3.1262 \(\int \frac{\left (a+b x+c x^2\right )^2}{(b d+2 c d x)^{5/2}} \, dx\)

Optimal. Leaf size=88 \[ -\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{8 c^3 d^3}-\frac{\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{5/2}}{80 c^3 d^5} \]

[Out]

-(b^2 - 4*a*c)^2/(48*c^3*d*(b*d + 2*c*d*x)^(3/2)) - ((b^2 - 4*a*c)*Sqrt[b*d + 2*
c*d*x])/(8*c^3*d^3) + (b*d + 2*c*d*x)^(5/2)/(80*c^3*d^5)

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Rubi [A]  time = 0.114136, 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 \[ -\frac{\left (b^2-4 a c\right ) \sqrt{b d+2 c d x}}{8 c^3 d^3}-\frac{\left (b^2-4 a c\right )^2}{48 c^3 d (b d+2 c d x)^{3/2}}+\frac{(b d+2 c d x)^{5/2}}{80 c^3 d^5} \]

Antiderivative was successfully verified.

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

[Out]

-(b^2 - 4*a*c)^2/(48*c^3*d*(b*d + 2*c*d*x)^(3/2)) - ((b^2 - 4*a*c)*Sqrt[b*d + 2*
c*d*x])/(8*c^3*d^3) + (b*d + 2*c*d*x)^(5/2)/(80*c^3*d^5)

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Rubi in Sympy [A]  time = 28.4452, size = 82, normalized size = 0.93 \[ - \frac{\left (- 4 a c + b^{2}\right )^{2}}{48 c^{3} d \left (b d + 2 c d x\right )^{\frac{3}{2}}} - \frac{\left (- 4 a c + b^{2}\right ) \sqrt{b d + 2 c d x}}{8 c^{3} d^{3}} + \frac{\left (b d + 2 c d x\right )^{\frac{5}{2}}}{80 c^{3} d^{5}} \]

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)**(5/2),x)

[Out]

-(-4*a*c + b**2)**2/(48*c**3*d*(b*d + 2*c*d*x)**(3/2)) - (-4*a*c + b**2)*sqrt(b*
d + 2*c*d*x)/(8*c**3*d**3) + (b*d + 2*c*d*x)**(5/2)/(80*c**3*d**5)

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Mathematica [A]  time = 0.179199, size = 73, normalized size = 0.83 \[ \frac{(b+2 c x)^3 \left (-\frac{5 \left (b^2-4 a c\right )^2}{(b+2 c x)^2}+3 \left (40 a c-9 b^2\right )+12 b c x+12 c^2 x^2\right )}{240 c^3 (d (b+2 c x))^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b + 2*c*x)^3*(3*(-9*b^2 + 40*a*c) + 12*b*c*x + 12*c^2*x^2 - (5*(b^2 - 4*a*c)^2
)/(b + 2*c*x)^2))/(240*c^3*(d*(b + 2*c*x))^(5/2))

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Maple [A]  time = 0.01, size = 96, normalized size = 1.1 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -3\,{c}^{4}{x}^{4}-6\,b{x}^{3}{c}^{3}-30\,a{c}^{3}{x}^{2}+3\,{b}^{2}{c}^{2}{x}^{2}-30\,ab{c}^{2}x+6\,{b}^{3}cx+5\,{a}^{2}{c}^{2}-10\,ac{b}^{2}+2\,{b}^{4} \right ) }{15\,{c}^{3}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}} \]

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)^(5/2),x)

[Out]

-1/15*(2*c*x+b)*(-3*c^4*x^4-6*b*c^3*x^3-30*a*c^3*x^2+3*b^2*c^2*x^2-30*a*b*c^2*x+
6*b^3*c*x+5*a^2*c^2-10*a*b^2*c+2*b^4)/c^3/(2*c*d*x+b*d)^(5/2)

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Maxima [A]  time = 0.685119, size = 122, normalized size = 1.39 \[ -\frac{\frac{5 \,{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{2}} + \frac{3 \,{\left (10 \, \sqrt{2 \, c d x + b d}{\left (b^{2} - 4 \, a c\right )} d^{2} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}}\right )}}{c^{2} d^{4}}}{240 \, c d} \]

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)^(5/2),x, algorithm="maxima")

[Out]

-1/240*(5*(b^4 - 8*a*b^2*c + 16*a^2*c^2)/((2*c*d*x + b*d)^(3/2)*c^2) + 3*(10*sqr
t(2*c*d*x + b*d)*(b^2 - 4*a*c)*d^2 - (2*c*d*x + b*d)^(5/2))/(c^2*d^4))/(c*d)

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Fricas [A]  time = 0.208888, size = 143, normalized size = 1.62 \[ \frac{3 \, c^{4} x^{4} + 6 \, b c^{3} x^{3} - 2 \, b^{4} + 10 \, a b^{2} c - 5 \, a^{2} c^{2} - 3 \,{\left (b^{2} c^{2} - 10 \, a c^{3}\right )} x^{2} - 6 \,{\left (b^{3} c - 5 \, a b c^{2}\right )} x}{15 \,{\left (2 \, c^{4} d^{2} x + b c^{3} d^{2}\right )} \sqrt{2 \, c d x + b d}} \]

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)^(5/2),x, algorithm="fricas")

[Out]

1/15*(3*c^4*x^4 + 6*b*c^3*x^3 - 2*b^4 + 10*a*b^2*c - 5*a^2*c^2 - 3*(b^2*c^2 - 10
*a*c^3)*x^2 - 6*(b^3*c - 5*a*b*c^2)*x)/((2*c^4*d^2*x + b*c^3*d^2)*sqrt(2*c*d*x +
b*d))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x + c x^{2}\right )^{2}}{\left (d \left (b + 2 c x\right )\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral((a + b*x + c*x**2)**2/(d*(b + 2*c*x))**(5/2), x)

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GIAC/XCAS [A]  time = 0.230382, size = 147, normalized size = 1.67 \[ -\frac{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}{48 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{3} d} - \frac{10 \, \sqrt{2 \, c d x + b d} b^{2} c^{12} d^{22} - 40 \, \sqrt{2 \, c d x + b d} a c^{13} d^{22} -{\left (2 \, c d x + b d\right )}^{\frac{5}{2}} c^{12} d^{20}}{80 \, c^{15} d^{25}} \]

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)^(5/2),x, algorithm="giac")

[Out]

-1/48*(b^4 - 8*a*b^2*c + 16*a^2*c^2)/((2*c*d*x + b*d)^(3/2)*c^3*d) - 1/80*(10*sq
rt(2*c*d*x + b*d)*b^2*c^12*d^22 - 40*sqrt(2*c*d*x + b*d)*a*c^13*d^22 - (2*c*d*x
+ b*d)^(5/2)*c^12*d^20)/(c^15*d^25)