3.1265 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{11/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac{1}{16 c^3 d^5 \sqrt{b d+2 c d x}} \]

[Out]

-(b^2 - 4*a*c)^2/(144*c^3*d*(b*d + 2*c*d*x)^(9/2)) + (b^2 - 4*a*c)/(40*c^3*d^3*(
b*d + 2*c*d*x)^(5/2)) - 1/(16*c^3*d^5*Sqrt[b*d + 2*c*d*x])

_______________________________________________________________________________________

Rubi [A]  time = 0.112918, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{b^2-4 a c}{40 c^3 d^3 (b d+2 c d x)^{5/2}}-\frac{\left (b^2-4 a c\right )^2}{144 c^3 d (b d+2 c d x)^{9/2}}-\frac{1}{16 c^3 d^5 \sqrt{b d+2 c d x}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(11/2),x]

[Out]

-(b^2 - 4*a*c)^2/(144*c^3*d*(b*d + 2*c*d*x)^(9/2)) + (b^2 - 4*a*c)/(40*c^3*d^3*(
b*d + 2*c*d*x)^(5/2)) - 1/(16*c^3*d^5*Sqrt[b*d + 2*c*d*x])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 28.7535, size = 83, normalized size = 0.94 \[ - \frac{\left (- 4 a c + b^{2}\right )^{2}}{144 c^{3} d \left (b d + 2 c d x\right )^{\frac{9}{2}}} + \frac{- 4 a c + b^{2}}{40 c^{3} d^{3} \left (b d + 2 c d x\right )^{\frac{5}{2}}} - \frac{1}{16 c^{3} d^{5} \sqrt{b d + 2 c d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**(11/2),x)

[Out]

-(-4*a*c + b**2)**2/(144*c**3*d*(b*d + 2*c*d*x)**(9/2)) + (-4*a*c + b**2)/(40*c*
*3*d**3*(b*d + 2*c*d*x)**(5/2)) - 1/(16*c**3*d**5*sqrt(b*d + 2*c*d*x))

_______________________________________________________________________________________

Mathematica [A]  time = 0.148812, size = 63, normalized size = 0.72 \[ \frac{18 \left (b^2-4 a c\right ) (b+2 c x)^2-5 \left (b^2-4 a c\right )^2-45 (b+2 c x)^4}{720 c^3 d (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(11/2),x]

[Out]

(-5*(b^2 - 4*a*c)^2 + 18*(b^2 - 4*a*c)*(b + 2*c*x)^2 - 45*(b + 2*c*x)^4)/(720*c^
3*d*(d*(b + 2*c*x))^(9/2))

_______________________________________________________________________________________

Maple [A]  time = 0.01, size = 96, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( 45\,{c}^{4}{x}^{4}+90\,b{x}^{3}{c}^{3}+18\,a{c}^{3}{x}^{2}+63\,{b}^{2}{c}^{2}{x}^{2}+18\,ab{c}^{2}x+18\,{b}^{3}cx+5\,{a}^{2}{c}^{2}+2\,ac{b}^{2}+2\,{b}^{4} \right ) }{45\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{11}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^2/(2*c*d*x+b*d)^(11/2),x)

[Out]

-1/45*(2*c*x+b)*(45*c^4*x^4+90*b*c^3*x^3+18*a*c^3*x^2+63*b^2*c^2*x^2+18*a*b*c^2*
x+18*b^3*c*x+5*a^2*c^2+2*a*b^2*c+2*b^4)/c^3/(2*c*d*x+b*d)^(11/2)

_______________________________________________________________________________________

Maxima [A]  time = 0.688649, size = 109, normalized size = 1.24 \[ \frac{18 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} d^{2} - 5 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{4} - 45 \,{\left (2 \, c d x + b d\right )}^{4}}{720 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^(11/2),x, algorithm="maxima")

[Out]

1/720*(18*(2*c*d*x + b*d)^2*(b^2 - 4*a*c)*d^2 - 5*(b^4 - 8*a*b^2*c + 16*a^2*c^2)
*d^4 - 45*(2*c*d*x + b*d)^4)/((2*c*d*x + b*d)^(9/2)*c^3*d^5)

_______________________________________________________________________________________

Fricas [A]  time = 0.210708, size = 200, normalized size = 2.27 \[ -\frac{45 \, c^{4} x^{4} + 90 \, b c^{3} x^{3} + 2 \, b^{4} + 2 \, a b^{2} c + 5 \, a^{2} c^{2} + 9 \,{\left (7 \, b^{2} c^{2} + 2 \, a c^{3}\right )} x^{2} + 18 \,{\left (b^{3} c + a b c^{2}\right )} x}{45 \,{\left (16 \, c^{7} d^{5} x^{4} + 32 \, b c^{6} d^{5} x^{3} + 24 \, b^{2} c^{5} d^{5} x^{2} + 8 \, b^{3} c^{4} d^{5} x + b^{4} c^{3} d^{5}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^(11/2),x, algorithm="fricas")

[Out]

-1/45*(45*c^4*x^4 + 90*b*c^3*x^3 + 2*b^4 + 2*a*b^2*c + 5*a^2*c^2 + 9*(7*b^2*c^2
+ 2*a*c^3)*x^2 + 18*(b^3*c + a*b*c^2)*x)/((16*c^7*d^5*x^4 + 32*b*c^6*d^5*x^3 + 2
4*b^2*c^5*d^5*x^2 + 8*b^3*c^4*d^5*x + b^4*c^3*d^5)*sqrt(2*c*d*x + b*d))

_______________________________________________________________________________________

Sympy [A]  time = 43.4441, size = 966, normalized size = 10.98 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)**2/(2*c*d*x+b*d)**(11/2),x)

[Out]

Piecewise((-5*a**2*c**2*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d
**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x
**4 + 1440*c**8*d**6*x**5) - 2*a*b**2*c*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 +
450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 36
00*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 18*a*b*c**2*x*sqrt(b*d + 2*c*d*x)/(
45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*
c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 18*a*c**3*x**2*s
qrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d*
*6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5
) - 2*b**4*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*
b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c*
*8*d**6*x**5) - 18*b**3*c*x*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c*
*4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d*
*6*x**4 + 1440*c**8*d**6*x**5) - 63*b**2*c**2*x**2*sqrt(b*d + 2*c*d*x)/(45*b**5*
c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2 + 3600*b**2*c**6*d**
6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 90*b*c**3*x**3*sqrt(b*d
+ 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b**3*c**5*d**6*x**2
+ 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8*d**6*x**5) - 45*c
**4*x**4*sqrt(b*d + 2*c*d*x)/(45*b**5*c**3*d**6 + 450*b**4*c**4*d**6*x + 1800*b*
*3*c**5*d**6*x**2 + 3600*b**2*c**6*d**6*x**3 + 3600*b*c**7*d**6*x**4 + 1440*c**8
*d**6*x**5), Ne(c, 0)), ((a**2*x + a*b*x**2 + b**2*x**3/3)/(b*d)**(11/2), True))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.227972, size = 134, normalized size = 1.52 \[ -\frac{5 \, b^{4} d^{4} - 40 \, a b^{2} c d^{4} + 80 \, a^{2} c^{2} d^{4} - 18 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} d^{2} + 72 \,{\left (2 \, c d x + b d\right )}^{2} a c d^{2} + 45 \,{\left (2 \, c d x + b d\right )}^{4}}{720 \,{\left (2 \, c d x + b d\right )}^{\frac{9}{2}} c^{3} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)^2/(2*c*d*x + b*d)^(11/2),x, algorithm="giac")

[Out]

-1/720*(5*b^4*d^4 - 40*a*b^2*c*d^4 + 80*a^2*c^2*d^4 - 18*(2*c*d*x + b*d)^2*b^2*d
^2 + 72*(2*c*d*x + b*d)^2*a*c*d^2 + 45*(2*c*d*x + b*d)^4)/((2*c*d*x + b*d)^(9/2)
*c^3*d^5)