3.1314 \(\int (A+B x) (d+e x)^4 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=334 \[ -\frac{c (d+e x)^8 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{8 e^8}+\frac{3 c^2 (d+e x)^{10} \left (a B e^2-2 A c d e+7 B c d^2\right )}{10 e^8}-\frac{c^2 (d+e x)^9 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}-\frac{3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8}+\frac{(d+e x)^6 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8}-\frac{(d+e x)^5 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8}-\frac{c^3 (d+e x)^{11} (7 B d-A e)}{11 e^8}+\frac{B c^3 (d+e x)^{12}}{12 e^8} \]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^5)/(5*e^8) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d
+ e*x)^6)/(6*e^8) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^7)/(7*e^
8) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^8)/(8*e^8) - (
c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^9)/(9*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*
e + a*B*e^2)*(d + e*x)^10)/(10*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^11)/(11*e^8) + (B*c^3*(d + e*x)^12)/(12*e^8
)

________________________________________________________________________________________

Rubi [A]  time = 0.448544, antiderivative size = 334, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {772} \[ -\frac{c (d+e x)^8 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{8 e^8}+\frac{3 c^2 (d+e x)^{10} \left (a B e^2-2 A c d e+7 B c d^2\right )}{10 e^8}-\frac{c^2 (d+e x)^9 \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{9 e^8}-\frac{3 c (d+e x)^7 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{7 e^8}+\frac{(d+e x)^6 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{6 e^8}-\frac{(d+e x)^5 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8}-\frac{c^3 (d+e x)^{11} (7 B d-A e)}{11 e^8}+\frac{B c^3 (d+e x)^{12}}{12 e^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^5)/(5*e^8) + ((c*d^2 + a*e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2)*(d
+ e*x)^6)/(6*e^8) - (3*c*(c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3)*(d + e*x)^7)/(7*e^
8) - (c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4))*(d + e*x)^8)/(8*e^8) - (
c^2*(35*B*c*d^3 - 15*A*c*d^2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^9)/(9*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*d*
e + a*B*e^2)*(d + e*x)^10)/(10*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^11)/(11*e^8) + (B*c^3*(d + e*x)^12)/(12*e^8
)

Rule 772

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int (A+B x) (d+e x)^4 \left (a+c x^2\right )^3 \, dx &=\int \left (\frac{(-B d+A e) \left (c d^2+a e^2\right )^3 (d+e x)^4}{e^7}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^5}{e^7}+\frac{3 c \left (c d^2+a e^2\right ) \left (-7 B c d^3+5 A c d^2 e-3 a B d e^2+a A e^3\right ) (d+e x)^6}{e^7}-\frac{c \left (-35 B c^2 d^4+20 A c^2 d^3 e-30 a B c d^2 e^2+12 a A c d e^3-3 a^2 B e^4\right ) (d+e x)^7}{e^7}+\frac{c^2 \left (-35 B c d^3+15 A c d^2 e-15 a B d e^2+3 a A e^3\right ) (d+e x)^8}{e^7}-\frac{3 c^2 \left (-7 B c d^2+2 A c d e-a B e^2\right ) (d+e x)^9}{e^7}+\frac{c^3 (-7 B d+A e) (d+e x)^{10}}{e^7}+\frac{B c^3 (d+e x)^{11}}{e^7}\right ) \, dx\\ &=-\frac{(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^5}{5 e^8}+\frac{\left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right ) (d+e x)^6}{6 e^8}-\frac{3 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right ) (d+e x)^7}{7 e^8}-\frac{c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) (d+e x)^8}{8 e^8}-\frac{c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^9}{9 e^8}+\frac{3 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{10}}{10 e^8}-\frac{c^3 (7 B d-A e) (d+e x)^{11}}{11 e^8}+\frac{B c^3 (d+e x)^{12}}{12 e^8}\\ \end{align*}

Mathematica [A]  time = 0.127092, size = 436, normalized size = 1.31 \[ \frac{1}{8} c x^8 \left (B \left (3 a^2 e^4+18 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (3 a e^2+c d^2\right )\right )+\frac{1}{7} c x^7 \left (A \left (3 a^2 e^4+18 a c d^2 e^2+c^2 d^4\right )+12 a B d e \left (a e^2+c d^2\right )\right )+\frac{1}{6} a x^6 \left (B \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+12 A c d e \left (a e^2+c d^2\right )\right )+\frac{1}{5} a x^5 \left (A \left (a^2 e^4+18 a c d^2 e^2+3 c^2 d^4\right )+4 a B d e \left (a e^2+3 c d^2\right )\right )+\frac{1}{4} a^2 d x^4 \left (4 a A e^3+6 a B d e^2+12 A c d^2 e+3 B c d^3\right )+\frac{1}{3} a^2 d^2 x^3 \left (6 a A e^2+4 a B d e+3 A c d^2\right )+\frac{1}{2} a^3 d^3 x^2 (4 A e+B d)+a^3 A d^4 x+\frac{1}{10} c^2 e^2 x^{10} \left (3 a B e^2+4 A c d e+6 B c d^2\right )+\frac{1}{9} c^2 e x^9 \left (3 a A e^3+12 a B d e^2+6 A c d^2 e+4 B c d^3\right )+\frac{1}{11} c^3 e^3 x^{11} (A e+4 B d)+\frac{1}{12} B c^3 e^4 x^{12} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a^3*A*d^4*x + (a^3*d^3*(B*d + 4*A*e)*x^2)/2 + (a^2*d^2*(3*A*c*d^2 + 4*a*B*d*e + 6*a*A*e^2)*x^3)/3 + (a^2*d*(3*
B*c*d^3 + 12*A*c*d^2*e + 6*a*B*d*e^2 + 4*a*A*e^3)*x^4)/4 + (a*(4*a*B*d*e*(3*c*d^2 + a*e^2) + A*(3*c^2*d^4 + 18
*a*c*d^2*e^2 + a^2*e^4))*x^5)/5 + (a*(12*A*c*d*e*(c*d^2 + a*e^2) + B*(3*c^2*d^4 + 18*a*c*d^2*e^2 + a^2*e^4))*x
^6)/6 + (c*(12*a*B*d*e*(c*d^2 + a*e^2) + A*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^7)/7 + (c*(4*A*c*d*e*(c*d
^2 + 3*a*e^2) + B*(c^2*d^4 + 18*a*c*d^2*e^2 + 3*a^2*e^4))*x^8)/8 + (c^2*e*(4*B*c*d^3 + 6*A*c*d^2*e + 12*a*B*d*
e^2 + 3*a*A*e^3)*x^9)/9 + (c^2*e^2*(6*B*c*d^2 + 4*A*c*d*e + 3*a*B*e^2)*x^10)/10 + (c^3*e^3*(4*B*d + A*e)*x^11)
/11 + (B*c^3*e^4*x^12)/12

________________________________________________________________________________________

Maple [A]  time = 0., size = 455, normalized size = 1.4 \begin{align*}{\frac{B{c}^{3}{e}^{4}{x}^{12}}{12}}+{\frac{ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){c}^{3}{x}^{11}}{11}}+{\frac{ \left ( \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){c}^{3}+3\,B{e}^{4}a{c}^{2} \right ){x}^{10}}{10}}+{\frac{ \left ( \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){c}^{3}+3\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ) a{c}^{2} \right ){x}^{9}}{9}}+{\frac{ \left ( \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){c}^{3}+3\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ) a{c}^{2}+3\,B{e}^{4}{a}^{2}c \right ){x}^{8}}{8}}+{\frac{ \left ( A{d}^{4}{c}^{3}+3\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ) a{c}^{2}+3\, \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{2}c \right ){x}^{7}}{7}}+{\frac{ \left ( 3\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ) a{c}^{2}+3\, \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{2}c+B{e}^{4}{a}^{3} \right ){x}^{6}}{6}}+{\frac{ \left ( 3\,A{d}^{4}a{c}^{2}+3\, \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{2}c+ \left ( A{e}^{4}+4\,Bd{e}^{3} \right ){a}^{3} \right ){x}^{5}}{5}}+{\frac{ \left ( 3\, \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{2}c+ \left ( 4\,Ad{e}^{3}+6\,B{d}^{2}{e}^{2} \right ){a}^{3} \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,A{d}^{4}{a}^{2}c+ \left ( 6\,A{d}^{2}{e}^{2}+4\,B{d}^{3}e \right ){a}^{3} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,A{d}^{3}e+B{d}^{4} \right ){a}^{3}{x}^{2}}{2}}+A{d}^{4}{a}^{3}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+a)^3,x)

[Out]

1/12*B*c^3*e^4*x^12+1/11*(A*e^4+4*B*d*e^3)*c^3*x^11+1/10*((4*A*d*e^3+6*B*d^2*e^2)*c^3+3*B*e^4*a*c^2)*x^10+1/9*
((6*A*d^2*e^2+4*B*d^3*e)*c^3+3*(A*e^4+4*B*d*e^3)*a*c^2)*x^9+1/8*((4*A*d^3*e+B*d^4)*c^3+3*(4*A*d*e^3+6*B*d^2*e^
2)*a*c^2+3*B*e^4*a^2*c)*x^8+1/7*(A*d^4*c^3+3*(6*A*d^2*e^2+4*B*d^3*e)*a*c^2+3*(A*e^4+4*B*d*e^3)*a^2*c)*x^7+1/6*
(3*(4*A*d^3*e+B*d^4)*a*c^2+3*(4*A*d*e^3+6*B*d^2*e^2)*a^2*c+B*e^4*a^3)*x^6+1/5*(3*A*d^4*a*c^2+3*(6*A*d^2*e^2+4*
B*d^3*e)*a^2*c+(A*e^4+4*B*d*e^3)*a^3)*x^5+1/4*(3*(4*A*d^3*e+B*d^4)*a^2*c+(4*A*d*e^3+6*B*d^2*e^2)*a^3)*x^4+1/3*
(3*A*d^4*a^2*c+(6*A*d^2*e^2+4*B*d^3*e)*a^3)*x^3+1/2*(4*A*d^3*e+B*d^4)*a^3*x^2+A*d^4*a^3*x

________________________________________________________________________________________

Maxima [A]  time = 1.09439, size = 645, normalized size = 1.93 \begin{align*} \frac{1}{12} \, B c^{3} e^{4} x^{12} + \frac{1}{11} \,{\left (4 \, B c^{3} d e^{3} + A c^{3} e^{4}\right )} x^{11} + \frac{1}{10} \,{\left (6 \, B c^{3} d^{2} e^{2} + 4 \, A c^{3} d e^{3} + 3 \, B a c^{2} e^{4}\right )} x^{10} + \frac{1}{9} \,{\left (4 \, B c^{3} d^{3} e + 6 \, A c^{3} d^{2} e^{2} + 12 \, B a c^{2} d e^{3} + 3 \, A a c^{2} e^{4}\right )} x^{9} + A a^{3} d^{4} x + \frac{1}{8} \,{\left (B c^{3} d^{4} + 4 \, A c^{3} d^{3} e + 18 \, B a c^{2} d^{2} e^{2} + 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} x^{8} + \frac{1}{7} \,{\left (A c^{3} d^{4} + 12 \, B a c^{2} d^{3} e + 18 \, A a c^{2} d^{2} e^{2} + 12 \, B a^{2} c d e^{3} + 3 \, A a^{2} c e^{4}\right )} x^{7} + \frac{1}{6} \,{\left (3 \, B a c^{2} d^{4} + 12 \, A a c^{2} d^{3} e + 18 \, B a^{2} c d^{2} e^{2} + 12 \, A a^{2} c d e^{3} + B a^{3} e^{4}\right )} x^{6} + \frac{1}{5} \,{\left (3 \, A a c^{2} d^{4} + 12 \, B a^{2} c d^{3} e + 18 \, A a^{2} c d^{2} e^{2} + 4 \, B a^{3} d e^{3} + A a^{3} e^{4}\right )} x^{5} + \frac{1}{4} \,{\left (3 \, B a^{2} c d^{4} + 12 \, A a^{2} c d^{3} e + 6 \, B a^{3} d^{2} e^{2} + 4 \, A a^{3} d e^{3}\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{2} c d^{4} + 4 \, B a^{3} d^{3} e + 6 \, A a^{3} d^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d^{4} + 4 \, A a^{3} d^{3} e\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+a)^3,x, algorithm="maxima")

[Out]

1/12*B*c^3*e^4*x^12 + 1/11*(4*B*c^3*d*e^3 + A*c^3*e^4)*x^11 + 1/10*(6*B*c^3*d^2*e^2 + 4*A*c^3*d*e^3 + 3*B*a*c^
2*e^4)*x^10 + 1/9*(4*B*c^3*d^3*e + 6*A*c^3*d^2*e^2 + 12*B*a*c^2*d*e^3 + 3*A*a*c^2*e^4)*x^9 + A*a^3*d^4*x + 1/8
*(B*c^3*d^4 + 4*A*c^3*d^3*e + 18*B*a*c^2*d^2*e^2 + 12*A*a*c^2*d*e^3 + 3*B*a^2*c*e^4)*x^8 + 1/7*(A*c^3*d^4 + 12
*B*a*c^2*d^3*e + 18*A*a*c^2*d^2*e^2 + 12*B*a^2*c*d*e^3 + 3*A*a^2*c*e^4)*x^7 + 1/6*(3*B*a*c^2*d^4 + 12*A*a*c^2*
d^3*e + 18*B*a^2*c*d^2*e^2 + 12*A*a^2*c*d*e^3 + B*a^3*e^4)*x^6 + 1/5*(3*A*a*c^2*d^4 + 12*B*a^2*c*d^3*e + 18*A*
a^2*c*d^2*e^2 + 4*B*a^3*d*e^3 + A*a^3*e^4)*x^5 + 1/4*(3*B*a^2*c*d^4 + 12*A*a^2*c*d^3*e + 6*B*a^3*d^2*e^2 + 4*A
*a^3*d*e^3)*x^4 + 1/3*(3*A*a^2*c*d^4 + 4*B*a^3*d^3*e + 6*A*a^3*d^2*e^2)*x^3 + 1/2*(B*a^3*d^4 + 4*A*a^3*d^3*e)*
x^2

________________________________________________________________________________________

Fricas [A]  time = 1.50519, size = 1204, normalized size = 3.6 \begin{align*} \frac{1}{12} x^{12} e^{4} c^{3} B + \frac{4}{11} x^{11} e^{3} d c^{3} B + \frac{1}{11} x^{11} e^{4} c^{3} A + \frac{3}{5} x^{10} e^{2} d^{2} c^{3} B + \frac{3}{10} x^{10} e^{4} c^{2} a B + \frac{2}{5} x^{10} e^{3} d c^{3} A + \frac{4}{9} x^{9} e d^{3} c^{3} B + \frac{4}{3} x^{9} e^{3} d c^{2} a B + \frac{2}{3} x^{9} e^{2} d^{2} c^{3} A + \frac{1}{3} x^{9} e^{4} c^{2} a A + \frac{1}{8} x^{8} d^{4} c^{3} B + \frac{9}{4} x^{8} e^{2} d^{2} c^{2} a B + \frac{3}{8} x^{8} e^{4} c a^{2} B + \frac{1}{2} x^{8} e d^{3} c^{3} A + \frac{3}{2} x^{8} e^{3} d c^{2} a A + \frac{12}{7} x^{7} e d^{3} c^{2} a B + \frac{12}{7} x^{7} e^{3} d c a^{2} B + \frac{1}{7} x^{7} d^{4} c^{3} A + \frac{18}{7} x^{7} e^{2} d^{2} c^{2} a A + \frac{3}{7} x^{7} e^{4} c a^{2} A + \frac{1}{2} x^{6} d^{4} c^{2} a B + 3 x^{6} e^{2} d^{2} c a^{2} B + \frac{1}{6} x^{6} e^{4} a^{3} B + 2 x^{6} e d^{3} c^{2} a A + 2 x^{6} e^{3} d c a^{2} A + \frac{12}{5} x^{5} e d^{3} c a^{2} B + \frac{4}{5} x^{5} e^{3} d a^{3} B + \frac{3}{5} x^{5} d^{4} c^{2} a A + \frac{18}{5} x^{5} e^{2} d^{2} c a^{2} A + \frac{1}{5} x^{5} e^{4} a^{3} A + \frac{3}{4} x^{4} d^{4} c a^{2} B + \frac{3}{2} x^{4} e^{2} d^{2} a^{3} B + 3 x^{4} e d^{3} c a^{2} A + x^{4} e^{3} d a^{3} A + \frac{4}{3} x^{3} e d^{3} a^{3} B + x^{3} d^{4} c a^{2} A + 2 x^{3} e^{2} d^{2} a^{3} A + \frac{1}{2} x^{2} d^{4} a^{3} B + 2 x^{2} e d^{3} a^{3} A + x d^{4} a^{3} 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+a)^3,x, algorithm="fricas")

[Out]

1/12*x^12*e^4*c^3*B + 4/11*x^11*e^3*d*c^3*B + 1/11*x^11*e^4*c^3*A + 3/5*x^10*e^2*d^2*c^3*B + 3/10*x^10*e^4*c^2
*a*B + 2/5*x^10*e^3*d*c^3*A + 4/9*x^9*e*d^3*c^3*B + 4/3*x^9*e^3*d*c^2*a*B + 2/3*x^9*e^2*d^2*c^3*A + 1/3*x^9*e^
4*c^2*a*A + 1/8*x^8*d^4*c^3*B + 9/4*x^8*e^2*d^2*c^2*a*B + 3/8*x^8*e^4*c*a^2*B + 1/2*x^8*e*d^3*c^3*A + 3/2*x^8*
e^3*d*c^2*a*A + 12/7*x^7*e*d^3*c^2*a*B + 12/7*x^7*e^3*d*c*a^2*B + 1/7*x^7*d^4*c^3*A + 18/7*x^7*e^2*d^2*c^2*a*A
+ 3/7*x^7*e^4*c*a^2*A + 1/2*x^6*d^4*c^2*a*B + 3*x^6*e^2*d^2*c*a^2*B + 1/6*x^6*e^4*a^3*B + 2*x^6*e*d^3*c^2*a*A
+ 2*x^6*e^3*d*c*a^2*A + 12/5*x^5*e*d^3*c*a^2*B + 4/5*x^5*e^3*d*a^3*B + 3/5*x^5*d^4*c^2*a*A + 18/5*x^5*e^2*d^2
*c*a^2*A + 1/5*x^5*e^4*a^3*A + 3/4*x^4*d^4*c*a^2*B + 3/2*x^4*e^2*d^2*a^3*B + 3*x^4*e*d^3*c*a^2*A + x^4*e^3*d*a
^3*A + 4/3*x^3*e*d^3*a^3*B + x^3*d^4*c*a^2*A + 2*x^3*e^2*d^2*a^3*A + 1/2*x^2*d^4*a^3*B + 2*x^2*e*d^3*a^3*A + x
*d^4*a^3*A

________________________________________________________________________________________

Sympy [A]  time = 0.203593, size = 564, normalized size = 1.69 \begin{align*} A a^{3} d^{4} x + \frac{B c^{3} e^{4} x^{12}}{12} + x^{11} \left (\frac{A c^{3} e^{4}}{11} + \frac{4 B c^{3} d e^{3}}{11}\right ) + x^{10} \left (\frac{2 A c^{3} d e^{3}}{5} + \frac{3 B a c^{2} e^{4}}{10} + \frac{3 B c^{3} d^{2} e^{2}}{5}\right ) + x^{9} \left (\frac{A a c^{2} e^{4}}{3} + \frac{2 A c^{3} d^{2} e^{2}}{3} + \frac{4 B a c^{2} d e^{3}}{3} + \frac{4 B c^{3} d^{3} e}{9}\right ) + x^{8} \left (\frac{3 A a c^{2} d e^{3}}{2} + \frac{A c^{3} d^{3} e}{2} + \frac{3 B a^{2} c e^{4}}{8} + \frac{9 B a c^{2} d^{2} e^{2}}{4} + \frac{B c^{3} d^{4}}{8}\right ) + x^{7} \left (\frac{3 A a^{2} c e^{4}}{7} + \frac{18 A a c^{2} d^{2} e^{2}}{7} + \frac{A c^{3} d^{4}}{7} + \frac{12 B a^{2} c d e^{3}}{7} + \frac{12 B a c^{2} d^{3} e}{7}\right ) + x^{6} \left (2 A a^{2} c d e^{3} + 2 A a c^{2} d^{3} e + \frac{B a^{3} e^{4}}{6} + 3 B a^{2} c d^{2} e^{2} + \frac{B a c^{2} d^{4}}{2}\right ) + x^{5} \left (\frac{A a^{3} e^{4}}{5} + \frac{18 A a^{2} c d^{2} e^{2}}{5} + \frac{3 A a c^{2} d^{4}}{5} + \frac{4 B a^{3} d e^{3}}{5} + \frac{12 B a^{2} c d^{3} e}{5}\right ) + x^{4} \left (A a^{3} d e^{3} + 3 A a^{2} c d^{3} e + \frac{3 B a^{3} d^{2} e^{2}}{2} + \frac{3 B a^{2} c d^{4}}{4}\right ) + x^{3} \left (2 A a^{3} d^{2} e^{2} + A a^{2} c d^{4} + \frac{4 B a^{3} d^{3} e}{3}\right ) + x^{2} \left (2 A a^{3} d^{3} e + \frac{B a^{3} 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+a)**3,x)

[Out]

A*a**3*d**4*x + B*c**3*e**4*x**12/12 + x**11*(A*c**3*e**4/11 + 4*B*c**3*d*e**3/11) + x**10*(2*A*c**3*d*e**3/5
+ 3*B*a*c**2*e**4/10 + 3*B*c**3*d**2*e**2/5) + x**9*(A*a*c**2*e**4/3 + 2*A*c**3*d**2*e**2/3 + 4*B*a*c**2*d*e**
3/3 + 4*B*c**3*d**3*e/9) + x**8*(3*A*a*c**2*d*e**3/2 + A*c**3*d**3*e/2 + 3*B*a**2*c*e**4/8 + 9*B*a*c**2*d**2*e
**2/4 + B*c**3*d**4/8) + x**7*(3*A*a**2*c*e**4/7 + 18*A*a*c**2*d**2*e**2/7 + A*c**3*d**4/7 + 12*B*a**2*c*d*e**
3/7 + 12*B*a*c**2*d**3*e/7) + x**6*(2*A*a**2*c*d*e**3 + 2*A*a*c**2*d**3*e + B*a**3*e**4/6 + 3*B*a**2*c*d**2*e*
*2 + B*a*c**2*d**4/2) + x**5*(A*a**3*e**4/5 + 18*A*a**2*c*d**2*e**2/5 + 3*A*a*c**2*d**4/5 + 4*B*a**3*d*e**3/5
+ 12*B*a**2*c*d**3*e/5) + x**4*(A*a**3*d*e**3 + 3*A*a**2*c*d**3*e + 3*B*a**3*d**2*e**2/2 + 3*B*a**2*c*d**4/4)
+ x**3*(2*A*a**3*d**2*e**2 + A*a**2*c*d**4 + 4*B*a**3*d**3*e/3) + x**2*(2*A*a**3*d**3*e + B*a**3*d**4/2)

________________________________________________________________________________________

Giac [A]  time = 1.15143, size = 702, normalized size = 2.1 \begin{align*} \frac{1}{12} \, B c^{3} x^{12} e^{4} + \frac{4}{11} \, B c^{3} d x^{11} e^{3} + \frac{3}{5} \, B c^{3} d^{2} x^{10} e^{2} + \frac{4}{9} \, B c^{3} d^{3} x^{9} e + \frac{1}{8} \, B c^{3} d^{4} x^{8} + \frac{1}{11} \, A c^{3} x^{11} e^{4} + \frac{2}{5} \, A c^{3} d x^{10} e^{3} + \frac{2}{3} \, A c^{3} d^{2} x^{9} e^{2} + \frac{1}{2} \, A c^{3} d^{3} x^{8} e + \frac{1}{7} \, A c^{3} d^{4} x^{7} + \frac{3}{10} \, B a c^{2} x^{10} e^{4} + \frac{4}{3} \, B a c^{2} d x^{9} e^{3} + \frac{9}{4} \, B a c^{2} d^{2} x^{8} e^{2} + \frac{12}{7} \, B a c^{2} d^{3} x^{7} e + \frac{1}{2} \, B a c^{2} d^{4} x^{6} + \frac{1}{3} \, A a c^{2} x^{9} e^{4} + \frac{3}{2} \, A a c^{2} d x^{8} e^{3} + \frac{18}{7} \, A a c^{2} d^{2} x^{7} e^{2} + 2 \, A a c^{2} d^{3} x^{6} e + \frac{3}{5} \, A a c^{2} d^{4} x^{5} + \frac{3}{8} \, B a^{2} c x^{8} e^{4} + \frac{12}{7} \, B a^{2} c d x^{7} e^{3} + 3 \, B a^{2} c d^{2} x^{6} e^{2} + \frac{12}{5} \, B a^{2} c d^{3} x^{5} e + \frac{3}{4} \, B a^{2} c d^{4} x^{4} + \frac{3}{7} \, A a^{2} c x^{7} e^{4} + 2 \, A a^{2} c d x^{6} e^{3} + \frac{18}{5} \, A a^{2} c d^{2} x^{5} e^{2} + 3 \, A a^{2} c d^{3} x^{4} e + A a^{2} c d^{4} x^{3} + \frac{1}{6} \, B a^{3} x^{6} e^{4} + \frac{4}{5} \, B a^{3} d x^{5} e^{3} + \frac{3}{2} \, B a^{3} d^{2} x^{4} e^{2} + \frac{4}{3} \, B a^{3} d^{3} x^{3} e + \frac{1}{2} \, B a^{3} d^{4} x^{2} + \frac{1}{5} \, A a^{3} x^{5} e^{4} + A a^{3} d x^{4} e^{3} + 2 \, A a^{3} d^{2} x^{3} e^{2} + 2 \, A a^{3} d^{3} x^{2} e + A a^{3} d^{4} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/12*B*c^3*x^12*e^4 + 4/11*B*c^3*d*x^11*e^3 + 3/5*B*c^3*d^2*x^10*e^2 + 4/9*B*c^3*d^3*x^9*e + 1/8*B*c^3*d^4*x^8
+ 1/11*A*c^3*x^11*e^4 + 2/5*A*c^3*d*x^10*e^3 + 2/3*A*c^3*d^2*x^9*e^2 + 1/2*A*c^3*d^3*x^8*e + 1/7*A*c^3*d^4*x^
7 + 3/10*B*a*c^2*x^10*e^4 + 4/3*B*a*c^2*d*x^9*e^3 + 9/4*B*a*c^2*d^2*x^8*e^2 + 12/7*B*a*c^2*d^3*x^7*e + 1/2*B*a
*c^2*d^4*x^6 + 1/3*A*a*c^2*x^9*e^4 + 3/2*A*a*c^2*d*x^8*e^3 + 18/7*A*a*c^2*d^2*x^7*e^2 + 2*A*a*c^2*d^3*x^6*e +
3/5*A*a*c^2*d^4*x^5 + 3/8*B*a^2*c*x^8*e^4 + 12/7*B*a^2*c*d*x^7*e^3 + 3*B*a^2*c*d^2*x^6*e^2 + 12/5*B*a^2*c*d^3*
x^5*e + 3/4*B*a^2*c*d^4*x^4 + 3/7*A*a^2*c*x^7*e^4 + 2*A*a^2*c*d*x^6*e^3 + 18/5*A*a^2*c*d^2*x^5*e^2 + 3*A*a^2*c
*d^3*x^4*e + A*a^2*c*d^4*x^3 + 1/6*B*a^3*x^6*e^4 + 4/5*B*a^3*d*x^5*e^3 + 3/2*B*a^3*d^2*x^4*e^2 + 4/3*B*a^3*d^3
*x^3*e + 1/2*B*a^3*d^4*x^2 + 1/5*A*a^3*x^5*e^4 + A*a^3*d*x^4*e^3 + 2*A*a^3*d^2*x^3*e^2 + 2*A*a^3*d^3*x^2*e + A
*a^3*d^4*x