3.1300 \(\int (A+B x) (d+e x)^2 \left (a+c x^2\right )^2 \, dx\)

Optimal. Leaf size=206 \[ \frac{c (d+e x)^6 \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}+\frac{(d+e x)^4 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6}-\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{2 c (d+e x)^5 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}-\frac{c^2 (d+e x)^7 (5 B d-A e)}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.445198, antiderivative size = 206, 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 \[ \frac{c (d+e x)^6 \left (a B e^2-2 A c d e+5 B c d^2\right )}{3 e^6}+\frac{(d+e x)^4 \left (a e^2+c d^2\right ) \left (a B e^2-4 A c d e+5 B c d^2\right )}{4 e^6}-\frac{(d+e x)^3 \left (a e^2+c d^2\right )^2 (B d-A e)}{3 e^6}-\frac{2 c (d+e x)^5 \left (-a A e^3+3 a B d e^2-3 A c d^2 e+5 B c d^3\right )}{5 e^6}-\frac{c^2 (d+e x)^7 (5 B d-A e)}{7 e^6}+\frac{B c^2 (d+e x)^8}{8 e^6} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0728201, size = 174, normalized size = 0.84 \[ \frac{1}{2} a^2 d x^2 (2 A e+B d)+a^2 A d^2 x+\frac{1}{6} c x^6 \left (2 a B e^2+2 A c d e+B c d^2\right )+\frac{1}{5} c x^5 \left (2 a A e^2+4 a B d e+A c d^2\right )+\frac{1}{4} a x^4 \left (a B e^2+4 A c d e+2 B c d^2\right )+\frac{1}{3} a x^3 \left (a A e^2+2 a B d e+2 A c d^2\right )+\frac{1}{7} c^2 e x^7 (A e+2 B d)+\frac{1}{8} B c^2 e^2 x^8 \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 177, normalized size = 0.9 \[{\frac{B{c}^{2}{e}^{2}{x}^{8}}{8}}+{\frac{ \left ( A{e}^{2}+2\,Bde \right ){c}^{2}{x}^{7}}{7}}+{\frac{ \left ( \left ( 2\,Ade+B{d}^{2} \right ){c}^{2}+2\,B{e}^{2}ac \right ){x}^{6}}{6}}+{\frac{ \left ( A{c}^{2}{d}^{2}+2\, \left ( A{e}^{2}+2\,Bde \right ) ac \right ){x}^{5}}{5}}+{\frac{ \left ( 2\, \left ( 2\,Ade+B{d}^{2} \right ) ac+{a}^{2}B{e}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ( 2\,A{d}^{2}ac+ \left ( A{e}^{2}+2\,Bde \right ){a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 2\,Ade+B{d}^{2} \right ){a}^{2}{x}^{2}}{2}}+A{d}^{2}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.70767, size = 248, normalized size = 1.2 \[ \frac{1}{8} \, B c^{2} e^{2} x^{8} + \frac{1}{7} \,{\left (2 \, B c^{2} d e + A c^{2} e^{2}\right )} x^{7} + \frac{1}{6} \,{\left (B c^{2} d^{2} + 2 \, A c^{2} d e + 2 \, B a c e^{2}\right )} x^{6} + A a^{2} d^{2} x + \frac{1}{5} \,{\left (A c^{2} d^{2} + 4 \, B a c d e + 2 \, A a c e^{2}\right )} x^{5} + \frac{1}{4} \,{\left (2 \, B a c d^{2} + 4 \, A a c d e + B a^{2} e^{2}\right )} x^{4} + \frac{1}{3} \,{\left (2 \, A a c d^{2} + 2 \, B a^{2} d e + A a^{2} e^{2}\right )} x^{3} + \frac{1}{2} \,{\left (B a^{2} d^{2} + 2 \, A a^{2} d e\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.249945, size = 1, normalized size = 0. \[ \frac{1}{8} x^{8} e^{2} c^{2} B + \frac{2}{7} x^{7} e d c^{2} B + \frac{1}{7} x^{7} e^{2} c^{2} A + \frac{1}{6} x^{6} d^{2} c^{2} B + \frac{1}{3} x^{6} e^{2} c a B + \frac{1}{3} x^{6} e d c^{2} A + \frac{4}{5} x^{5} e d c a B + \frac{1}{5} x^{5} d^{2} c^{2} A + \frac{2}{5} x^{5} e^{2} c a A + \frac{1}{2} x^{4} d^{2} c a B + \frac{1}{4} x^{4} e^{2} a^{2} B + x^{4} e d c a A + \frac{2}{3} x^{3} e d a^{2} B + \frac{2}{3} x^{3} d^{2} c a A + \frac{1}{3} x^{3} e^{2} a^{2} A + \frac{1}{2} x^{2} d^{2} a^{2} B + x^{2} e d a^{2} A + x d^{2} a^{2} A \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 0.212429, size = 211, normalized size = 1.02 \[ A a^{2} d^{2} x + \frac{B c^{2} e^{2} x^{8}}{8} + x^{7} \left (\frac{A c^{2} e^{2}}{7} + \frac{2 B c^{2} d e}{7}\right ) + x^{6} \left (\frac{A c^{2} d e}{3} + \frac{B a c e^{2}}{3} + \frac{B c^{2} d^{2}}{6}\right ) + x^{5} \left (\frac{2 A a c e^{2}}{5} + \frac{A c^{2} d^{2}}{5} + \frac{4 B a c d e}{5}\right ) + x^{4} \left (A a c d e + \frac{B a^{2} e^{2}}{4} + \frac{B a c d^{2}}{2}\right ) + x^{3} \left (\frac{A a^{2} e^{2}}{3} + \frac{2 A a c d^{2}}{3} + \frac{2 B a^{2} d e}{3}\right ) + x^{2} \left (A a^{2} d e + \frac{B a^{2} d^{2}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.279994, size = 270, normalized size = 1.31 \[ \frac{1}{8} \, B c^{2} x^{8} e^{2} + \frac{2}{7} \, B c^{2} d x^{7} e + \frac{1}{6} \, B c^{2} d^{2} x^{6} + \frac{1}{7} \, A c^{2} x^{7} e^{2} + \frac{1}{3} \, A c^{2} d x^{6} e + \frac{1}{5} \, A c^{2} d^{2} x^{5} + \frac{1}{3} \, B a c x^{6} e^{2} + \frac{4}{5} \, B a c d x^{5} e + \frac{1}{2} \, B a c d^{2} x^{4} + \frac{2}{5} \, A a c x^{5} e^{2} + A a c d x^{4} e + \frac{2}{3} \, A a c d^{2} x^{3} + \frac{1}{4} \, B a^{2} x^{4} e^{2} + \frac{2}{3} \, B a^{2} d x^{3} e + \frac{1}{2} \, B a^{2} d^{2} x^{2} + \frac{1}{3} \, A a^{2} x^{3} e^{2} + A a^{2} d x^{2} e + A a^{2} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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