3.2833 \(\int \sqrt{\frac{c}{(a+b x)^3}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]

[Out]

(-2*Sqrt[c/(a + b*x)^3]*(a + b*x))/b

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Rubi [A]  time = 0.0076797, antiderivative size = 23, 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} \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c/(a + b*x)^3],x]

[Out]

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

Mathematica [A]  time = 0.0077133, size = 23, normalized size = 1. \[ -\frac{2 (a+b x) \sqrt{\frac{c}{(a+b x)^3}}}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c/(a + b*x)^3],x]

[Out]

(-2*Sqrt[c/(a + b*x)^3]*(a + b*x))/b

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Maple [A]  time = 0.002, size = 22, normalized size = 1. \begin{align*} -2\,{\frac{bx+a}{b}\sqrt{{\frac{c}{ \left ( bx+a \right ) ^{3}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c/(b*x+a)^3)^(1/2),x)

[Out]

-2*(b*x+a)*(c/(b*x+a)^3)^(1/2)/b

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Maxima [A]  time = 1.45024, size = 32, normalized size = 1.39 \begin{align*} -\frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a)^3)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.34563, size = 89, normalized size = 3.87 \begin{align*} -\frac{2 \,{\left (b x + a\right )} \sqrt{\frac{c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a)^3)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.944818, size = 97, normalized size = 4.22 \begin{align*} \begin{cases} - \frac{2 a \sqrt{c} \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}}}{b} - 2 \sqrt{c} x \sqrt{\frac{1}{a^{3} + 3 a^{2} b x + 3 a b^{2} x^{2} + b^{3} x^{3}}} & \text{for}\: b \neq 0 \\x \sqrt{\frac{c}{a^{3}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c/(b*x+a)**3)**(1/2),x)

[Out]

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

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Giac [B]  time = 1.34431, size = 68, normalized size = 2.96 \begin{align*} -\frac{2 \, c \mathrm{sgn}\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right ) \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)^(1/2),x, algorithm="giac")

[Out]

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