3.1264 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{9/2}} \, dx\)

Optimal. Leaf size=88 \[ \frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5} \]

[Out]

-(b^2 - 4*a*c)^2/(112*c^3*d*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(24*c^3*d^3*(
b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(16*c^3*d^5)

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Rubi [A]  time = 0.113455, 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 \[ \frac{b^2-4 a c}{24 c^3 d^3 (b d+2 c d x)^{3/2}}-\frac{\left (b^2-4 a c\right )^2}{112 c^3 d (b d+2 c d x)^{7/2}}+\frac{\sqrt{b d+2 c d x}}{16 c^3 d^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(9/2),x]

[Out]

-(b^2 - 4*a*c)^2/(112*c^3*d*(b*d + 2*c*d*x)^(7/2)) + (b^2 - 4*a*c)/(24*c^3*d^3*(
b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(16*c^3*d^5)

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Rubi in Sympy [A]  time = 28.4673, size = 82, normalized size = 0.93 \[ - \frac{\left (- 4 a c + b^{2}\right )^{2}}{112 c^{3} d \left (b d + 2 c d x\right )^{\frac{7}{2}}} + \frac{- 4 a c + b^{2}}{24 c^{3} d^{3} \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{b d + 2 c d x}}{16 c^{3} d^{5}} \]

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)**(9/2),x)

[Out]

-(-4*a*c + b**2)**2/(112*c**3*d*(b*d + 2*c*d*x)**(7/2)) + (-4*a*c + b**2)/(24*c*
*3*d**3*(b*d + 2*c*d*x)**(3/2)) + sqrt(b*d + 2*c*d*x)/(16*c**3*d**5)

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Mathematica [A]  time = 0.177509, size = 67, normalized size = 0.76 \[ \frac{(b+2 c x)^5 \left (-\frac{3 \left (b^2-4 a c\right )^2}{(b+2 c x)^4}+\frac{14 \left (b^2-4 a c\right )}{(b+2 c x)^2}+21\right )}{336 c^3 (d (b+2 c x))^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)^2/(b*d + 2*c*d*x)^(9/2),x]

[Out]

((b + 2*c*x)^5*(21 - (3*(b^2 - 4*a*c)^2)/(b + 2*c*x)^4 + (14*(b^2 - 4*a*c))/(b +
2*c*x)^2))/(336*c^3*(d*(b + 2*c*x))^(9/2))

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Maple [A]  time = 0.01, size = 96, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -21\,{c}^{4}{x}^{4}-42\,b{x}^{3}{c}^{3}+14\,a{c}^{3}{x}^{2}-35\,{b}^{2}{c}^{2}{x}^{2}+14\,ab{c}^{2}x-14\,{b}^{3}cx+3\,{a}^{2}{c}^{2}+2\,ac{b}^{2}-2\,{b}^{4} \right ) }{21\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{9}{2}}}} \]

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)^(9/2),x)

[Out]

-1/21*(2*c*x+b)*(-21*c^4*x^4-42*b*c^3*x^3+14*a*c^3*x^2-35*b^2*c^2*x^2+14*a*b*c^2
*x-14*b^3*c*x+3*a^2*c^2+2*a*b^2*c-2*b^4)/c^3/(2*c*d*x+b*d)^(9/2)

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Maxima [A]  time = 0.687424, size = 124, normalized size = 1.41 \[ \frac{\frac{21 \, \sqrt{2 \, c d x + b d}}{c^{2} d^{4}} + \frac{14 \,{\left (2 \, c d x + b d\right )}^{2}{\left (b^{2} - 4 \, a c\right )} - 3 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} d^{2}}{{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{2} d^{2}}}{336 \, c d} \]

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)^(9/2),x, algorithm="maxima")

[Out]

1/336*(21*sqrt(2*c*d*x + b*d)/(c^2*d^4) + (14*(2*c*d*x + b*d)^2*(b^2 - 4*a*c) -
3*(b^4 - 8*a*b^2*c + 16*a^2*c^2)*d^2)/((2*c*d*x + b*d)^(7/2)*c^2*d^2))/(c*d)

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Fricas [A]  time = 0.208014, size = 182, normalized size = 2.07 \[ \frac{21 \, c^{4} x^{4} + 42 \, b c^{3} x^{3} + 2 \, b^{4} - 2 \, a b^{2} c - 3 \, a^{2} c^{2} + 7 \,{\left (5 \, b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} + 14 \,{\left (b^{3} c - a b c^{2}\right )} x}{21 \,{\left (8 \, c^{6} d^{4} x^{3} + 12 \, b c^{5} d^{4} x^{2} + 6 \, b^{2} c^{4} d^{4} x + b^{3} c^{3} d^{4}\right )} \sqrt{2 \, c d x + b d}} \]

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)^(9/2),x, algorithm="fricas")

[Out]

1/21*(21*c^4*x^4 + 42*b*c^3*x^3 + 2*b^4 - 2*a*b^2*c - 3*a^2*c^2 + 7*(5*b^2*c^2 -
2*a*c^3)*x^2 + 14*(b^3*c - a*b*c^2)*x)/((8*c^6*d^4*x^3 + 12*b*c^5*d^4*x^2 + 6*b
^2*c^4*d^4*x + b^3*c^3*d^4)*sqrt(2*c*d*x + b*d))

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Sympy [A]  time = 22.1841, size = 826, normalized size = 9.39 \[ \begin{cases} - \frac{3 a^{2} c^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{2 a b^{2} c \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{14 a b c^{2} x \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} - \frac{14 a c^{3} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{2 b^{4} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{14 b^{3} c x \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{35 b^{2} c^{2} x^{2} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{42 b c^{3} x^{3} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} + \frac{21 c^{4} x^{4} \sqrt{b d + 2 c d x}}{21 b^{4} c^{3} d^{5} + 168 b^{3} c^{4} d^{5} x + 504 b^{2} c^{5} d^{5} x^{2} + 672 b c^{6} d^{5} x^{3} + 336 c^{7} d^{5} x^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3}}{\left (b d\right )^{\frac{9}{2}}} & \text{otherwise} \end{cases} \]

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)**(9/2),x)

[Out]

Piecewise((-3*a**2*c**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d
**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) - 2
*a*b**2*c*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b*
*2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) - 14*a*b*c**2*x*s
qrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**
5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) - 14*a*c**3*x**2*sqrt(b*d +
2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 6
72*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) + 2*b**4*sqrt(b*d + 2*c*d*x)/(21*b**4*
c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**
3 + 336*c**7*d**5*x**4) + 14*b**3*c*x*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 1
68*b**3*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*
d**5*x**4) + 35*b**2*c**2*x**2*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3
*c**4*d**5*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x*
*4) + 42*b*c**3*x**3*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5
*x + 504*b**2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4) + 21*c
**4*x**4*sqrt(b*d + 2*c*d*x)/(21*b**4*c**3*d**5 + 168*b**3*c**4*d**5*x + 504*b**
2*c**5*d**5*x**2 + 672*b*c**6*d**5*x**3 + 336*c**7*d**5*x**4), Ne(c, 0)), ((a**2
*x + a*b*x**2 + b**2*x**3/3)/(b*d)**(9/2), True))

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GIAC/XCAS [A]  time = 0.234428, size = 135, normalized size = 1.53 \[ \frac{\sqrt{2 \, c d x + b d}}{16 \, c^{3} d^{5}} - \frac{3 \, b^{4} d^{2} - 24 \, a b^{2} c d^{2} + 48 \, a^{2} c^{2} d^{2} - 14 \,{\left (2 \, c d x + b d\right )}^{2} b^{2} + 56 \,{\left (2 \, c d x + b d\right )}^{2} a c}{336 \,{\left (2 \, c d x + b d\right )}^{\frac{7}{2}} c^{3} d^{3}} \]

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)^(9/2),x, algorithm="giac")

[Out]

1/16*sqrt(2*c*d*x + b*d)/(c^3*d^5) - 1/336*(3*b^4*d^2 - 24*a*b^2*c*d^2 + 48*a^2*
c^2*d^2 - 14*(2*c*d*x + b*d)^2*b^2 + 56*(2*c*d*x + b*d)^2*a*c)/((2*c*d*x + b*d)^
(7/2)*c^3*d^3)