3.2820 \(\int (\frac{c}{a+b x})^{3/2} \, dx\)

Optimal. Leaf size=19 \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

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Rubi [A]  time = 0.0060305, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {247, 15, 30} \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c/(a + b*x))^(3/2),x]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

Rule 247

Int[((a_.) + (b_.)*(v_)^(n_))^(p_), x_Symbol] :> Dist[1/Coefficient[v, x, 1], Subst[Int[(a + b*x^n)^p, x], x,
v], x] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && NeQ[v, x]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \left (\frac{c}{a+b x}\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{c}{x}\right )^{3/2} \, dx,x,a+b x\right )}{b}\\ &=\frac{\left (c \sqrt{\frac{c}{a+b x}} \sqrt{a+b x}\right ) \operatorname{Subst}\left (\int \frac{1}{x^{3/2}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{2 c \sqrt{\frac{c}{a+b x}}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0061148, size = 19, normalized size = 1. \[ -\frac{2 c \sqrt{\frac{c}{a+b x}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c/(a + b*x))^(3/2),x]

[Out]

(-2*c*Sqrt[c/(a + b*x)])/b

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Maple [A]  time = 0.002, size = 22, normalized size = 1.2 \begin{align*} -2\,{\frac{bx+a}{b} \left ({\frac{c}{bx+a}} \right ) ^{3/2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/(b*x+a))^(3/2),x)

[Out]

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

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Maxima [A]  time = 0.956174, size = 23, normalized size = 1.21 \begin{align*} -\frac{2 \, c \sqrt{\frac{c}{b x + a}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*sqrt(c/(b*x + a))/b

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Fricas [A]  time = 1.33943, size = 35, normalized size = 1.84 \begin{align*} -\frac{2 \, c \sqrt{\frac{c}{b x + a}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c*sqrt(c/(b*x + a))/b

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Sympy [A]  time = 0.842649, size = 48, normalized size = 2.53 \begin{align*} \begin{cases} - \frac{2 a c^{\frac{3}{2}} \left (\frac{1}{a + b x}\right )^{\frac{3}{2}}}{b} - 2 c^{\frac{3}{2}} x \left (\frac{1}{a + b x}\right )^{\frac{3}{2}} & \text{for}\: b \neq 0 \\x \left (\frac{c}{a}\right )^{\frac{3}{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a))**(3/2),x)

[Out]

Piecewise((-2*a*c**(3/2)*(1/(a + b*x))**(3/2)/b - 2*c**(3/2)*x*(1/(a + b*x))**(3/2), Ne(b, 0)), (x*(c/a)**(3/2
), True))

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Giac [A]  time = 1.09958, size = 32, normalized size = 1.68 \begin{align*} -\frac{2 \, c^{2} \mathrm{sgn}\left (b x + a\right )}{\sqrt{b c x + a c} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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