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 = 0.0142636, 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, Rules used = {43} \[ \frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2} \]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{a+b x}{(c+d x)^{3/2}} \, dx &=\int \left (\frac{-b c+a d}{d (c+d x)^{3/2}}+\frac{b}{d \sqrt{c+d x}}\right ) \, dx\\ &=\frac{2 (b c-a d)}{d^2 \sqrt{c+d x}}+\frac{2 b \sqrt{c+d x}}{d^2}\\ \end{align*}

Mathematica [A]  time = 0.0193362, 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 = 0.003, size = 26, normalized size = 0.7 \begin{align*} -2\,{\frac{-bdx+ad-2\,bc}{\sqrt{dx+c}{d}^{2}}} \end{align*}

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 = 0.946742, size = 50, normalized size = 1.32 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{d x + c} b}{d} + \frac{b c - a d}{\sqrt{d x + c} d}\right )}}{d} \end{align*}

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 = 1.97996, size = 74, normalized size = 1.95 \begin{align*} \frac{2 \,{\left (b d x + 2 \, b c - a d\right )} \sqrt{d x + c}}{d^{3} x + c d^{2}} \end{align*}

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^3*x + c*d^2)

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Sympy [A]  time = 0.577191, size = 60, normalized size = 1.58 \begin{align*} \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} \end{align*}

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 [A]  time = 1.06666, size = 46, normalized size = 1.21 \begin{align*} \frac{2 \, \sqrt{d x + c} b}{d^{2}} + \frac{2 \,{\left (b c - a d\right )}}{\sqrt{d x + c} d^{2}} \end{align*}

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)