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

Optimal. Leaf size=29 \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]

[Out]

-(a + b*x)^2/(2*a*c*x*Sqrt[c*x^2])

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Rubi [A]  time = 0.0043576, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {15, 37} \[ -\frac{(a+b x)^2}{2 a c x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(c*x^2)^(3/2),x]

[Out]

-(a + b*x)^2/(2*a*c*x*Sqrt[c*x^2])

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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0027527, size = 22, normalized size = 0.76 \[ \frac{x (-a-2 b x)}{2 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(c*x^2)^(3/2),x]

[Out]

(x*(-a - 2*b*x))/(2*(c*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 17, normalized size = 0.6 \begin{align*} -{\frac{x \left ( 2\,bx+a \right ) }{2} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.06174, size = 31, normalized size = 1.07 \begin{align*} -\frac{b}{\sqrt{c x^{2}} c} - \frac{a}{2 \, c^{\frac{3}{2}} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.62874, size = 54, normalized size = 1.86 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )}}{2 \, c^{2} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.505787, size = 34, normalized size = 1.17 \begin{align*} - \frac{a x}{2 c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} - \frac{b x^{2}}{c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sage0*x