3.1428 \(\int \frac{a+b x}{(c+d x)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2} \]

[Out]

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

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Rubi [A]  time = 0.0431119, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.96988, size = 36, normalized size = 0.95 \[ \frac{2 b \sqrt{c + d x}}{d^{2}} - \frac{2 \left (a d - b c\right )}{d^{2} \sqrt{c + d x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0252015, size = 27, normalized size = 0.71 \[ \frac{2 (-a d+2 b c+b d x)}{d^2 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 26, normalized size = 0.7 \[ -2\,{\frac{-bdx+ad-2\,bc}{\sqrt{dx+c}{d}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(d*x+c)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.33906, size = 50, normalized size = 1.32 \[ \frac{2 \,{\left (\frac{\sqrt{d x + c} b}{d} + \frac{b c - a d}{\sqrt{d x + c} d}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.202807, size = 34, normalized size = 0.89 \[ \frac{2 \,{\left (b d x + 2 \, b c - a d\right )}}{\sqrt{d x + c} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.983132, size = 60, normalized size = 1.58 \[ \begin{cases} - \frac{2 a}{d \sqrt{c + d x}} + \frac{4 b c}{d^{2} \sqrt{c + d x}} + \frac{2 b x}{d \sqrt{c + d x}} & \text{for}\: d \neq 0 \\\frac{a x + \frac{b x^{2}}{2}}{c^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217007, size = 46, normalized size = 1.21 \[ \frac{2 \, \sqrt{d x + c} b}{d^{2}} + \frac{2 \,{\left (b c - a d\right )}}{\sqrt{d x + c} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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