2} x^{10} + \frac{1}{9} \,{\left (2 \, B c^{3} d e +{\left (3 \, B b c^{2} + A c^{3}\right )} e^{2}\right )} x^{9} + \frac{1}{8} \,{\left (B c^{3} d^{2} + 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^{2}\right )} x^{8} + \frac{1}{7} \,{\left ({\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} + 6 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} d e +{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} e^{2}\right )} x^{7} + A a^{3} d^{2} x + \frac{1}{6} \,{\left (3 \,{\left (B b^{2} c +{\left (B a + A b\right )} c^{2}\right )} d^{2} + 2 \,{\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} d e +{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} e^{2}\right )} x^{6} + \frac{1}{5} \,{\left ({\left (B b^{3} + 3 \, A a c^{2} + 3 \,{\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} + 2 \,{\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} d e + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} e^{2}\right )} x^{5} + \frac{1}{4} \,{\left ({\left (3 \, B a b^{2} + A b^{3} + 3 \,{\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} + 6 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d e +{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (A a^{3} e^{2} + 3 \,{\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e\right )} x^{3} + \frac{1}{2} \,{\left (2 \, A a^{3} d e +{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*B*c^3*e^2*x^10 + 1/9*(2*B*c^3*d*e + (3*B*b*c^2 + A*c^3)*e^2)*x^9 + 1/8*(B*c^3*d^2 + 2*(3*B*b*c^2 + A*c^3)
*d*e + 3*(B*b^2*c + (B*a + A*b)*c^2)*e^2)*x^8 + 1/7*((3*B*b*c^2 + A*c^3)*d^2 + 6*(B*b^2*c + (B*a + A*b)*c^2)*d
*e + (B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*e^2)*x^7 + A*a^3*d^2*x + 1/6*(3*(B*b^2*c + (B*a + A*b)*c^2)*d
^2 + 2*(B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d*e + (3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*e^2)*x^6
+ 1/5*((B*b^3 + 3*A*a*c^2 + 3*(2*B*a*b + A*b^2)*c)*d^2 + 2*(3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d*e + 3
*(B*a^2*b + A*a*b^2 + A*a^2*c)*e^2)*x^5 + 1/4*((3*B*a*b^2 + A*b^3 + 3*(B*a^2 + 2*A*a*b)*c)*d^2 + 6*(B*a^2*b +
A*a*b^2 + A*a^2*c)*d*e + (B*a^3 + 3*A*a^2*b)*e^2)*x^4 + 1/3*(A*a^3*e^2 + 3*(B*a^2*b + A*a*b^2 + A*a^2*c)*d^2 +
2*(B*a^3 + 3*A*a^2*b)*d*e)*x^3 + 1/2*(2*A*a^3*d*e + (B*a^3 + 3*A*a^2*b)*d^2)*x^2

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Fricas [A]  time = 0.886668, size = 1685, normalized size = 3.04 \begin{align*} \frac{1}{10} x^{10} e^{2} c^{3} B + \frac{2}{9} x^{9} e d c^{3} B + \frac{1}{3} x^{9} e^{2} c^{2} b B + \frac{1}{9} x^{9} e^{2} c^{3} A + \frac{1}{8} x^{8} d^{2} c^{3} B + \frac{3}{4} x^{8} e d c^{2} b B + \frac{3}{8} x^{8} e^{2} c b^{2} B + \frac{3}{8} x^{8} e^{2} c^{2} a B + \frac{1}{4} x^{8} e d c^{3} A + \frac{3}{8} x^{8} e^{2} c^{2} b A + \frac{3}{7} x^{7} d^{2} c^{2} b B + \frac{6}{7} x^{7} e d c b^{2} B + \frac{1}{7} x^{7} e^{2} b^{3} B + \frac{6}{7} x^{7} e d c^{2} a B + \frac{6}{7} x^{7} e^{2} c b a B + \frac{1}{7} x^{7} d^{2} c^{3} A + \frac{6}{7} x^{7} e d c^{2} b A + \frac{3}{7} x^{7} e^{2} c b^{2} A + \frac{3}{7} x^{7} e^{2} c^{2} a A + \frac{1}{2} x^{6} d^{2} c b^{2} B + \frac{1}{3} x^{6} e d b^{3} B + \frac{1}{2} x^{6} d^{2} c^{2} a B + 2 x^{6} e d c b a B + \frac{1}{2} x^{6} e^{2} b^{2} a B + \frac{1}{2} x^{6} e^{2} c a^{2} B + \frac{1}{2} x^{6} d^{2} c^{2} b A + x^{6} e d c b^{2} A + \frac{1}{6} x^{6} e^{2} b^{3} A + x^{6} e d c^{2} a A + x^{6} e^{2} c b a A + \frac{1}{5} x^{5} d^{2} b^{3} B + \frac{6}{5} x^{5} d^{2} c b a B + \frac{6}{5} x^{5} e d b^{2} a B + \frac{6}{5} x^{5} e d c a^{2} B + \frac{3}{5} x^{5} e^{2} b a^{2} B + \frac{3}{5} x^{5} d^{2} c b^{2} A + \frac{2}{5} x^{5} e d b^{3} A + \frac{3}{5} x^{5} d^{2} c^{2} a A + \frac{12}{5} x^{5} e d c b a A + \frac{3}{5} x^{5} e^{2} b^{2} a A + \frac{3}{5} x^{5} e^{2} c a^{2} A + \frac{3}{4} x^{4} d^{2} b^{2} a B + \frac{3}{4} x^{4} d^{2} c a^{2} B + \frac{3}{2} x^{4} e d b a^{2} B + \frac{1}{4} x^{4} e^{2} a^{3} B + \frac{1}{4} x^{4} d^{2} b^{3} A + \frac{3}{2} x^{4} d^{2} c b a A + \frac{3}{2} x^{4} e d b^{2} a A + \frac{3}{2} x^{4} e d c a^{2} A + \frac{3}{4} x^{4} e^{2} b a^{2} A + x^{3} d^{2} b a^{2} B + \frac{2}{3} x^{3} e d a^{3} B + x^{3} d^{2} b^{2} a A + x^{3} d^{2} c a^{2} A + 2 x^{3} e d b a^{2} A + \frac{1}{3} x^{3} e^{2} a^{3} A + \frac{1}{2} x^{2} d^{2} a^{3} B + \frac{3}{2} x^{2} d^{2} b a^{2} A + x^{2} e d a^{3} A + x d^{2} a^{3} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*x^10*e^2*c^3*B + 2/9*x^9*e*d*c^3*B + 1/3*x^9*e^2*c^2*b*B + 1/9*x^9*e^2*c^3*A + 1/8*x^8*d^2*c^3*B + 3/4*x^
8*e*d*c^2*b*B + 3/8*x^8*e^2*c*b^2*B + 3/8*x^8*e^2*c^2*a*B + 1/4*x^8*e*d*c^3*A + 3/8*x^8*e^2*c^2*b*A + 3/7*x^7*
d^2*c^2*b*B + 6/7*x^7*e*d*c*b^2*B + 1/7*x^7*e^2*b^3*B + 6/7*x^7*e*d*c^2*a*B + 6/7*x^7*e^2*c*b*a*B + 1/7*x^7*d^
2*c^3*A + 6/7*x^7*e*d*c^2*b*A + 3/7*x^7*e^2*c*b^2*A + 3/7*x^7*e^2*c^2*a*A + 1/2*x^6*d^2*c*b^2*B + 1/3*x^6*e*d*
b^3*B + 1/2*x^6*d^2*c^2*a*B + 2*x^6*e*d*c*b*a*B + 1/2*x^6*e^2*b^2*a*B + 1/2*x^6*e^2*c*a^2*B + 1/2*x^6*d^2*c^2*
b*A + x^6*e*d*c*b^2*A + 1/6*x^6*e^2*b^3*A + x^6*e*d*c^2*a*A + x^6*e^2*c*b*a*A + 1/5*x^5*d^2*b^3*B + 6/5*x^5*d^
2*c*b*a*B + 6/5*x^5*e*d*b^2*a*B + 6/5*x^5*e*d*c*a^2*B + 3/5*x^5*e^2*b*a^2*B + 3/5*x^5*d^2*c*b^2*A + 2/5*x^5*e*
d*b^3*A + 3/5*x^5*d^2*c^2*a*A + 12/5*x^5*e*d*c*b*a*A + 3/5*x^5*e^2*b^2*a*A + 3/5*x^5*e^2*c*a^2*A + 3/4*x^4*d^2
*b^2*a*B + 3/4*x^4*d^2*c*a^2*B + 3/2*x^4*e*d*b*a^2*B + 1/4*x^4*e^2*a^3*B + 1/4*x^4*d^2*b^3*A + 3/2*x^4*d^2*c*b
*a*A + 3/2*x^4*e*d*b^2*a*A + 3/2*x^4*e*d*c*a^2*A + 3/4*x^4*e^2*b*a^2*A + x^3*d^2*b*a^2*B + 2/3*x^3*e*d*a^3*B +
x^3*d^2*b^2*a*A + x^3*d^2*c*a^2*A + 2*x^3*e*d*b*a^2*A + 1/3*x^3*e^2*a^3*A + 1/2*x^2*d^2*a^3*B + 3/2*x^2*d^2*b
*a^2*A + x^2*e*d*a^3*A + x*d^2*a^3*A

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Sympy [A]  time = 0.159893, size = 753, normalized size = 1.36 \begin{align*} A a^{3} d^{2} x + \frac{B c^{3} e^{2} x^{10}}{10} + x^{9} \left (\frac{A c^{3} e^{2}}{9} + \frac{B b c^{2} e^{2}}{3} + \frac{2 B c^{3} d e}{9}\right ) + x^{8} \left (\frac{3 A b c^{2} e^{2}}{8} + \frac{A c^{3} d e}{4} + \frac{3 B a c^{2} e^{2}}{8} + \frac{3 B b^{2} c e^{2}}{8} + \frac{3 B b c^{2} d e}{4} + \frac{B c^{3} d^{2}}{8}\right ) + x^{7} \left (\frac{3 A a c^{2} e^{2}}{7} + \frac{3 A b^{2} c e^{2}}{7} + \frac{6 A b c^{2} d e}{7} + \frac{A c^{3} d^{2}}{7} + \frac{6 B a b c e^{2}}{7} + \frac{6 B a c^{2} d e}{7} + \frac{B b^{3} e^{2}}{7} + \frac{6 B b^{2} c d e}{7} + \frac{3 B b c^{2} d^{2}}{7}\right ) + x^{6} \left (A a b c e^{2} + A a c^{2} d e + \frac{A b^{3} e^{2}}{6} + A b^{2} c d e + \frac{A b c^{2} d^{2}}{2} + \frac{B a^{2} c e^{2}}{2} + \frac{B a b^{2} e^{2}}{2} + 2 B a b c d e + \frac{B a c^{2} d^{2}}{2} + \frac{B b^{3} d e}{3} + \frac{B b^{2} c d^{2}}{2}\right ) + x^{5} \left (\frac{3 A a^{2} c e^{2}}{5} + \frac{3 A a b^{2} e^{2}}{5} + \frac{12 A a b c d e}{5} + \frac{3 A a c^{2} d^{2}}{5} + \frac{2 A b^{3} d e}{5} + \frac{3 A b^{2} c d^{2}}{5} + \frac{3 B a^{2} b e^{2}}{5} + \frac{6 B a^{2} c d e}{5} + \frac{6 B a b^{2} d e}{5} + \frac{6 B a b c d^{2}}{5} + \frac{B b^{3} d^{2}}{5}\right ) + x^{4} \left (\frac{3 A a^{2} b e^{2}}{4} + \frac{3 A a^{2} c d e}{2} + \frac{3 A a b^{2} d e}{2} + \frac{3 A a b c d^{2}}{2} + \frac{A b^{3} d^{2}}{4} + \frac{B a^{3} e^{2}}{4} + \frac{3 B a^{2} b d e}{2} + \frac{3 B a^{2} c d^{2}}{4} + \frac{3 B a b^{2} d^{2}}{4}\right ) + x^{3} \left (\frac{A a^{3} e^{2}}{3} + 2 A a^{2} b d e + A a^{2} c d^{2} + A a b^{2} d^{2} + \frac{2 B a^{3} d e}{3} + B a^{2} b d^{2}\right ) + x^{2} \left (A a^{3} d e + \frac{3 A a^{2} b d^{2}}{2} + \frac{B a^{3} d^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*a**3*d**2*x + B*c**3*e**2*x**10/10 + x**9*(A*c**3*e**2/9 + B*b*c**2*e**2/3 + 2*B*c**3*d*e/9) + x**8*(3*A*b*c
**2*e**2/8 + A*c**3*d*e/4 + 3*B*a*c**2*e**2/8 + 3*B*b**2*c*e**2/8 + 3*B*b*c**2*d*e/4 + B*c**3*d**2/8) + x**7*(
3*A*a*c**2*e**2/7 + 3*A*b**2*c*e**2/7 + 6*A*b*c**2*d*e/7 + A*c**3*d**2/7 + 6*B*a*b*c*e**2/7 + 6*B*a*c**2*d*e/7
+ B*b**3*e**2/7 + 6*B*b**2*c*d*e/7 + 3*B*b*c**2*d**2/7) + x**6*(A*a*b*c*e**2 + A*a*c**2*d*e + A*b**3*e**2/6 +
A*b**2*c*d*e + A*b*c**2*d**2/2 + B*a**2*c*e**2/2 + B*a*b**2*e**2/2 + 2*B*a*b*c*d*e + B*a*c**2*d**2/2 + B*b**3
*d*e/3 + B*b**2*c*d**2/2) + x**5*(3*A*a**2*c*e**2/5 + 3*A*a*b**2*e**2/5 + 12*A*a*b*c*d*e/5 + 3*A*a*c**2*d**2/5
+ 2*A*b**3*d*e/5 + 3*A*b**2*c*d**2/5 + 3*B*a**2*b*e**2/5 + 6*B*a**2*c*d*e/5 + 6*B*a*b**2*d*e/5 + 6*B*a*b*c*d*
*2/5 + B*b**3*d**2/5) + x**4*(3*A*a**2*b*e**2/4 + 3*A*a**2*c*d*e/2 + 3*A*a*b**2*d*e/2 + 3*A*a*b*c*d**2/2 + A*b
**3*d**2/4 + B*a**3*e**2/4 + 3*B*a**2*b*d*e/2 + 3*B*a**2*c*d**2/4 + 3*B*a*b**2*d**2/4) + x**3*(A*a**3*e**2/3 +
2*A*a**2*b*d*e + A*a**2*c*d**2 + A*a*b**2*d**2 + 2*B*a**3*d*e/3 + B*a**2*b*d**2) + x**2*(A*a**3*d*e + 3*A*a**
2*b*d**2/2 + B*a**3*d**2/2)

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Giac [A]  time = 1.10903, size = 981, normalized size = 1.77 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*B*c^3*x^10*e^2 + 2/9*B*c^3*d*x^9*e + 1/8*B*c^3*d^2*x^8 + 1/3*B*b*c^2*x^9*e^2 + 1/9*A*c^3*x^9*e^2 + 3/4*B*
b*c^2*d*x^8*e + 1/4*A*c^3*d*x^8*e + 3/7*B*b*c^2*d^2*x^7 + 1/7*A*c^3*d^2*x^7 + 3/8*B*b^2*c*x^8*e^2 + 3/8*B*a*c^
2*x^8*e^2 + 3/8*A*b*c^2*x^8*e^2 + 6/7*B*b^2*c*d*x^7*e + 6/7*B*a*c^2*d*x^7*e + 6/7*A*b*c^2*d*x^7*e + 1/2*B*b^2*
c*d^2*x^6 + 1/2*B*a*c^2*d^2*x^6 + 1/2*A*b*c^2*d^2*x^6 + 1/7*B*b^3*x^7*e^2 + 6/7*B*a*b*c*x^7*e^2 + 3/7*A*b^2*c*
x^7*e^2 + 3/7*A*a*c^2*x^7*e^2 + 1/3*B*b^3*d*x^6*e + 2*B*a*b*c*d*x^6*e + A*b^2*c*d*x^6*e + A*a*c^2*d*x^6*e + 1/
5*B*b^3*d^2*x^5 + 6/5*B*a*b*c*d^2*x^5 + 3/5*A*b^2*c*d^2*x^5 + 3/5*A*a*c^2*d^2*x^5 + 1/2*B*a*b^2*x^6*e^2 + 1/6*
A*b^3*x^6*e^2 + 1/2*B*a^2*c*x^6*e^2 + A*a*b*c*x^6*e^2 + 6/5*B*a*b^2*d*x^5*e + 2/5*A*b^3*d*x^5*e + 6/5*B*a^2*c*
d*x^5*e + 12/5*A*a*b*c*d*x^5*e + 3/4*B*a*b^2*d^2*x^4 + 1/4*A*b^3*d^2*x^4 + 3/4*B*a^2*c*d^2*x^4 + 3/2*A*a*b*c*d
^2*x^4 + 3/5*B*a^2*b*x^5*e^2 + 3/5*A*a*b^2*x^5*e^2 + 3/5*A*a^2*c*x^5*e^2 + 3/2*B*a^2*b*d*x^4*e + 3/2*A*a*b^2*d
*x^4*e + 3/2*A*a^2*c*d*x^4*e + B*a^2*b*d^2*x^3 + A*a*b^2*d^2*x^3 + A*a^2*c*d^2*x^3 + 1/4*B*a^3*x^4*e^2 + 3/4*A
*a^2*b*x^4*e^2 + 2/3*B*a^3*d*x^3*e + 2*A*a^2*b*d*x^3*e + 1/2*B*a^3*d^2*x^2 + 3/2*A*a^2*b*d^2*x^2 + 1/3*A*a^3*x
^3*e^2 + A*a^3*d*x^2*e + A*a^3*d^2*x