3.2319 \(\int (A+B x) (d+e x)^4 (a+b x+c x^2)^2 \, dx\)

Optimal. Leaf size=304 \[ -\frac{(d+e x)^8 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{8 e^6}-\frac{(d+e x)^7 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{7 e^6}-\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (2 A e (2 c d-b e)-B \left (5 c d^2-e (3 b d-a e)\right )\right )}{6 e^6}-\frac{(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac{c (d+e x)^9 (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6} \]

[Out]

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

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Rubi [A]  time = 0.628602, antiderivative size = 302, normalized size of antiderivative = 0.99, 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} \[ -\frac{(d+e x)^8 \left (2 A c e (2 c d-b e)-B \left (-2 c e (4 b d-a e)+b^2 e^2+10 c^2 d^2\right )\right )}{8 e^6}-\frac{(d+e x)^7 \left (B \left (-6 c d e (2 b d-a e)+b e^2 (3 b d-2 a e)+10 c^2 d^3\right )-A e \left (-2 c e (3 b d-a e)+b^2 e^2+6 c^2 d^2\right )\right )}{7 e^6}+\frac{(d+e x)^6 \left (a e^2-b d e+c d^2\right ) \left (-B e (3 b d-a e)-2 A e (2 c d-b e)+5 B c d^2\right )}{6 e^6}-\frac{(d+e x)^5 (B d-A e) \left (a e^2-b d e+c d^2\right )^2}{5 e^6}-\frac{c (d+e x)^9 (-A c e-2 b B e+5 B c d)}{9 e^6}+\frac{B c^2 (d+e x)^{10}}{10 e^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.267507, size = 550, normalized size = 1.81 \[ \frac{1}{6} x^6 \left (B \left (e^2 \left (a^2 e^2+8 a b d e+6 b^2 d^2\right )+4 c d^2 e (3 a e+2 b d)+c^2 d^4\right )+2 A e \left (2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right )\right )+\frac{1}{5} x^5 \left (A \left (a^2 e^4+12 a c d^2 e^2+c^2 d^4\right )+2 b d \left (4 a A e^3+6 a B d e^2+4 A c d^2 e+B c d^3\right )+4 a B d e \left (a e^2+2 c d^2\right )+2 b^2 d^2 e (3 A e+2 B d)\right )+a^2 A d^4 x+\frac{1}{8} e^2 x^8 \left (B \left (2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+2 A c e (b e+2 c d)\right )+\frac{1}{7} e x^7 \left (A e \left (2 c e (a e+4 b d)+b^2 e^2+6 c^2 d^2\right )+2 B \left (2 c d e (2 a e+3 b d)+b e^2 (a e+2 b d)+2 c^2 d^3\right )\right )+\frac{1}{4} d x^4 \left (2 b d \left (6 a A e^2+4 a B d e+A c d^2\right )+2 a \left (2 a A e^3+3 a B d e^2+4 A c d^2 e+B c d^3\right )+b^2 d^2 (4 A e+B d)\right )+\frac{1}{3} d^2 x^3 \left (A \left (8 a b d e+2 a \left (3 a e^2+c d^2\right )+b^2 d^2\right )+2 a B d (2 a e+b d)\right )+\frac{1}{2} a d^3 x^2 (4 a A e+a B d+2 A b d)+\frac{1}{9} c e^3 x^9 (A c e+2 b B e+4 B c d)+\frac{1}{10} B c^2 e^4 x^{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 545, normalized size = 1.8 \begin{align*}{\frac{B{e}^{4}{c}^{2}{x}^{10}}{10}}+{\frac{ \left ( \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){c}^{2}+2\,B{e}^{4}bc \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){c}^{2}+2\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) bc+B{e}^{4} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){c}^{2}+2\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) bc+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\,B{e}^{4}ab \right ){x}^{7}}{7}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){c}^{2}+2\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) bc+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) ab+B{a}^{2}{e}^{4} \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}{d}^{4}+2\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) bc+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) ab+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,A{d}^{4}bc+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) ab+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( A{d}^{4} \left ( 2\,ac+{b}^{2} \right ) +2\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) ab+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,A{d}^{4}ab+ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2} \right ){x}^{2}}{2}}+A{d}^{4}{a}^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 0.98435, size = 718, normalized size = 2.36 \begin{align*} \frac{1}{10} \, B c^{2} e^{4} x^{10} + \frac{1}{9} \,{\left (4 \, B c^{2} d e^{3} +{\left (2 \, B b c + A c^{2}\right )} e^{4}\right )} x^{9} + \frac{1}{8} \,{\left (6 \, B c^{2} d^{2} e^{2} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} e^{4}\right )} x^{8} + A a^{2} d^{4} x + \frac{1}{7} \,{\left (4 \, B c^{2} d^{3} e + 6 \,{\left (2 \, B b c + A c^{2}\right )} d^{2} e^{2} + 4 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d e^{3} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{4} + 4 \,{\left (2 \, B b c + A c^{2}\right )} d^{3} e + 6 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{2} e^{2} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d e^{3} +{\left (B a^{2} + 2 \, A a b\right )} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (A a^{2} e^{4} +{\left (2 \, B b c + A c^{2}\right )} d^{4} + 4 \,{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{3} e + 6 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{2} e^{2} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} d e^{3}\right )} x^{5} + \frac{1}{4} \,{\left (4 \, A a^{2} d e^{3} +{\left (B b^{2} + 2 \,{\left (B a + A b\right )} c\right )} d^{4} + 4 \,{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{3} e + 6 \,{\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (6 \, A a^{2} d^{2} e^{2} +{\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} d^{4} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} d^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (4 \, A a^{2} d^{3} e +{\left (B a^{2} + 2 \, A a b\right )} d^{4}\right )} x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.10265, size = 1701, normalized size = 5.6 \begin{align*} \frac{1}{10} x^{10} e^{4} c^{2} B + \frac{4}{9} x^{9} e^{3} d c^{2} B + \frac{2}{9} x^{9} e^{4} c b B + \frac{1}{9} x^{9} e^{4} c^{2} A + \frac{3}{4} x^{8} e^{2} d^{2} c^{2} B + x^{8} e^{3} d c b B + \frac{1}{8} x^{8} e^{4} b^{2} B + \frac{1}{4} x^{8} e^{4} c a B + \frac{1}{2} x^{8} e^{3} d c^{2} A + \frac{1}{4} x^{8} e^{4} c b A + \frac{4}{7} x^{7} e d^{3} c^{2} B + \frac{12}{7} x^{7} e^{2} d^{2} c b B + \frac{4}{7} x^{7} e^{3} d b^{2} B + \frac{8}{7} x^{7} e^{3} d c a B + \frac{2}{7} x^{7} e^{4} b a B + \frac{6}{7} x^{7} e^{2} d^{2} c^{2} A + \frac{8}{7} x^{7} e^{3} d c b A + \frac{1}{7} x^{7} e^{4} b^{2} A + \frac{2}{7} x^{7} e^{4} c a A + \frac{1}{6} x^{6} d^{4} c^{2} B + \frac{4}{3} x^{6} e d^{3} c b B + x^{6} e^{2} d^{2} b^{2} B + 2 x^{6} e^{2} d^{2} c a B + \frac{4}{3} x^{6} e^{3} d b a B + \frac{1}{6} x^{6} e^{4} a^{2} B + \frac{2}{3} x^{6} e d^{3} c^{2} A + 2 x^{6} e^{2} d^{2} c b A + \frac{2}{3} x^{6} e^{3} d b^{2} A + \frac{4}{3} x^{6} e^{3} d c a A + \frac{1}{3} x^{6} e^{4} b a A + \frac{2}{5} x^{5} d^{4} c b B + \frac{4}{5} x^{5} e d^{3} b^{2} B + \frac{8}{5} x^{5} e d^{3} c a B + \frac{12}{5} x^{5} e^{2} d^{2} b a B + \frac{4}{5} x^{5} e^{3} d a^{2} B + \frac{1}{5} x^{5} d^{4} c^{2} A + \frac{8}{5} x^{5} e d^{3} c b A + \frac{6}{5} x^{5} e^{2} d^{2} b^{2} A + \frac{12}{5} x^{5} e^{2} d^{2} c a A + \frac{8}{5} x^{5} e^{3} d b a A + \frac{1}{5} x^{5} e^{4} a^{2} A + \frac{1}{4} x^{4} d^{4} b^{2} B + \frac{1}{2} x^{4} d^{4} c a B + 2 x^{4} e d^{3} b a B + \frac{3}{2} x^{4} e^{2} d^{2} a^{2} B + \frac{1}{2} x^{4} d^{4} c b A + x^{4} e d^{3} b^{2} A + 2 x^{4} e d^{3} c a A + 3 x^{4} e^{2} d^{2} b a A + x^{4} e^{3} d a^{2} A + \frac{2}{3} x^{3} d^{4} b a B + \frac{4}{3} x^{3} e d^{3} a^{2} B + \frac{1}{3} x^{3} d^{4} b^{2} A + \frac{2}{3} x^{3} d^{4} c a A + \frac{8}{3} x^{3} e d^{3} b a A + 2 x^{3} e^{2} d^{2} a^{2} A + \frac{1}{2} x^{2} d^{4} a^{2} B + x^{2} d^{4} b a A + 2 x^{2} e d^{3} a^{2} A + x d^{4} a^{2} A \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+b*x+a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 0.151386, size = 765, normalized size = 2.52 \begin{align*} A a^{2} d^{4} x + \frac{B c^{2} e^{4} x^{10}}{10} + x^{9} \left (\frac{A c^{2} e^{4}}{9} + \frac{2 B b c e^{4}}{9} + \frac{4 B c^{2} d e^{3}}{9}\right ) + x^{8} \left (\frac{A b c e^{4}}{4} + \frac{A c^{2} d e^{3}}{2} + \frac{B a c e^{4}}{4} + \frac{B b^{2} e^{4}}{8} + B b c d e^{3} + \frac{3 B c^{2} d^{2} e^{2}}{4}\right ) + x^{7} \left (\frac{2 A a c e^{4}}{7} + \frac{A b^{2} e^{4}}{7} + \frac{8 A b c d e^{3}}{7} + \frac{6 A c^{2} d^{2} e^{2}}{7} + \frac{2 B a b e^{4}}{7} + \frac{8 B a c d e^{3}}{7} + \frac{4 B b^{2} d e^{3}}{7} + \frac{12 B b c d^{2} e^{2}}{7} + \frac{4 B c^{2} d^{3} e}{7}\right ) + x^{6} \left (\frac{A a b e^{4}}{3} + \frac{4 A a c d e^{3}}{3} + \frac{2 A b^{2} d e^{3}}{3} + 2 A b c d^{2} e^{2} + \frac{2 A c^{2} d^{3} e}{3} + \frac{B a^{2} e^{4}}{6} + \frac{4 B a b d e^{3}}{3} + 2 B a c d^{2} e^{2} + B b^{2} d^{2} e^{2} + \frac{4 B b c d^{3} e}{3} + \frac{B c^{2} d^{4}}{6}\right ) + x^{5} \left (\frac{A a^{2} e^{4}}{5} + \frac{8 A a b d e^{3}}{5} + \frac{12 A a c d^{2} e^{2}}{5} + \frac{6 A b^{2} d^{2} e^{2}}{5} + \frac{8 A b c d^{3} e}{5} + \frac{A c^{2} d^{4}}{5} + \frac{4 B a^{2} d e^{3}}{5} + \frac{12 B a b d^{2} e^{2}}{5} + \frac{8 B a c d^{3} e}{5} + \frac{4 B b^{2} d^{3} e}{5} + \frac{2 B b c d^{4}}{5}\right ) + x^{4} \left (A a^{2} d e^{3} + 3 A a b d^{2} e^{2} + 2 A a c d^{3} e + A b^{2} d^{3} e + \frac{A b c d^{4}}{2} + \frac{3 B a^{2} d^{2} e^{2}}{2} + 2 B a b d^{3} e + \frac{B a c d^{4}}{2} + \frac{B b^{2} d^{4}}{4}\right ) + x^{3} \left (2 A a^{2} d^{2} e^{2} + \frac{8 A a b d^{3} e}{3} + \frac{2 A a c d^{4}}{3} + \frac{A b^{2} d^{4}}{3} + \frac{4 B a^{2} d^{3} e}{3} + \frac{2 B a b d^{4}}{3}\right ) + x^{2} \left (2 A a^{2} d^{3} e + A a b d^{4} + \frac{B a^{2} d^{4}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4*(c*x**2+b*x+a)**2,x)

[Out]

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

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Giac [B]  time = 1.14867, size = 971, normalized size = 3.19 \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)^4*(c*x^2+b*x+a)^2,x, algorithm="giac")

[Out]

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