3.1113 \(\int (b d+2 c d x)^2 \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=73 \[ -\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3} \]

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3)/(96*c^3) - ((b^2 - 4*a*c)*d^2*(b + 2*c*x)^5)
/(80*c^3) + (d^2*(b + 2*c*x)^7)/(224*c^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.199835, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{d^2 \left (b^2-4 a c\right ) (b+2 c x)^5}{80 c^3}+\frac{d^2 \left (b^2-4 a c\right )^2 (b+2 c x)^3}{96 c^3}+\frac{d^2 (b+2 c x)^7}{224 c^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*d^2*(b + 2*c*x)^3)/(96*c^3) - ((b^2 - 4*a*c)*d^2*(b + 2*c*x)^5)
/(80*c^3) + (d^2*(b + 2*c*x)^7)/(224*c^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 38.9677, size = 68, normalized size = 0.93 \[ \frac{d^{2} \left (b + 2 c x\right )^{7}}{224 c^{3}} - \frac{d^{2} \left (b + 2 c x\right )^{5} \left (- 4 a c + b^{2}\right )}{80 c^{3}} + \frac{d^{2} \left (b + 2 c x\right )^{3} \left (- 4 a c + b^{2}\right )^{2}}{96 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a)**2,x)

[Out]

d**2*(b + 2*c*x)**7/(224*c**3) - d**2*(b + 2*c*x)**5*(-4*a*c + b**2)/(80*c**3) +
d**2*(b + 2*c*x)**3*(-4*a*c + b**2)**2/(96*c**3)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0367894, size = 111, normalized size = 1.52 \[ d^2 \left (a^2 b^2 x+\frac{1}{3} x^3 \left (4 a^2 c^2+10 a b^2 c+b^4\right )+\frac{1}{5} c^2 x^5 \left (8 a c+13 b^2\right )+\frac{1}{2} b c x^4 \left (8 a c+3 b^2\right )+a b x^2 \left (2 a c+b^2\right )+2 b c^3 x^6+\frac{4 c^4 x^7}{7}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^2*(a + b*x + c*x^2)^2,x]

[Out]

d^2*(a^2*b^2*x + a*b*(b^2 + 2*a*c)*x^2 + ((b^4 + 10*a*b^2*c + 4*a^2*c^2)*x^3)/3
+ (b*c*(3*b^2 + 8*a*c)*x^4)/2 + (c^2*(13*b^2 + 8*a*c)*x^5)/5 + 2*b*c^3*x^6 + (4*
c^4*x^7)/7)

_______________________________________________________________________________________

Maple [B]  time = 0.002, size = 176, normalized size = 2.4 \[{\frac{4\,{c}^{4}{d}^{2}{x}^{7}}{7}}+2\,b{d}^{2}{c}^{3}{x}^{6}+{\frac{ \left ( 9\,{b}^{2}{d}^{2}{c}^{2}+4\,{c}^{2}{d}^{2} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{3}c{d}^{2}+4\,b{d}^{2}c \left ( 2\,ac+{b}^{2} \right ) +8\,{c}^{2}{d}^{2}ab \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{2}{d}^{2} \left ( 2\,ac+{b}^{2} \right ) +8\,{b}^{2}{d}^{2}ca+4\,{a}^{2}{c}^{2}{d}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 4\,{a}^{2}c{d}^{2}b+2\,{b}^{3}{d}^{2}a \right ){x}^{2}}{2}}+{b}^{2}{d}^{2}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^2*(c*x^2+b*x+a)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.675885, size = 171, normalized size = 2.34 \[ \frac{4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac{1}{5} \,{\left (13 \, b^{2} c^{2} + 8 \, a c^{3}\right )} d^{2} x^{5} + a^{2} b^{2} d^{2} x + \frac{1}{2} \,{\left (3 \, b^{3} c + 8 \, a b c^{2}\right )} d^{2} x^{4} + \frac{1}{3} \,{\left (b^{4} + 10 \, a b^{2} c + 4 \, a^{2} c^{2}\right )} d^{2} x^{3} +{\left (a b^{3} + 2 \, a^{2} b c\right )} d^{2} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 1/5*(13*b^2*c^2 + 8*a*c^3)*d^2*x^5 + a^2*b^2
*d^2*x + 1/2*(3*b^3*c + 8*a*b*c^2)*d^2*x^4 + 1/3*(b^4 + 10*a*b^2*c + 4*a^2*c^2)*
d^2*x^3 + (a*b^3 + 2*a^2*b*c)*d^2*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.194654, size = 1, normalized size = 0.01 \[ \frac{4}{7} x^{7} d^{2} c^{4} + 2 x^{6} d^{2} c^{3} b + \frac{13}{5} x^{5} d^{2} c^{2} b^{2} + \frac{8}{5} x^{5} d^{2} c^{3} a + \frac{3}{2} x^{4} d^{2} c b^{3} + 4 x^{4} d^{2} c^{2} b a + \frac{1}{3} x^{3} d^{2} b^{4} + \frac{10}{3} x^{3} d^{2} c b^{2} a + \frac{4}{3} x^{3} d^{2} c^{2} a^{2} + x^{2} d^{2} b^{3} a + 2 x^{2} d^{2} c b a^{2} + x d^{2} b^{2} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/7*x^7*d^2*c^4 + 2*x^6*d^2*c^3*b + 13/5*x^5*d^2*c^2*b^2 + 8/5*x^5*d^2*c^3*a + 3
/2*x^4*d^2*c*b^3 + 4*x^4*d^2*c^2*b*a + 1/3*x^3*d^2*b^4 + 10/3*x^3*d^2*c*b^2*a +
4/3*x^3*d^2*c^2*a^2 + x^2*d^2*b^3*a + 2*x^2*d^2*c*b*a^2 + x*d^2*b^2*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.176665, size = 156, normalized size = 2.14 \[ a^{2} b^{2} d^{2} x + 2 b c^{3} d^{2} x^{6} + \frac{4 c^{4} d^{2} x^{7}}{7} + x^{5} \left (\frac{8 a c^{3} d^{2}}{5} + \frac{13 b^{2} c^{2} d^{2}}{5}\right ) + x^{4} \left (4 a b c^{2} d^{2} + \frac{3 b^{3} c d^{2}}{2}\right ) + x^{3} \left (\frac{4 a^{2} c^{2} d^{2}}{3} + \frac{10 a b^{2} c d^{2}}{3} + \frac{b^{4} d^{2}}{3}\right ) + x^{2} \left (2 a^{2} b c d^{2} + a b^{3} d^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**2*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b**2*d**2*x + 2*b*c**3*d**2*x**6 + 4*c**4*d**2*x**7/7 + x**5*(8*a*c**3*d**2
/5 + 13*b**2*c**2*d**2/5) + x**4*(4*a*b*c**2*d**2 + 3*b**3*c*d**2/2) + x**3*(4*a
**2*c**2*d**2/3 + 10*a*b**2*c*d**2/3 + b**4*d**2/3) + x**2*(2*a**2*b*c*d**2 + a*
b**3*d**2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.214444, size = 200, normalized size = 2.74 \[ \frac{4}{7} \, c^{4} d^{2} x^{7} + 2 \, b c^{3} d^{2} x^{6} + \frac{13}{5} \, b^{2} c^{2} d^{2} x^{5} + \frac{8}{5} \, a c^{3} d^{2} x^{5} + \frac{3}{2} \, b^{3} c d^{2} x^{4} + 4 \, a b c^{2} d^{2} x^{4} + \frac{1}{3} \, b^{4} d^{2} x^{3} + \frac{10}{3} \, a b^{2} c d^{2} x^{3} + \frac{4}{3} \, a^{2} c^{2} d^{2} x^{3} + a b^{3} d^{2} x^{2} + 2 \, a^{2} b c d^{2} x^{2} + a^{2} b^{2} d^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/7*c^4*d^2*x^7 + 2*b*c^3*d^2*x^6 + 13/5*b^2*c^2*d^2*x^5 + 8/5*a*c^3*d^2*x^5 + 3
/2*b^3*c*d^2*x^4 + 4*a*b*c^2*d^2*x^4 + 1/3*b^4*d^2*x^3 + 10/3*a*b^2*c*d^2*x^3 +
4/3*a^2*c^2*d^2*x^3 + a*b^3*d^2*x^2 + 2*a^2*b*c*d^2*x^2 + a^2*b^2*d^2*x