3.426 \(\int \frac{A+B x}{(a+b x)^{3/2}} \, dx\)

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

[Out]

(-2*(A*b - a*B))/(b^2*Sqrt[a + b*x]) + (2*B*Sqrt[a + b*x])/b^2

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Rubi [A]  time = 0.0435026, 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 \[ \frac{2 B \sqrt{a+b x}}{b^2}-\frac{2 (A b-a B)}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(A*b - a*B))/(b^2*Sqrt[a + b*x]) + (2*B*Sqrt[a + b*x])/b^2

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Rubi in Sympy [A]  time = 7.57542, size = 36, normalized size = 0.95 \[ \frac{2 B \sqrt{a + b x}}{b^{2}} - \frac{2 \left (A b - B a\right )}{b^{2} \sqrt{a + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(3/2),x)

[Out]

2*B*sqrt(a + b*x)/b**2 - 2*(A*b - B*a)/(b**2*sqrt(a + b*x))

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Mathematica [A]  time = 0.0266815, size = 27, normalized size = 0.71 \[ \frac{2 (2 a B-A b+b B x)}{b^2 \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-(A*b) + 2*a*B + b*B*x))/(b^2*Sqrt[a + b*x])

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Maple [A]  time = 0.006, size = 26, normalized size = 0.7 \[ -2\,{\frac{-bBx+Ab-2\,Ba}{\sqrt{bx+a}{b}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/(b*x+a)^(3/2),x)

[Out]

-2/(b*x+a)^(1/2)*(-B*b*x+A*b-2*B*a)/b^2

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Maxima [A]  time = 1.33647, size = 50, normalized size = 1.32 \[ \frac{2 \,{\left (\frac{\sqrt{b x + a} B}{b} + \frac{B a - A b}{\sqrt{b x + a} b}\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(sqrt(b*x + a)*B/b + (B*a - A*b)/(sqrt(b*x + a)*b))/b

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Fricas [A]  time = 0.205247, size = 34, normalized size = 0.89 \[ \frac{2 \,{\left (B b x + 2 \, B a - A b\right )}}{\sqrt{b x + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(B*b*x + 2*B*a - A*b)/(sqrt(b*x + a)*b^2)

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Sympy [A]  time = 2.21517, size = 60, normalized size = 1.58 \[ \begin{cases} - \frac{2 A}{b \sqrt{a + b x}} + \frac{4 B a}{b^{2} \sqrt{a + b x}} + \frac{2 B x}{b \sqrt{a + b x}} & \text{for}\: b \neq 0 \\\frac{A x + \frac{B x^{2}}{2}}{a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/(b*x+a)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.249777, size = 46, normalized size = 1.21 \[ \frac{2 \, \sqrt{b x + a} B}{b^{2}} + \frac{2 \,{\left (B a - A b\right )}}{\sqrt{b x + a} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(b*x + a)*B/b^2 + 2*(B*a - A*b)/(sqrt(b*x + a)*b^2)