3.1255 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^{5/2}} \, dx\)

Optimal. Leaf size=55 \[ \frac{b^2-4 a c}{12 c^2 d (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{4 c^2 d^3} \]

[Out]

(b^2 - 4*a*c)/(12*c^2*d*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(4*c^2*d^3)

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Rubi [A]  time = 0.0729056, antiderivative size = 55, 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 \[ \frac{b^2-4 a c}{12 c^2 d (b d+2 c d x)^{3/2}}+\frac{\sqrt{b d+2 c d x}}{4 c^2 d^3} \]

Antiderivative was successfully verified.

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

[Out]

(b^2 - 4*a*c)/(12*c^2*d*(b*d + 2*c*d*x)^(3/2)) + Sqrt[b*d + 2*c*d*x]/(4*c^2*d^3)

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Rubi in Sympy [A]  time = 14.313, size = 49, normalized size = 0.89 \[ \frac{- a c + \frac{b^{2}}{4}}{3 c^{2} d \left (b d + 2 c d x\right )^{\frac{3}{2}}} + \frac{\sqrt{b d + 2 c d x}}{4 c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-a*c + b**2/4)/(3*c**2*d*(b*d + 2*c*d*x)**(3/2)) + sqrt(b*d + 2*c*d*x)/(4*c**2*
d**3)

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Mathematica [A]  time = 0.0416627, size = 43, normalized size = 0.78 \[ \frac{c \left (3 c x^2-a\right )+b^2+3 b c x}{3 c^2 d (d (b+2 c x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(b^2 + 3*b*c*x + c*(-a + 3*c*x^2))/(3*c^2*d*(d*(b + 2*c*x))^(3/2))

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Maple [A]  time = 0.005, size = 45, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -3\,{c}^{2}{x}^{2}-3\,bxc+ac-{b}^{2} \right ) }{3\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(2*c*x+b)*(-3*c^2*x^2-3*b*c*x+a*c-b^2)/c^2/(2*c*d*x+b*d)^(5/2)

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Maxima [A]  time = 0.682727, size = 69, normalized size = 1.25 \[ \frac{\frac{b^{2} - 4 \, a c}{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c} + \frac{3 \, \sqrt{2 \, c d x + b d}}{c d^{2}}}{12 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*((b^2 - 4*a*c)/((2*c*d*x + b*d)^(3/2)*c) + 3*sqrt(2*c*d*x + b*d)/(c*d^2))/(
c*d)

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Fricas [A]  time = 0.20896, size = 73, normalized size = 1.33 \[ \frac{3 \, c^{2} x^{2} + 3 \, b c x + b^{2} - a c}{3 \,{\left (2 \, c^{3} d^{2} x + b c^{2} d^{2}\right )} \sqrt{2 \, c d x + b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*c^2*x^2 + 3*b*c*x + b^2 - a*c)/((2*c^3*d^2*x + b*c^2*d^2)*sqrt(2*c*d*x +
b*d))

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Sympy [A]  time = 3.87223, size = 235, normalized size = 4.27 \[ \begin{cases} - \frac{a c \sqrt{b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} + \frac{b^{2} \sqrt{b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} + \frac{3 b c x \sqrt{b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} + \frac{3 c^{2} x^{2} \sqrt{b d + 2 c d x}}{3 b^{2} c^{2} d^{3} + 12 b c^{3} d^{3} x + 12 c^{4} d^{3} x^{2}} & \text{for}\: c \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{\left (b d\right )^{\frac{5}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Piecewise((-a*c*sqrt(b*d + 2*c*d*x)/(3*b**2*c**2*d**3 + 12*b*c**3*d**3*x + 12*c*
*4*d**3*x**2) + b**2*sqrt(b*d + 2*c*d*x)/(3*b**2*c**2*d**3 + 12*b*c**3*d**3*x +
12*c**4*d**3*x**2) + 3*b*c*x*sqrt(b*d + 2*c*d*x)/(3*b**2*c**2*d**3 + 12*b*c**3*d
**3*x + 12*c**4*d**3*x**2) + 3*c**2*x**2*sqrt(b*d + 2*c*d*x)/(3*b**2*c**2*d**3 +
12*b*c**3*d**3*x + 12*c**4*d**3*x**2), Ne(c, 0)), ((a*x + b*x**2/2)/(b*d)**(5/2
), True))

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GIAC/XCAS [A]  time = 0.227693, size = 63, normalized size = 1.15 \[ \frac{b^{2} - 4 \, a c}{12 \,{\left (2 \, c d x + b d\right )}^{\frac{3}{2}} c^{2} d} + \frac{\sqrt{2 \, c d x + b d}}{4 \, c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*(b^2 - 4*a*c)/((2*c*d*x + b*d)^(3/2)*c^2*d) + 1/4*sqrt(2*c*d*x + b*d)/(c^2*
d^3)