3.41 \(\int \frac{1}{(c+d (a+b x))^{3/2}} \, dx\)

Optimal. Leaf size=21 \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

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Rubi [A]  time = 0.0242598, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*(a + b*x))^(-3/2),x]

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

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Rubi in Sympy [A]  time = 2.18362, size = 17, normalized size = 0.81 \[ - \frac{2}{b d \sqrt{c + d \left (a + b x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(c+d*(b*x+a))**(3/2),x)

[Out]

-2/(b*d*sqrt(c + d*(a + b*x)))

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Mathematica [A]  time = 0.0129472, size = 21, normalized size = 1. \[ -\frac{2}{b d \sqrt{d (a+b x)+c}} \]

Antiderivative was successfully verified.

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

[Out]

-2/(b*d*Sqrt[c + d*(a + b*x)])

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Maple [A]  time = 0.003, size = 20, normalized size = 1. \[ -2\,{\frac{1}{\sqrt{bdx+ad+c}db}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(c+d*(b*x+a))^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.34253, size = 26, normalized size = 1.24 \[ -\frac{2}{\sqrt{{\left (b x + a\right )} d + c} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt((b*x + a)*d + c)*b*d)

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Fricas [A]  time = 0.201832, size = 26, normalized size = 1.24 \[ -\frac{2}{\sqrt{b d x + a d + c} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt(b*d*x + a*d + c)*b*d)

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Sympy [A]  time = 4.44539, size = 58, normalized size = 2.76 \[ \begin{cases} \frac{x}{c^{\frac{3}{2}}} & \text{for}\: b = 0 \wedge d = 0 \\\frac{x}{\left (a d + c\right )^{\frac{3}{2}}} & \text{for}\: b = 0 \\\frac{x}{c^{\frac{3}{2}}} & \text{for}\: d = 0 \\- \frac{2 \sqrt{a d + b d x + c}}{a b d^{2} + b^{2} d^{2} x + b c d} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(c+d*(b*x+a))**(3/2),x)

[Out]

Piecewise((x/c**(3/2), Eq(b, 0) & Eq(d, 0)), (x/(a*d + c)**(3/2), Eq(b, 0)), (x/
c**(3/2), Eq(d, 0)), (-2*sqrt(a*d + b*d*x + c)/(a*b*d**2 + b**2*d**2*x + b*c*d),
True))

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GIAC/XCAS [A]  time = 0.211734, size = 26, normalized size = 1.24 \[ -\frac{2}{\sqrt{{\left (b x + a\right )} d + c} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/(sqrt((b*x + a)*d + c)*b*d)