3.12.39 \(\int (A+B x) (d+e x)^5 (a+c x^2)^3 \, dx\)

Optimal. Leaf size=334 \[ -\frac {c (d+e x)^9 \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{9 e^8}+\frac {3 c^2 (d+e x)^{11} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac {c^2 (d+e x)^{10} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{10 e^8}+\frac {(d+e x)^7 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{7 e^8}-\frac {(d+e x)^6 \left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8}-\frac {3 c (d+e x)^8 \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{8 e^8}-\frac {c^3 (d+e x)^{12} (7 B d-A e)}{12 e^8}+\frac {B c^3 (d+e x)^{13}}{13 e^8} \]

________________________________________________________________________________________

Rubi [A]  time = 0.60, antiderivative size = 334, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {c (d+e x)^9 \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 )}{9 e^8}+\frac {3 c^2 (d+e x)^{11} \left (a B e^2-2 A c d e+7 B c d^2\right )}{11 e^8}-\frac {c^2 (d+e x)^{10} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{10 e^8}-\frac {3 c (d+e x)^8 \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 )}{8 e^8}+\frac {(d+e x)^7 \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 )}{7 e^8}-\frac {(d+e x)^6 \left (a e^2+c d^2\right )^3 (B d-A e)}{6 e^8}-\frac {c^3 (d+e x)^{12} (7 B d-A e)}{12 e^8}+\frac {B c^3 (d+e x)^{13}}{13 e^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

-((B*d - A*e)*(c*d^2 + a*e^2)^3*(d + e*x)^6)/(6*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)^7)/(7*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)^8)/(8*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)^9)/(9*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)^10)/(10*e^8) + (3*c^2*(7*B*c*d^2 - 2*A*c*
d*e + a*B*e^2)*(d + e*x)^11)/(11*e^8) - (c^3*(7*B*d - A*e)*(d + e*x)^12)/(12*e^8) + (B*c^3*(d + e*x)^13)/(13*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)^5 \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)^5}{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)^6}{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)^7}{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)^8}{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)^9}{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)^{10}}{e^7}+\frac {c^3 (-7 B d+A e) (d+e x)^{11}}{e^7}+\frac {B c^3 (d+e x)^{12}}{e^7}\right ) \, dx\\ &=-\frac {(B d-A e) \left (c d^2+a e^2\right )^3 (d+e x)^6}{6 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)^7}{7 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)^8}{8 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)^9}{9 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)^{10}}{10 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)^{11}}{11 e^8}-\frac {c^3 (7 B d-A e) (d+e x)^{12}}{12 e^8}+\frac {B c^3 (d+e x)^{13}}{13 e^8}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 542, normalized size = 1.62 \begin {gather*} \frac {1}{2} a^3 d^4 x^2 (5 A e+B d)+a^3 A d^5 x+\frac {1}{9} c e x^9 \left (B \left (3 a^2 e^4+30 a c d^2 e^2+5 c^2 d^4\right )+5 A c d e \left (3 a e^2+2 c d^2\right )\right )+\frac {1}{7} x^7 \left (A c d \left (15 a^2 e^4+30 a c d^2 e^2+c^2 d^4\right )+a B e \left (a^2 e^4+30 a c d^2 e^2+15 c^2 d^4\right )\right )+\frac {1}{5} a d x^5 \left (A \left (5 a^2 e^4+30 a c d^2 e^2+3 c^2 d^4\right )+5 a B d e \left (2 a e^2+3 c d^2\right )\right )+\frac {1}{8} c x^8 \left (A e \left (3 a^2 e^4+30 a c d^2 e^2+5 c^2 d^4\right )+B \left (15 a^2 d e^4+30 a c d^3 e^2+c^2 d^5\right )\right )+\frac {1}{6} a x^6 \left (A e \left (a^2 e^4+30 a c d^2 e^2+15 c^2 d^4\right )+B \left (5 a^2 d e^4+30 a c d^3 e^2+3 c^2 d^5\right )\right )+\frac {1}{3} a^2 d^3 x^3 \left (10 a A e^2+5 a B d e+3 A c d^2\right )+\frac {1}{4} a^2 d^2 x^4 \left (10 a A e^3+10 a B d e^2+15 A c d^2 e+3 B c d^3\right )+\frac {1}{11} c^2 e^3 x^{11} \left (3 a B e^2+5 A c d e+10 B c d^2\right )+\frac {1}{10} c^2 e^2 x^{10} \left (3 a A e^3+15 a B d e^2+10 A c d^2 e+10 B c d^3\right )+\frac {1}{12} c^3 e^4 x^{12} (A e+5 B d)+\frac {1}{13} B c^3 e^5 x^{13} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

a^3*A*d^5*x + (a^3*d^4*(B*d + 5*A*e)*x^2)/2 + (a^2*d^3*(3*A*c*d^2 + 5*a*B*d*e + 10*a*A*e^2)*x^3)/3 + (a^2*d^2*
(3*B*c*d^3 + 15*A*c*d^2*e + 10*a*B*d*e^2 + 10*a*A*e^3)*x^4)/4 + (a*d*(5*a*B*d*e*(3*c*d^2 + 2*a*e^2) + A*(3*c^2
*d^4 + 30*a*c*d^2*e^2 + 5*a^2*e^4))*x^5)/5 + (a*(A*e*(15*c^2*d^4 + 30*a*c*d^2*e^2 + a^2*e^4) + B*(3*c^2*d^5 +
30*a*c*d^3*e^2 + 5*a^2*d*e^4))*x^6)/6 + ((a*B*e*(15*c^2*d^4 + 30*a*c*d^2*e^2 + a^2*e^4) + A*c*d*(c^2*d^4 + 30*
a*c*d^2*e^2 + 15*a^2*e^4))*x^7)/7 + (c*(A*e*(5*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4) + B*(c^2*d^5 + 30*a*c*d^3
*e^2 + 15*a^2*d*e^4))*x^8)/8 + (c*e*(5*A*c*d*e*(2*c*d^2 + 3*a*e^2) + B*(5*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4
))*x^9)/9 + (c^2*e^2*(10*B*c*d^3 + 10*A*c*d^2*e + 15*a*B*d*e^2 + 3*a*A*e^3)*x^10)/10 + (c^2*e^3*(10*B*c*d^2 +
5*A*c*d*e + 3*a*B*e^2)*x^11)/11 + (c^3*e^4*(5*B*d + A*e)*x^12)/12 + (B*c^3*e^5*x^13)/13

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (A+B x) (d+e x)^5 \left (a+c x^2\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^5*(a + c*x^2)^3, x]

________________________________________________________________________________________

fricas [B]  time = 0.34, size = 658, normalized size = 1.97 \begin {gather*} \frac {1}{13} x^{13} e^{5} c^{3} B + \frac {5}{12} x^{12} e^{4} d c^{3} B + \frac {1}{12} x^{12} e^{5} c^{3} A + \frac {10}{11} x^{11} e^{3} d^{2} c^{3} B + \frac {3}{11} x^{11} e^{5} c^{2} a B + \frac {5}{11} x^{11} e^{4} d c^{3} A + x^{10} e^{2} d^{3} c^{3} B + \frac {3}{2} x^{10} e^{4} d c^{2} a B + x^{10} e^{3} d^{2} c^{3} A + \frac {3}{10} x^{10} e^{5} c^{2} a A + \frac {5}{9} x^{9} e d^{4} c^{3} B + \frac {10}{3} x^{9} e^{3} d^{2} c^{2} a B + \frac {1}{3} x^{9} e^{5} c a^{2} B + \frac {10}{9} x^{9} e^{2} d^{3} c^{3} A + \frac {5}{3} x^{9} e^{4} d c^{2} a A + \frac {1}{8} x^{8} d^{5} c^{3} B + \frac {15}{4} x^{8} e^{2} d^{3} c^{2} a B + \frac {15}{8} x^{8} e^{4} d c a^{2} B + \frac {5}{8} x^{8} e d^{4} c^{3} A + \frac {15}{4} x^{8} e^{3} d^{2} c^{2} a A + \frac {3}{8} x^{8} e^{5} c a^{2} A + \frac {15}{7} x^{7} e d^{4} c^{2} a B + \frac {30}{7} x^{7} e^{3} d^{2} c a^{2} B + \frac {1}{7} x^{7} e^{5} a^{3} B + \frac {1}{7} x^{7} d^{5} c^{3} A + \frac {30}{7} x^{7} e^{2} d^{3} c^{2} a A + \frac {15}{7} x^{7} e^{4} d c a^{2} A + \frac {1}{2} x^{6} d^{5} c^{2} a B + 5 x^{6} e^{2} d^{3} c a^{2} B + \frac {5}{6} x^{6} e^{4} d a^{3} B + \frac {5}{2} x^{6} e d^{4} c^{2} a A + 5 x^{6} e^{3} d^{2} c a^{2} A + \frac {1}{6} x^{6} e^{5} a^{3} A + 3 x^{5} e d^{4} c a^{2} B + 2 x^{5} e^{3} d^{2} a^{3} B + \frac {3}{5} x^{5} d^{5} c^{2} a A + 6 x^{5} e^{2} d^{3} c a^{2} A + x^{5} e^{4} d a^{3} A + \frac {3}{4} x^{4} d^{5} c a^{2} B + \frac {5}{2} x^{4} e^{2} d^{3} a^{3} B + \frac {15}{4} x^{4} e d^{4} c a^{2} A + \frac {5}{2} x^{4} e^{3} d^{2} a^{3} A + \frac {5}{3} x^{3} e d^{4} a^{3} B + x^{3} d^{5} c a^{2} A + \frac {10}{3} x^{3} e^{2} d^{3} a^{3} A + \frac {1}{2} x^{2} d^{5} a^{3} B + \frac {5}{2} x^{2} e d^{4} a^{3} A + x d^{5} a^{3} A \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/13*x^13*e^5*c^3*B + 5/12*x^12*e^4*d*c^3*B + 1/12*x^12*e^5*c^3*A + 10/11*x^11*e^3*d^2*c^3*B + 3/11*x^11*e^5*c
^2*a*B + 5/11*x^11*e^4*d*c^3*A + x^10*e^2*d^3*c^3*B + 3/2*x^10*e^4*d*c^2*a*B + x^10*e^3*d^2*c^3*A + 3/10*x^10*
e^5*c^2*a*A + 5/9*x^9*e*d^4*c^3*B + 10/3*x^9*e^3*d^2*c^2*a*B + 1/3*x^9*e^5*c*a^2*B + 10/9*x^9*e^2*d^3*c^3*A +
5/3*x^9*e^4*d*c^2*a*A + 1/8*x^8*d^5*c^3*B + 15/4*x^8*e^2*d^3*c^2*a*B + 15/8*x^8*e^4*d*c*a^2*B + 5/8*x^8*e*d^4*
c^3*A + 15/4*x^8*e^3*d^2*c^2*a*A + 3/8*x^8*e^5*c*a^2*A + 15/7*x^7*e*d^4*c^2*a*B + 30/7*x^7*e^3*d^2*c*a^2*B + 1
/7*x^7*e^5*a^3*B + 1/7*x^7*d^5*c^3*A + 30/7*x^7*e^2*d^3*c^2*a*A + 15/7*x^7*e^4*d*c*a^2*A + 1/2*x^6*d^5*c^2*a*B
+ 5*x^6*e^2*d^3*c*a^2*B + 5/6*x^6*e^4*d*a^3*B + 5/2*x^6*e*d^4*c^2*a*A + 5*x^6*e^3*d^2*c*a^2*A + 1/6*x^6*e^5*a
^3*A + 3*x^5*e*d^4*c*a^2*B + 2*x^5*e^3*d^2*a^3*B + 3/5*x^5*d^5*c^2*a*A + 6*x^5*e^2*d^3*c*a^2*A + x^5*e^4*d*a^3
*A + 3/4*x^4*d^5*c*a^2*B + 5/2*x^4*e^2*d^3*a^3*B + 15/4*x^4*e*d^4*c*a^2*A + 5/2*x^4*e^3*d^2*a^3*A + 5/3*x^3*e*
d^4*a^3*B + x^3*d^5*c*a^2*A + 10/3*x^3*e^2*d^3*a^3*A + 1/2*x^2*d^5*a^3*B + 5/2*x^2*e*d^4*a^3*A + x*d^5*a^3*A

________________________________________________________________________________________

giac [A]  time = 0.19, size = 634, normalized size = 1.90 \begin {gather*} \frac {1}{13} \, B c^{3} x^{13} e^{5} + \frac {5}{12} \, B c^{3} d x^{12} e^{4} + \frac {10}{11} \, B c^{3} d^{2} x^{11} e^{3} + B c^{3} d^{3} x^{10} e^{2} + \frac {5}{9} \, B c^{3} d^{4} x^{9} e + \frac {1}{8} \, B c^{3} d^{5} x^{8} + \frac {1}{12} \, A c^{3} x^{12} e^{5} + \frac {5}{11} \, A c^{3} d x^{11} e^{4} + A c^{3} d^{2} x^{10} e^{3} + \frac {10}{9} \, A c^{3} d^{3} x^{9} e^{2} + \frac {5}{8} \, A c^{3} d^{4} x^{8} e + \frac {1}{7} \, A c^{3} d^{5} x^{7} + \frac {3}{11} \, B a c^{2} x^{11} e^{5} + \frac {3}{2} \, B a c^{2} d x^{10} e^{4} + \frac {10}{3} \, B a c^{2} d^{2} x^{9} e^{3} + \frac {15}{4} \, B a c^{2} d^{3} x^{8} e^{2} + \frac {15}{7} \, B a c^{2} d^{4} x^{7} e + \frac {1}{2} \, B a c^{2} d^{5} x^{6} + \frac {3}{10} \, A a c^{2} x^{10} e^{5} + \frac {5}{3} \, A a c^{2} d x^{9} e^{4} + \frac {15}{4} \, A a c^{2} d^{2} x^{8} e^{3} + \frac {30}{7} \, A a c^{2} d^{3} x^{7} e^{2} + \frac {5}{2} \, A a c^{2} d^{4} x^{6} e + \frac {3}{5} \, A a c^{2} d^{5} x^{5} + \frac {1}{3} \, B a^{2} c x^{9} e^{5} + \frac {15}{8} \, B a^{2} c d x^{8} e^{4} + \frac {30}{7} \, B a^{2} c d^{2} x^{7} e^{3} + 5 \, B a^{2} c d^{3} x^{6} e^{2} + 3 \, B a^{2} c d^{4} x^{5} e + \frac {3}{4} \, B a^{2} c d^{5} x^{4} + \frac {3}{8} \, A a^{2} c x^{8} e^{5} + \frac {15}{7} \, A a^{2} c d x^{7} e^{4} + 5 \, A a^{2} c d^{2} x^{6} e^{3} + 6 \, A a^{2} c d^{3} x^{5} e^{2} + \frac {15}{4} \, A a^{2} c d^{4} x^{4} e + A a^{2} c d^{5} x^{3} + \frac {1}{7} \, B a^{3} x^{7} e^{5} + \frac {5}{6} \, B a^{3} d x^{6} e^{4} + 2 \, B a^{3} d^{2} x^{5} e^{3} + \frac {5}{2} \, B a^{3} d^{3} x^{4} e^{2} + \frac {5}{3} \, B a^{3} d^{4} x^{3} e + \frac {1}{2} \, B a^{3} d^{5} x^{2} + \frac {1}{6} \, A a^{3} x^{6} e^{5} + A a^{3} d x^{5} e^{4} + \frac {5}{2} \, A a^{3} d^{2} x^{4} e^{3} + \frac {10}{3} \, A a^{3} d^{3} x^{3} e^{2} + \frac {5}{2} \, A a^{3} d^{4} x^{2} e + A a^{3} d^{5} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/13*B*c^3*x^13*e^5 + 5/12*B*c^3*d*x^12*e^4 + 10/11*B*c^3*d^2*x^11*e^3 + B*c^3*d^3*x^10*e^2 + 5/9*B*c^3*d^4*x^
9*e + 1/8*B*c^3*d^5*x^8 + 1/12*A*c^3*x^12*e^5 + 5/11*A*c^3*d*x^11*e^4 + A*c^3*d^2*x^10*e^3 + 10/9*A*c^3*d^3*x^
9*e^2 + 5/8*A*c^3*d^4*x^8*e + 1/7*A*c^3*d^5*x^7 + 3/11*B*a*c^2*x^11*e^5 + 3/2*B*a*c^2*d*x^10*e^4 + 10/3*B*a*c^
2*d^2*x^9*e^3 + 15/4*B*a*c^2*d^3*x^8*e^2 + 15/7*B*a*c^2*d^4*x^7*e + 1/2*B*a*c^2*d^5*x^6 + 3/10*A*a*c^2*x^10*e^
5 + 5/3*A*a*c^2*d*x^9*e^4 + 15/4*A*a*c^2*d^2*x^8*e^3 + 30/7*A*a*c^2*d^3*x^7*e^2 + 5/2*A*a*c^2*d^4*x^6*e + 3/5*
A*a*c^2*d^5*x^5 + 1/3*B*a^2*c*x^9*e^5 + 15/8*B*a^2*c*d*x^8*e^4 + 30/7*B*a^2*c*d^2*x^7*e^3 + 5*B*a^2*c*d^3*x^6*
e^2 + 3*B*a^2*c*d^4*x^5*e + 3/4*B*a^2*c*d^5*x^4 + 3/8*A*a^2*c*x^8*e^5 + 15/7*A*a^2*c*d*x^7*e^4 + 5*A*a^2*c*d^2
*x^6*e^3 + 6*A*a^2*c*d^3*x^5*e^2 + 15/4*A*a^2*c*d^4*x^4*e + A*a^2*c*d^5*x^3 + 1/7*B*a^3*x^7*e^5 + 5/6*B*a^3*d*
x^6*e^4 + 2*B*a^3*d^2*x^5*e^3 + 5/2*B*a^3*d^3*x^4*e^2 + 5/3*B*a^3*d^4*x^3*e + 1/2*B*a^3*d^5*x^2 + 1/6*A*a^3*x^
6*e^5 + A*a^3*d*x^5*e^4 + 5/2*A*a^3*d^2*x^4*e^3 + 10/3*A*a^3*d^3*x^3*e^2 + 5/2*A*a^3*d^4*x^2*e + A*a^3*d^5*x

________________________________________________________________________________________

maple [A]  time = 0.05, size = 557, normalized size = 1.67 \begin {gather*} \frac {B \,c^{3} e^{5} x^{13}}{13}+\frac {\left (A \,e^{5}+5 B d \,e^{4}\right ) c^{3} x^{12}}{12}+\frac {\left (3 B a \,c^{2} e^{5}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) c^{3}\right ) x^{11}}{11}+A \,a^{3} d^{5} x +\frac {\left (3 \left (A \,e^{5}+5 B d \,e^{4}\right ) a \,c^{2}+\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) c^{3}\right ) x^{10}}{10}+\frac {\left (3 B \,a^{2} c \,e^{5}+3 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a \,c^{2}+\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) c^{3}\right ) x^{9}}{9}+\frac {\left (3 \left (A \,e^{5}+5 B d \,e^{4}\right ) a^{2} c +3 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a \,c^{2}+\left (5 A \,d^{4} e +B \,d^{5}\right ) c^{3}\right ) x^{8}}{8}+\frac {\left (A \,c^{3} d^{5}+B \,a^{3} e^{5}+3 \left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{2} c +3 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a \,c^{2}\right ) x^{7}}{7}+\frac {\left (\left (A \,e^{5}+5 B d \,e^{4}\right ) a^{3}+3 \left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{2} c +3 \left (5 A \,d^{4} e +B \,d^{5}\right ) a \,c^{2}\right ) x^{6}}{6}+\frac {\left (5 A \,d^{4} e +B \,d^{5}\right ) a^{3} x^{2}}{2}+\frac {\left (3 A a \,c^{2} d^{5}+\left (5 A d \,e^{4}+10 B \,d^{2} e^{3}\right ) a^{3}+3 \left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{2} c \right ) x^{5}}{5}+\frac {\left (\left (10 A \,d^{2} e^{3}+10 B \,d^{3} e^{2}\right ) a^{3}+3 \left (5 A \,d^{4} e +B \,d^{5}\right ) a^{2} c \right ) x^{4}}{4}+\frac {\left (3 A \,a^{2} c \,d^{5}+\left (10 A \,d^{3} e^{2}+5 B \,d^{4} e \right ) a^{3}\right ) x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x)

[Out]

1/13*B*e^5*c^3*x^13+1/12*(A*e^5+5*B*d*e^4)*c^3*x^12+1/11*((5*A*d*e^4+10*B*d^2*e^3)*c^3+3*B*e^5*a*c^2)*x^11+1/1
0*((10*A*d^2*e^3+10*B*d^3*e^2)*c^3+3*(A*e^5+5*B*d*e^4)*a*c^2)*x^10+1/9*((10*A*d^3*e^2+5*B*d^4*e)*c^3+3*(5*A*d*
e^4+10*B*d^2*e^3)*a*c^2+3*B*e^5*a^2*c)*x^9+1/8*((5*A*d^4*e+B*d^5)*c^3+3*(10*A*d^2*e^3+10*B*d^3*e^2)*a*c^2+3*(A
*e^5+5*B*d*e^4)*a^2*c)*x^8+1/7*(A*d^5*c^3+3*(10*A*d^3*e^2+5*B*d^4*e)*a*c^2+3*(5*A*d*e^4+10*B*d^2*e^3)*a^2*c+B*
e^5*a^3)*x^7+1/6*(3*(5*A*d^4*e+B*d^5)*a*c^2+3*(10*A*d^2*e^3+10*B*d^3*e^2)*a^2*c+(A*e^5+5*B*d*e^4)*a^3)*x^6+1/5
*(3*A*d^5*a*c^2+3*(10*A*d^3*e^2+5*B*d^4*e)*a^2*c+(5*A*d*e^4+10*B*d^2*e^3)*a^3)*x^5+1/4*(3*(5*A*d^4*e+B*d^5)*a^
2*c+(10*A*d^2*e^3+10*B*d^3*e^2)*a^3)*x^4+1/3*(3*A*d^5*a^2*c+(10*A*d^3*e^2+5*B*d^4*e)*a^3)*x^3+1/2*(5*A*d^4*e+B
*d^5)*a^3*x^2+A*d^5*a^3*x

________________________________________________________________________________________

maxima [A]  time = 0.66, size = 584, normalized size = 1.75 \begin {gather*} \frac {1}{13} \, B c^{3} e^{5} x^{13} + \frac {1}{12} \, {\left (5 \, B c^{3} d e^{4} + A c^{3} e^{5}\right )} x^{12} + \frac {1}{11} \, {\left (10 \, B c^{3} d^{2} e^{3} + 5 \, A c^{3} d e^{4} + 3 \, B a c^{2} e^{5}\right )} x^{11} + \frac {1}{10} \, {\left (10 \, B c^{3} d^{3} e^{2} + 10 \, A c^{3} d^{2} e^{3} + 15 \, B a c^{2} d e^{4} + 3 \, A a c^{2} e^{5}\right )} x^{10} + A a^{3} d^{5} x + \frac {1}{9} \, {\left (5 \, B c^{3} d^{4} e + 10 \, A c^{3} d^{3} e^{2} + 30 \, B a c^{2} d^{2} e^{3} + 15 \, A a c^{2} d e^{4} + 3 \, B a^{2} c e^{5}\right )} x^{9} + \frac {1}{8} \, {\left (B c^{3} d^{5} + 5 \, A c^{3} d^{4} e + 30 \, B a c^{2} d^{3} e^{2} + 30 \, A a c^{2} d^{2} e^{3} + 15 \, B a^{2} c d e^{4} + 3 \, A a^{2} c e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (A c^{3} d^{5} + 15 \, B a c^{2} d^{4} e + 30 \, A a c^{2} d^{3} e^{2} + 30 \, B a^{2} c d^{2} e^{3} + 15 \, A a^{2} c d e^{4} + B a^{3} e^{5}\right )} x^{7} + \frac {1}{6} \, {\left (3 \, B a c^{2} d^{5} + 15 \, A a c^{2} d^{4} e + 30 \, B a^{2} c d^{3} e^{2} + 30 \, A a^{2} c d^{2} e^{3} + 5 \, B a^{3} d e^{4} + A a^{3} e^{5}\right )} x^{6} + \frac {1}{5} \, {\left (3 \, A a c^{2} d^{5} + 15 \, B a^{2} c d^{4} e + 30 \, A a^{2} c d^{3} e^{2} + 10 \, B a^{3} d^{2} e^{3} + 5 \, A a^{3} d e^{4}\right )} x^{5} + \frac {1}{4} \, {\left (3 \, B a^{2} c d^{5} + 15 \, A a^{2} c d^{4} e + 10 \, B a^{3} d^{3} e^{2} + 10 \, A a^{3} d^{2} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (3 \, A a^{2} c d^{5} + 5 \, B a^{3} d^{4} e + 10 \, A a^{3} d^{3} e^{2}\right )} x^{3} + \frac {1}{2} \, {\left (B a^{3} d^{5} + 5 \, A a^{3} d^{4} e\right )} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/13*B*c^3*e^5*x^13 + 1/12*(5*B*c^3*d*e^4 + A*c^3*e^5)*x^12 + 1/11*(10*B*c^3*d^2*e^3 + 5*A*c^3*d*e^4 + 3*B*a*c
^2*e^5)*x^11 + 1/10*(10*B*c^3*d^3*e^2 + 10*A*c^3*d^2*e^3 + 15*B*a*c^2*d*e^4 + 3*A*a*c^2*e^5)*x^10 + A*a^3*d^5*
x + 1/9*(5*B*c^3*d^4*e + 10*A*c^3*d^3*e^2 + 30*B*a*c^2*d^2*e^3 + 15*A*a*c^2*d*e^4 + 3*B*a^2*c*e^5)*x^9 + 1/8*(
B*c^3*d^5 + 5*A*c^3*d^4*e + 30*B*a*c^2*d^3*e^2 + 30*A*a*c^2*d^2*e^3 + 15*B*a^2*c*d*e^4 + 3*A*a^2*c*e^5)*x^8 +
1/7*(A*c^3*d^5 + 15*B*a*c^2*d^4*e + 30*A*a*c^2*d^3*e^2 + 30*B*a^2*c*d^2*e^3 + 15*A*a^2*c*d*e^4 + B*a^3*e^5)*x^
7 + 1/6*(3*B*a*c^2*d^5 + 15*A*a*c^2*d^4*e + 30*B*a^2*c*d^3*e^2 + 30*A*a^2*c*d^2*e^3 + 5*B*a^3*d*e^4 + A*a^3*e^
5)*x^6 + 1/5*(3*A*a*c^2*d^5 + 15*B*a^2*c*d^4*e + 30*A*a^2*c*d^3*e^2 + 10*B*a^3*d^2*e^3 + 5*A*a^3*d*e^4)*x^5 +
1/4*(3*B*a^2*c*d^5 + 15*A*a^2*c*d^4*e + 10*B*a^3*d^3*e^2 + 10*A*a^3*d^2*e^3)*x^4 + 1/3*(3*A*a^2*c*d^5 + 5*B*a^
3*d^4*e + 10*A*a^3*d^3*e^2)*x^3 + 1/2*(B*a^3*d^5 + 5*A*a^3*d^4*e)*x^2

________________________________________________________________________________________

mupad [B]  time = 1.86, size = 542, normalized size = 1.62 \begin {gather*} x^6\,\left (\frac {5\,B\,a^3\,d\,e^4}{6}+\frac {A\,a^3\,e^5}{6}+5\,B\,a^2\,c\,d^3\,e^2+5\,A\,a^2\,c\,d^2\,e^3+\frac {B\,a\,c^2\,d^5}{2}+\frac {5\,A\,a\,c^2\,d^4\,e}{2}\right )+x^7\,\left (\frac {B\,a^3\,e^5}{7}+\frac {30\,B\,a^2\,c\,d^2\,e^3}{7}+\frac {15\,A\,a^2\,c\,d\,e^4}{7}+\frac {15\,B\,a\,c^2\,d^4\,e}{7}+\frac {30\,A\,a\,c^2\,d^3\,e^2}{7}+\frac {A\,c^3\,d^5}{7}\right )+x^8\,\left (\frac {15\,B\,a^2\,c\,d\,e^4}{8}+\frac {3\,A\,a^2\,c\,e^5}{8}+\frac {15\,B\,a\,c^2\,d^3\,e^2}{4}+\frac {15\,A\,a\,c^2\,d^2\,e^3}{4}+\frac {B\,c^3\,d^5}{8}+\frac {5\,A\,c^3\,d^4\,e}{8}\right )+x^5\,\left (2\,B\,a^3\,d^2\,e^3+A\,a^3\,d\,e^4+3\,B\,a^2\,c\,d^4\,e+6\,A\,a^2\,c\,d^3\,e^2+\frac {3\,A\,a\,c^2\,d^5}{5}\right )+x^9\,\left (\frac {B\,a^2\,c\,e^5}{3}+\frac {10\,B\,a\,c^2\,d^2\,e^3}{3}+\frac {5\,A\,a\,c^2\,d\,e^4}{3}+\frac {5\,B\,c^3\,d^4\,e}{9}+\frac {10\,A\,c^3\,d^3\,e^2}{9}\right )+\frac {c^2\,e^2\,x^{10}\,\left (10\,B\,c\,d^3+10\,A\,c\,d^2\,e+15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{10}+\frac {a^3\,d^4\,x^2\,\left (5\,A\,e+B\,d\right )}{2}+\frac {c^3\,e^4\,x^{12}\,\left (A\,e+5\,B\,d\right )}{12}+\frac {a^2\,d^3\,x^3\,\left (3\,A\,c\,d^2+5\,B\,a\,d\,e+10\,A\,a\,e^2\right )}{3}+\frac {c^2\,e^3\,x^{11}\,\left (10\,B\,c\,d^2+5\,A\,c\,d\,e+3\,B\,a\,e^2\right )}{11}+A\,a^3\,d^5\,x+\frac {a^2\,d^2\,x^4\,\left (3\,B\,c\,d^3+15\,A\,c\,d^2\,e+10\,B\,a\,d\,e^2+10\,A\,a\,e^3\right )}{4}+\frac {B\,c^3\,e^5\,x^{13}}{13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(A + B*x)*(d + e*x)^5,x)

[Out]

x^6*((A*a^3*e^5)/6 + (B*a*c^2*d^5)/2 + (5*B*a^3*d*e^4)/6 + 5*A*a^2*c*d^2*e^3 + 5*B*a^2*c*d^3*e^2 + (5*A*a*c^2*
d^4*e)/2) + x^7*((A*c^3*d^5)/7 + (B*a^3*e^5)/7 + (30*A*a*c^2*d^3*e^2)/7 + (30*B*a^2*c*d^2*e^3)/7 + (15*A*a^2*c
*d*e^4)/7 + (15*B*a*c^2*d^4*e)/7) + x^8*((B*c^3*d^5)/8 + (3*A*a^2*c*e^5)/8 + (5*A*c^3*d^4*e)/8 + (15*A*a*c^2*d
^2*e^3)/4 + (15*B*a*c^2*d^3*e^2)/4 + (15*B*a^2*c*d*e^4)/8) + x^5*((3*A*a*c^2*d^5)/5 + A*a^3*d*e^4 + 2*B*a^3*d^
2*e^3 + 6*A*a^2*c*d^3*e^2 + 3*B*a^2*c*d^4*e) + x^9*((B*a^2*c*e^5)/3 + (5*B*c^3*d^4*e)/9 + (10*A*c^3*d^3*e^2)/9
+ (10*B*a*c^2*d^2*e^3)/3 + (5*A*a*c^2*d*e^4)/3) + (c^2*e^2*x^10*(3*A*a*e^3 + 10*B*c*d^3 + 15*B*a*d*e^2 + 10*A
*c*d^2*e))/10 + (a^3*d^4*x^2*(5*A*e + B*d))/2 + (c^3*e^4*x^12*(A*e + 5*B*d))/12 + (a^2*d^3*x^3*(10*A*a*e^2 + 3
*A*c*d^2 + 5*B*a*d*e))/3 + (c^2*e^3*x^11*(3*B*a*e^2 + 10*B*c*d^2 + 5*A*c*d*e))/11 + A*a^3*d^5*x + (a^2*d^2*x^4
*(10*A*a*e^3 + 3*B*c*d^3 + 10*B*a*d*e^2 + 15*A*c*d^2*e))/4 + (B*c^3*e^5*x^13)/13

________________________________________________________________________________________

sympy [B]  time = 0.16, size = 694, normalized size = 2.08 \begin {gather*} A a^{3} d^{5} x + \frac {B c^{3} e^{5} x^{13}}{13} + x^{12} \left (\frac {A c^{3} e^{5}}{12} + \frac {5 B c^{3} d e^{4}}{12}\right ) + x^{11} \left (\frac {5 A c^{3} d e^{4}}{11} + \frac {3 B a c^{2} e^{5}}{11} + \frac {10 B c^{3} d^{2} e^{3}}{11}\right ) + x^{10} \left (\frac {3 A a c^{2} e^{5}}{10} + A c^{3} d^{2} e^{3} + \frac {3 B a c^{2} d e^{4}}{2} + B c^{3} d^{3} e^{2}\right ) + x^{9} \left (\frac {5 A a c^{2} d e^{4}}{3} + \frac {10 A c^{3} d^{3} e^{2}}{9} + \frac {B a^{2} c e^{5}}{3} + \frac {10 B a c^{2} d^{2} e^{3}}{3} + \frac {5 B c^{3} d^{4} e}{9}\right ) + x^{8} \left (\frac {3 A a^{2} c e^{5}}{8} + \frac {15 A a c^{2} d^{2} e^{3}}{4} + \frac {5 A c^{3} d^{4} e}{8} + \frac {15 B a^{2} c d e^{4}}{8} + \frac {15 B a c^{2} d^{3} e^{2}}{4} + \frac {B c^{3} d^{5}}{8}\right ) + x^{7} \left (\frac {15 A a^{2} c d e^{4}}{7} + \frac {30 A a c^{2} d^{3} e^{2}}{7} + \frac {A c^{3} d^{5}}{7} + \frac {B a^{3} e^{5}}{7} + \frac {30 B a^{2} c d^{2} e^{3}}{7} + \frac {15 B a c^{2} d^{4} e}{7}\right ) + x^{6} \left (\frac {A a^{3} e^{5}}{6} + 5 A a^{2} c d^{2} e^{3} + \frac {5 A a c^{2} d^{4} e}{2} + \frac {5 B a^{3} d e^{4}}{6} + 5 B a^{2} c d^{3} e^{2} + \frac {B a c^{2} d^{5}}{2}\right ) + x^{5} \left (A a^{3} d e^{4} + 6 A a^{2} c d^{3} e^{2} + \frac {3 A a c^{2} d^{5}}{5} + 2 B a^{3} d^{2} e^{3} + 3 B a^{2} c d^{4} e\right ) + x^{4} \left (\frac {5 A a^{3} d^{2} e^{3}}{2} + \frac {15 A a^{2} c d^{4} e}{4} + \frac {5 B a^{3} d^{3} e^{2}}{2} + \frac {3 B a^{2} c d^{5}}{4}\right ) + x^{3} \left (\frac {10 A a^{3} d^{3} e^{2}}{3} + A a^{2} c d^{5} + \frac {5 B a^{3} d^{4} e}{3}\right ) + x^{2} \left (\frac {5 A a^{3} d^{4} e}{2} + \frac {B a^{3} d^{5}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**5*(c*x**2+a)**3,x)

[Out]

A*a**3*d**5*x + B*c**3*e**5*x**13/13 + x**12*(A*c**3*e**5/12 + 5*B*c**3*d*e**4/12) + x**11*(5*A*c**3*d*e**4/11
+ 3*B*a*c**2*e**5/11 + 10*B*c**3*d**2*e**3/11) + x**10*(3*A*a*c**2*e**5/10 + A*c**3*d**2*e**3 + 3*B*a*c**2*d*
e**4/2 + B*c**3*d**3*e**2) + x**9*(5*A*a*c**2*d*e**4/3 + 10*A*c**3*d**3*e**2/9 + B*a**2*c*e**5/3 + 10*B*a*c**2
*d**2*e**3/3 + 5*B*c**3*d**4*e/9) + x**8*(3*A*a**2*c*e**5/8 + 15*A*a*c**2*d**2*e**3/4 + 5*A*c**3*d**4*e/8 + 15
*B*a**2*c*d*e**4/8 + 15*B*a*c**2*d**3*e**2/4 + B*c**3*d**5/8) + x**7*(15*A*a**2*c*d*e**4/7 + 30*A*a*c**2*d**3*
e**2/7 + A*c**3*d**5/7 + B*a**3*e**5/7 + 30*B*a**2*c*d**2*e**3/7 + 15*B*a*c**2*d**4*e/7) + x**6*(A*a**3*e**5/6
+ 5*A*a**2*c*d**2*e**3 + 5*A*a*c**2*d**4*e/2 + 5*B*a**3*d*e**4/6 + 5*B*a**2*c*d**3*e**2 + B*a*c**2*d**5/2) +
x**5*(A*a**3*d*e**4 + 6*A*a**2*c*d**3*e**2 + 3*A*a*c**2*d**5/5 + 2*B*a**3*d**2*e**3 + 3*B*a**2*c*d**4*e) + x**
4*(5*A*a**3*d**2*e**3/2 + 15*A*a**2*c*d**4*e/4 + 5*B*a**3*d**3*e**2/2 + 3*B*a**2*c*d**5/4) + x**3*(10*A*a**3*d
**3*e**2/3 + A*a**2*c*d**5 + 5*B*a**3*d**4*e/3) + x**2*(5*A*a**3*d**4*e/2 + B*a**3*d**5/2)

________________________________________________________________________________________