3.1110 \(\int (b d+2 c d x)^5 \left (a+b x+c x^2\right )^2 \, dx\)

Optimal. Leaf size=73 \[ -\frac{d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3} \]

[Out]

((b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^6)/(192*c^3) - ((b^2 - 4*a*c)*d^5*(b + 2*c*x)^8
)/(128*c^3) + (d^5*(b + 2*c*x)^10)/(320*c^3)

_______________________________________________________________________________________

Rubi [A]  time = 0.371214, 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 \[ -\frac{d^5 \left (b^2-4 a c\right ) (b+2 c x)^8}{128 c^3}+\frac{d^5 \left (b^2-4 a c\right )^2 (b+2 c x)^6}{192 c^3}+\frac{d^5 (b+2 c x)^{10}}{320 c^3} \]

Antiderivative was successfully verified.

[In]  Int[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

((b^2 - 4*a*c)^2*d^5*(b + 2*c*x)^6)/(192*c^3) - ((b^2 - 4*a*c)*d^5*(b + 2*c*x)^8
)/(128*c^3) + (d^5*(b + 2*c*x)^10)/(320*c^3)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 49.359, size = 68, normalized size = 0.93 \[ \frac{d^{5} \left (b + 2 c x\right )^{10}}{320 c^{3}} - \frac{d^{5} \left (b + 2 c x\right )^{8} \left (- 4 a c + b^{2}\right )}{128 c^{3}} + \frac{d^{5} \left (b + 2 c x\right )^{6} \left (- 4 a c + b^{2}\right )^{2}}{192 c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**2,x)

[Out]

d**5*(b + 2*c*x)**10/(320*c**3) - d**5*(b + 2*c*x)**8*(-4*a*c + b**2)/(128*c**3)
+ d**5*(b + 2*c*x)**6*(-4*a*c + b**2)**2/(192*c**3)

_______________________________________________________________________________________

Mathematica [B]  time = 0.115849, size = 168, normalized size = 2.3 \[ \frac{1}{15} d^5 x (b+c x) \left (5 a^2 \left (3 b^4+12 b^3 c x+28 b^2 c^2 x^2+32 b c^3 x^3+16 c^4 x^4\right )+5 a x \left (3 b^5+19 b^4 c x+56 b^3 c^2 x^2+88 b^2 c^3 x^3+72 b c^4 x^4+24 c^5 x^5\right )+x^2 (b+c x)^2 \left (5 b^4+30 b^3 c x+78 b^2 c^2 x^2+96 b c^3 x^3+48 c^4 x^4\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(b*d + 2*c*d*x)^5*(a + b*x + c*x^2)^2,x]

[Out]

(d^5*x*(b + c*x)*(5*a^2*(3*b^4 + 12*b^3*c*x + 28*b^2*c^2*x^2 + 32*b*c^3*x^3 + 16
*c^4*x^4) + x^2*(b + c*x)^2*(5*b^4 + 30*b^3*c*x + 78*b^2*c^2*x^2 + 96*b*c^3*x^3
+ 48*c^4*x^4) + 5*a*x*(3*b^5 + 19*b^4*c*x + 56*b^3*c^2*x^2 + 88*b^2*c^3*x^3 + 72
*b*c^4*x^4 + 24*c^5*x^5)))/15

_______________________________________________________________________________________

Maple [B]  time = 0.003, size = 362, normalized size = 5. \[{\frac{16\,{c}^{7}{d}^{5}{x}^{10}}{5}}+16\,b{c}^{6}{d}^{5}{x}^{9}+{\frac{ \left ( 240\,{b}^{2}{d}^{5}{c}^{5}+32\,{c}^{5}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) \right ){x}^{8}}{8}}+{\frac{ \left ( 200\,{b}^{3}{c}^{4}{d}^{5}+80\,b{c}^{4}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) +64\,{c}^{5}{d}^{5}ab \right ){x}^{7}}{7}}+{\frac{ \left ( 90\,{b}^{4}{d}^{5}{c}^{3}+80\,{b}^{2}{d}^{5}{c}^{3} \left ( 2\,ac+{b}^{2} \right ) +160\,{b}^{2}{c}^{4}{d}^{5}a+32\,{c}^{5}{d}^{5}{a}^{2} \right ){x}^{6}}{6}}+{\frac{ \left ( 21\,{b}^{5}{d}^{5}{c}^{2}+40\,{b}^{3}{d}^{5}{c}^{2} \left ( 2\,ac+{b}^{2} \right ) +160\,{b}^{3}{d}^{5}{c}^{3}a+80\,b{c}^{4}{d}^{5}{a}^{2} \right ){x}^{5}}{5}}+{\frac{ \left ( 2\,{b}^{6}{d}^{5}c+10\,{b}^{4}{d}^{5}c \left ( 2\,ac+{b}^{2} \right ) +80\,{b}^{4}{d}^{5}{c}^{2}a+80\,{b}^{2}{d}^{5}{c}^{3}{a}^{2} \right ){x}^{4}}{4}}+{\frac{ \left ({b}^{5}{d}^{5} \left ( 2\,ac+{b}^{2} \right ) +20\,{b}^{5}{d}^{5}ca+40\,{b}^{3}{d}^{5}{c}^{2}{a}^{2} \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,{b}^{4}{d}^{5}c{a}^{2}+2\,{b}^{6}{d}^{5}a \right ){x}^{2}}{2}}+{b}^{5}{d}^{5}{a}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*c*d*x+b*d)^5*(c*x^2+b*x+a)^2,x)

[Out]

16/5*c^7*d^5*x^10+16*b*c^6*d^5*x^9+1/8*(240*b^2*d^5*c^5+32*c^5*d^5*(2*a*c+b^2))*
x^8+1/7*(200*b^3*c^4*d^5+80*b*c^4*d^5*(2*a*c+b^2)+64*c^5*d^5*a*b)*x^7+1/6*(90*b^
4*d^5*c^3+80*b^2*d^5*c^3*(2*a*c+b^2)+160*b^2*c^4*d^5*a+32*c^5*d^5*a^2)*x^6+1/5*(
21*b^5*d^5*c^2+40*b^3*d^5*c^2*(2*a*c+b^2)+160*b^3*d^5*c^3*a+80*b*c^4*d^5*a^2)*x^
5+1/4*(2*b^6*d^5*c+10*b^4*d^5*c*(2*a*c+b^2)+80*b^4*d^5*c^2*a+80*b^2*d^5*c^3*a^2)
*x^4+1/3*(b^5*d^5*(2*a*c+b^2)+20*b^5*d^5*c*a+40*b^3*d^5*c^2*a^2)*x^3+1/2*(10*a^2
*b^4*c*d^5+2*a*b^6*d^5)*x^2+b^5*d^5*a^2*x

_______________________________________________________________________________________

Maxima [A]  time = 0.679035, size = 320, normalized size = 4.38 \[ \frac{16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 2 \,{\left (17 \, b^{2} c^{5} + 4 \, a c^{6}\right )} d^{5} x^{8} + a^{2} b^{5} d^{5} x + 8 \,{\left (5 \, b^{3} c^{4} + 4 \, a b c^{5}\right )} d^{5} x^{7} + \frac{1}{3} \,{\left (85 \, b^{4} c^{3} + 160 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} d^{5} x^{6} + \frac{1}{5} \,{\left (61 \, b^{5} c^{2} + 240 \, a b^{3} c^{3} + 80 \, a^{2} b c^{4}\right )} d^{5} x^{5} +{\left (3 \, b^{6} c + 25 \, a b^{4} c^{2} + 20 \, a^{2} b^{2} c^{3}\right )} d^{5} x^{4} + \frac{1}{3} \,{\left (b^{7} + 22 \, a b^{5} c + 40 \, a^{2} b^{3} c^{2}\right )} d^{5} x^{3} +{\left (a b^{6} + 5 \, a^{2} b^{4} c\right )} d^{5} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5*(c*x^2 + b*x + a)^2,x, algorithm="maxima")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 2*(17*b^2*c^5 + 4*a*c^6)*d^5*x^8 + a^2*b^
5*d^5*x + 8*(5*b^3*c^4 + 4*a*b*c^5)*d^5*x^7 + 1/3*(85*b^4*c^3 + 160*a*b^2*c^4 +
16*a^2*c^5)*d^5*x^6 + 1/5*(61*b^5*c^2 + 240*a*b^3*c^3 + 80*a^2*b*c^4)*d^5*x^5 +
(3*b^6*c + 25*a*b^4*c^2 + 20*a^2*b^2*c^3)*d^5*x^4 + 1/3*(b^7 + 22*a*b^5*c + 40*a
^2*b^3*c^2)*d^5*x^3 + (a*b^6 + 5*a^2*b^4*c)*d^5*x^2

_______________________________________________________________________________________

Fricas [A]  time = 0.186717, size = 1, normalized size = 0.01 \[ \frac{16}{5} x^{10} d^{5} c^{7} + 16 x^{9} d^{5} c^{6} b + 34 x^{8} d^{5} c^{5} b^{2} + 8 x^{8} d^{5} c^{6} a + 40 x^{7} d^{5} c^{4} b^{3} + 32 x^{7} d^{5} c^{5} b a + \frac{85}{3} x^{6} d^{5} c^{3} b^{4} + \frac{160}{3} x^{6} d^{5} c^{4} b^{2} a + \frac{16}{3} x^{6} d^{5} c^{5} a^{2} + \frac{61}{5} x^{5} d^{5} c^{2} b^{5} + 48 x^{5} d^{5} c^{3} b^{3} a + 16 x^{5} d^{5} c^{4} b a^{2} + 3 x^{4} d^{5} c b^{6} + 25 x^{4} d^{5} c^{2} b^{4} a + 20 x^{4} d^{5} c^{3} b^{2} a^{2} + \frac{1}{3} x^{3} d^{5} b^{7} + \frac{22}{3} x^{3} d^{5} c b^{5} a + \frac{40}{3} x^{3} d^{5} c^{2} b^{3} a^{2} + x^{2} d^{5} b^{6} a + 5 x^{2} d^{5} c b^{4} a^{2} + x d^{5} b^{5} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5*(c*x^2 + b*x + a)^2,x, algorithm="fricas")

[Out]

16/5*x^10*d^5*c^7 + 16*x^9*d^5*c^6*b + 34*x^8*d^5*c^5*b^2 + 8*x^8*d^5*c^6*a + 40
*x^7*d^5*c^4*b^3 + 32*x^7*d^5*c^5*b*a + 85/3*x^6*d^5*c^3*b^4 + 160/3*x^6*d^5*c^4
*b^2*a + 16/3*x^6*d^5*c^5*a^2 + 61/5*x^5*d^5*c^2*b^5 + 48*x^5*d^5*c^3*b^3*a + 16
*x^5*d^5*c^4*b*a^2 + 3*x^4*d^5*c*b^6 + 25*x^4*d^5*c^2*b^4*a + 20*x^4*d^5*c^3*b^2
*a^2 + 1/3*x^3*d^5*b^7 + 22/3*x^3*d^5*c*b^5*a + 40/3*x^3*d^5*c^2*b^3*a^2 + x^2*d
^5*b^6*a + 5*x^2*d^5*c*b^4*a^2 + x*d^5*b^5*a^2

_______________________________________________________________________________________

Sympy [A]  time = 0.262746, size = 291, normalized size = 3.99 \[ a^{2} b^{5} d^{5} x + 16 b c^{6} d^{5} x^{9} + \frac{16 c^{7} d^{5} x^{10}}{5} + x^{8} \left (8 a c^{6} d^{5} + 34 b^{2} c^{5} d^{5}\right ) + x^{7} \left (32 a b c^{5} d^{5} + 40 b^{3} c^{4} d^{5}\right ) + x^{6} \left (\frac{16 a^{2} c^{5} d^{5}}{3} + \frac{160 a b^{2} c^{4} d^{5}}{3} + \frac{85 b^{4} c^{3} d^{5}}{3}\right ) + x^{5} \left (16 a^{2} b c^{4} d^{5} + 48 a b^{3} c^{3} d^{5} + \frac{61 b^{5} c^{2} d^{5}}{5}\right ) + x^{4} \left (20 a^{2} b^{2} c^{3} d^{5} + 25 a b^{4} c^{2} d^{5} + 3 b^{6} c d^{5}\right ) + x^{3} \left (\frac{40 a^{2} b^{3} c^{2} d^{5}}{3} + \frac{22 a b^{5} c d^{5}}{3} + \frac{b^{7} d^{5}}{3}\right ) + x^{2} \left (5 a^{2} b^{4} c d^{5} + a b^{6} d^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x+b*d)**5*(c*x**2+b*x+a)**2,x)

[Out]

a**2*b**5*d**5*x + 16*b*c**6*d**5*x**9 + 16*c**7*d**5*x**10/5 + x**8*(8*a*c**6*d
**5 + 34*b**2*c**5*d**5) + x**7*(32*a*b*c**5*d**5 + 40*b**3*c**4*d**5) + x**6*(1
6*a**2*c**5*d**5/3 + 160*a*b**2*c**4*d**5/3 + 85*b**4*c**3*d**5/3) + x**5*(16*a*
*2*b*c**4*d**5 + 48*a*b**3*c**3*d**5 + 61*b**5*c**2*d**5/5) + x**4*(20*a**2*b**2
*c**3*d**5 + 25*a*b**4*c**2*d**5 + 3*b**6*c*d**5) + x**3*(40*a**2*b**3*c**2*d**5
/3 + 22*a*b**5*c*d**5/3 + b**7*d**5/3) + x**2*(5*a**2*b**4*c*d**5 + a*b**6*d**5)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.212425, size = 386, normalized size = 5.29 \[ \frac{16}{5} \, c^{7} d^{5} x^{10} + 16 \, b c^{6} d^{5} x^{9} + 34 \, b^{2} c^{5} d^{5} x^{8} + 8 \, a c^{6} d^{5} x^{8} + 40 \, b^{3} c^{4} d^{5} x^{7} + 32 \, a b c^{5} d^{5} x^{7} + \frac{85}{3} \, b^{4} c^{3} d^{5} x^{6} + \frac{160}{3} \, a b^{2} c^{4} d^{5} x^{6} + \frac{16}{3} \, a^{2} c^{5} d^{5} x^{6} + \frac{61}{5} \, b^{5} c^{2} d^{5} x^{5} + 48 \, a b^{3} c^{3} d^{5} x^{5} + 16 \, a^{2} b c^{4} d^{5} x^{5} + 3 \, b^{6} c d^{5} x^{4} + 25 \, a b^{4} c^{2} d^{5} x^{4} + 20 \, a^{2} b^{2} c^{3} d^{5} x^{4} + \frac{1}{3} \, b^{7} d^{5} x^{3} + \frac{22}{3} \, a b^{5} c d^{5} x^{3} + \frac{40}{3} \, a^{2} b^{3} c^{2} d^{5} x^{3} + a b^{6} d^{5} x^{2} + 5 \, a^{2} b^{4} c d^{5} x^{2} + a^{2} b^{5} d^{5} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*d*x + b*d)^5*(c*x^2 + b*x + a)^2,x, algorithm="giac")

[Out]

16/5*c^7*d^5*x^10 + 16*b*c^6*d^5*x^9 + 34*b^2*c^5*d^5*x^8 + 8*a*c^6*d^5*x^8 + 40
*b^3*c^4*d^5*x^7 + 32*a*b*c^5*d^5*x^7 + 85/3*b^4*c^3*d^5*x^6 + 160/3*a*b^2*c^4*d
^5*x^6 + 16/3*a^2*c^5*d^5*x^6 + 61/5*b^5*c^2*d^5*x^5 + 48*a*b^3*c^3*d^5*x^5 + 16
*a^2*b*c^4*d^5*x^5 + 3*b^6*c*d^5*x^4 + 25*a*b^4*c^2*d^5*x^4 + 20*a^2*b^2*c^3*d^5
*x^4 + 1/3*b^7*d^5*x^3 + 22/3*a*b^5*c*d^5*x^3 + 40/3*a^2*b^3*c^2*d^5*x^3 + a*b^6
*d^5*x^2 + 5*a^2*b^4*c*d^5*x^2 + a^2*b^5*d^5*x