3.1317 \(\int (A+B x) (d+e x) \left (a+c x^2\right )^3 \, dx\)

Optimal. Leaf size=148 \[ \frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{3}{4} a^2 c x^4 (A e+B d)+\frac{1}{3} a^2 x^3 (a B e+3 A c d)+\frac{1}{7} c^2 x^7 (3 a B e+A c d)+\frac{1}{2} a c^2 x^6 (A e+B d)+\frac{3}{5} a c x^5 (a B e+A c d)+\frac{1}{8} c^3 x^8 (A e+B d)+\frac{1}{9} B c^3 e x^9 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.421518, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{1}{2} a^3 x^2 (A e+B d)+a^3 A d x+\frac{3}{4} a^2 c x^4 (A e+B d)+\frac{1}{3} a^2 x^3 (a B e+3 A c d)+\frac{1}{7} c^2 x^7 (3 a B e+A c d)+\frac{1}{2} a c^2 x^6 (A e+B d)+\frac{3}{5} a c x^5 (a B e+A c d)+\frac{1}{8} c^3 x^8 (A e+B d)+\frac{1}{9} B c^3 e x^9 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{B c^{3} e x^{9}}{9} + a^{3} d \int A\, dx + a^{3} \left (A e + B d\right ) \int x\, dx + \frac{3 a^{2} c x^{4} \left (A e + B d\right )}{4} + \frac{a^{2} x^{3} \left (3 A c d + B a e\right )}{3} + \frac{a c^{2} x^{6} \left (A e + B d\right )}{2} + \frac{3 a c x^{5} \left (A c d + B a e\right )}{5} + \frac{c^{3} x^{8} \left (A e + B d\right )}{8} + \frac{c^{2} x^{7} \left (A c d + 3 B a e\right )}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)*(c*x**2+a)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0807071, size = 135, normalized size = 0.91 \[ \frac{1}{6} a^3 x (3 A (2 d+e x)+B x (3 d+2 e x))+\frac{1}{20} a^2 c x^3 (5 A (4 d+3 e x)+3 B x (5 d+4 e x))+\frac{1}{70} a c^2 x^5 (7 A (6 d+5 e x)+5 B x (7 d+6 e x))+\frac{1}{504} c^3 x^7 (9 A (8 d+7 e x)+7 B x (9 d+8 e x)) \]

Antiderivative was successfully verified.

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

[Out]

(a^3*x*(3*A*(2*d + e*x) + B*x*(3*d + 2*e*x)))/6 + (a^2*c*x^3*(5*A*(4*d + 3*e*x)
+ 3*B*x*(5*d + 4*e*x)))/20 + (a*c^2*x^5*(7*A*(6*d + 5*e*x) + 5*B*x*(7*d + 6*e*x)
))/70 + (c^3*x^7*(9*A*(8*d + 7*e*x) + 7*B*x*(9*d + 8*e*x)))/504

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 143, normalized size = 1. \[{\frac{B{c}^{3}e{x}^{9}}{9}}+{\frac{{c}^{3} \left ( Ae+Bd \right ){x}^{8}}{8}}+{\frac{ \left ( Ad{c}^{3}+3\,Bea{c}^{2} \right ){x}^{7}}{7}}+{\frac{a{c}^{2} \left ( Ae+Bd \right ){x}^{6}}{2}}+{\frac{ \left ( 3\,Ada{c}^{2}+3\,Be{a}^{2}c \right ){x}^{5}}{5}}+{\frac{3\,{a}^{2}c \left ( Ae+Bd \right ){x}^{4}}{4}}+{\frac{ \left ( 3\,Ad{a}^{2}c+Be{a}^{3} \right ){x}^{3}}{3}}+{\frac{{a}^{3} \left ( Ae+Bd \right ){x}^{2}}{2}}+{a}^{3}Adx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.722457, size = 208, normalized size = 1.41 \[ \frac{1}{9} \, B c^{3} e x^{9} + \frac{1}{8} \,{\left (B c^{3} d + A c^{3} e\right )} x^{8} + \frac{1}{7} \,{\left (A c^{3} d + 3 \, B a c^{2} e\right )} x^{7} + \frac{1}{2} \,{\left (B a c^{2} d + A a c^{2} e\right )} x^{6} + A a^{3} d x + \frac{3}{5} \,{\left (A a c^{2} d + B a^{2} c e\right )} x^{5} + \frac{3}{4} \,{\left (B a^{2} c d + A a^{2} c e\right )} x^{4} + \frac{1}{3} \,{\left (3 \, A a^{2} c d + B a^{3} e\right )} x^{3} + \frac{1}{2} \,{\left (B a^{3} d + A a^{3} e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.251761, size = 1, normalized size = 0.01 \[ \frac{1}{9} x^{9} e c^{3} B + \frac{1}{8} x^{8} d c^{3} B + \frac{1}{8} x^{8} e c^{3} A + \frac{3}{7} x^{7} e c^{2} a B + \frac{1}{7} x^{7} d c^{3} A + \frac{1}{2} x^{6} d c^{2} a B + \frac{1}{2} x^{6} e c^{2} a A + \frac{3}{5} x^{5} e c a^{2} B + \frac{3}{5} x^{5} d c^{2} a A + \frac{3}{4} x^{4} d c a^{2} B + \frac{3}{4} x^{4} e c a^{2} A + \frac{1}{3} x^{3} e a^{3} B + x^{3} d c a^{2} A + \frac{1}{2} x^{2} d a^{3} B + \frac{1}{2} x^{2} e a^{3} A + x d a^{3} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.191956, size = 182, normalized size = 1.23 \[ A a^{3} d x + \frac{B c^{3} e x^{9}}{9} + x^{8} \left (\frac{A c^{3} e}{8} + \frac{B c^{3} d}{8}\right ) + x^{7} \left (\frac{A c^{3} d}{7} + \frac{3 B a c^{2} e}{7}\right ) + x^{6} \left (\frac{A a c^{2} e}{2} + \frac{B a c^{2} d}{2}\right ) + x^{5} \left (\frac{3 A a c^{2} d}{5} + \frac{3 B a^{2} c e}{5}\right ) + x^{4} \left (\frac{3 A a^{2} c e}{4} + \frac{3 B a^{2} c d}{4}\right ) + x^{3} \left (A a^{2} c d + \frac{B a^{3} e}{3}\right ) + x^{2} \left (\frac{A a^{3} e}{2} + \frac{B a^{3} d}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(e*x+d)*(c*x**2+a)**3,x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.285684, size = 234, normalized size = 1.58 \[ \frac{1}{9} \, B c^{3} x^{9} e + \frac{1}{8} \, B c^{3} d x^{8} + \frac{1}{8} \, A c^{3} x^{8} e + \frac{1}{7} \, A c^{3} d x^{7} + \frac{3}{7} \, B a c^{2} x^{7} e + \frac{1}{2} \, B a c^{2} d x^{6} + \frac{1}{2} \, A a c^{2} x^{6} e + \frac{3}{5} \, A a c^{2} d x^{5} + \frac{3}{5} \, B a^{2} c x^{5} e + \frac{3}{4} \, B a^{2} c d x^{4} + \frac{3}{4} \, A a^{2} c x^{4} e + A a^{2} c d x^{3} + \frac{1}{3} \, B a^{3} x^{3} e + \frac{1}{2} \, B a^{3} d x^{2} + \frac{1}{2} \, A a^{3} x^{2} e + A a^{3} d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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