3.1254 \(\int \frac{a+b x+c x^2}{(b d+2 c d x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0725177, 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 \[ \frac{b^2-4 a c}{4 c^2 d \sqrt{b d+2 c d x}}+\frac{(b d+2 c d x)^{3/2}}{12 c^2 d^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.2419, size = 48, normalized size = 0.87 \[ \frac{- a c + \frac{b^{2}}{4}}{c^{2} d \sqrt{b d + 2 c d x}} + \frac{\left (b d + 2 c d x\right )^{\frac{3}{2}}}{12 c^{2} d^{3}} \]

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)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0358615, size = 41, normalized size = 0.75 \[ \frac{c \left (c x^2-3 a\right )+b^2+b c x}{3 c^2 d \sqrt{d (b+2 c x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x + c*x^2)/(b*d + 2*c*d*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.006, size = 46, normalized size = 0.8 \[ -{\frac{ \left ( 2\,cx+b \right ) \left ( -{c}^{2}{x}^{2}-bxc+3\,ac-{b}^{2} \right ) }{3\,{c}^{2}} \left ( 2\,cdx+bd \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)/(2*c*d*x+b*d)^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.685951, size = 69, normalized size = 1.25 \[ \frac{\frac{3 \,{\left (b^{2} - 4 \, a c\right )}}{\sqrt{2 \, c d x + b d} c} + \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{c d^{2}}}{12 \, c d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.209416, size = 51, normalized size = 0.93 \[ \frac{c^{2} x^{2} + b c x + b^{2} - 3 \, a c}{3 \, \sqrt{2 \, c d x + b d} c^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 13.7456, size = 840, normalized size = 15.27 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x+a)/(2*c*d*x+b*d)**(3/2),x)

[Out]

-a/(c*d*sqrt(b*d + 2*c*d*x)) + b*Piecewise((b*sqrt(b*d + 2*c*d*x)/(b*c**2*d**2 +
2*c**3*d**2*x) + c*x*sqrt(b*d + 2*c*d*x)/(b*c**2*d**2 + 2*c**3*d**2*x), Ne(b, 0
)), (sqrt(2)*sqrt(x)/(2*c**(3/2)*d**(3/2)), True)) + c*(-2*b**(19/2)*sqrt(1 + 2*
c*x/b)/(3*b**8*c**3*d**(3/2) + 18*b**7*c**4*d**(3/2)*x + 36*b**6*c**5*d**(3/2)*x
**2 + 24*b**5*c**6*d**(3/2)*x**3) + 2*b**(19/2)/(3*b**8*c**3*d**(3/2) + 18*b**7*
c**4*d**(3/2)*x + 36*b**6*c**5*d**(3/2)*x**2 + 24*b**5*c**6*d**(3/2)*x**3) - 10*
b**(17/2)*c*x*sqrt(1 + 2*c*x/b)/(3*b**8*c**3*d**(3/2) + 18*b**7*c**4*d**(3/2)*x
+ 36*b**6*c**5*d**(3/2)*x**2 + 24*b**5*c**6*d**(3/2)*x**3) + 12*b**(17/2)*c*x/(3
*b**8*c**3*d**(3/2) + 18*b**7*c**4*d**(3/2)*x + 36*b**6*c**5*d**(3/2)*x**2 + 24*
b**5*c**6*d**(3/2)*x**3) - 15*b**(15/2)*c**2*x**2*sqrt(1 + 2*c*x/b)/(3*b**8*c**3
*d**(3/2) + 18*b**7*c**4*d**(3/2)*x + 36*b**6*c**5*d**(3/2)*x**2 + 24*b**5*c**6*
d**(3/2)*x**3) + 24*b**(15/2)*c**2*x**2/(3*b**8*c**3*d**(3/2) + 18*b**7*c**4*d**
(3/2)*x + 36*b**6*c**5*d**(3/2)*x**2 + 24*b**5*c**6*d**(3/2)*x**3) - 4*b**(13/2)
*c**3*x**3*sqrt(1 + 2*c*x/b)/(3*b**8*c**3*d**(3/2) + 18*b**7*c**4*d**(3/2)*x + 3
6*b**6*c**5*d**(3/2)*x**2 + 24*b**5*c**6*d**(3/2)*x**3) + 16*b**(13/2)*c**3*x**3
/(3*b**8*c**3*d**(3/2) + 18*b**7*c**4*d**(3/2)*x + 36*b**6*c**5*d**(3/2)*x**2 +
24*b**5*c**6*d**(3/2)*x**3) + 4*b**(11/2)*c**4*x**4*sqrt(1 + 2*c*x/b)/(3*b**8*c*
*3*d**(3/2) + 18*b**7*c**4*d**(3/2)*x + 36*b**6*c**5*d**(3/2)*x**2 + 24*b**5*c**
6*d**(3/2)*x**3))

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GIAC/XCAS [A]  time = 0.231606, size = 63, normalized size = 1.15 \[ \frac{b^{2} - 4 \, a c}{4 \, \sqrt{2 \, c d x + b d} c^{2} d} + \frac{{\left (2 \, c d x + b d\right )}^{\frac{3}{2}}}{12 \, c^{2} d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)/(2*c*d*x + b*d)^(3/2),x, algorithm="giac")

[Out]

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