3.2816 \(\int \frac{1}{\sqrt{c (a+b x)^3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0110628, 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} \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[c*(a + b*x)^3],x]

[Out]

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

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

Mathematica [A]  time = 0.0089655, size = 23, normalized size = 1. \[ -\frac{2 (a+b x)}{b \sqrt{c (a+b x)^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[c*(a + b*x)^3],x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01131, size = 36, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (b \sqrt{c} x + a \sqrt{c}\right )}}{{\left (b x + a\right )}^{\frac{3}{2}} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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*c)

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Fricas [B]  time = 1.25826, size = 127, normalized size = 5.52 \begin{align*} -\frac{2 \, \sqrt{b^{3} c x^{3} + 3 \, a b^{2} c x^{2} + 3 \, a^{2} b c x + a^{3} c}}{b^{3} c x^{2} + 2 \, a b^{2} c x + a^{2} b c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \left (a + b x\right )^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(c*(a + b*x)**3), x)

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Giac [A]  time = 1.10042, size = 31, normalized size = 1.35 \begin{align*} -\frac{2}{\sqrt{b c x + a c} b \mathrm{sgn}\left (b x + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(sqrt(b*c*x + a*c)*b*sgn(b*x + a))