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

Optimal. Leaf size=35 \[ -\frac{a}{3 x^2 \sqrt{c x^2}}-\frac{b}{2 x \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0067594, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {15, 43} \[ -\frac{a}{3 x^2 \sqrt{c x^2}}-\frac{b}{2 x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^3*Sqrt[c*x^2]),x]

[Out]

-a/(3*x^2*Sqrt[c*x^2]) - b/(2*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 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}{x^3 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{a+b x}{x^4} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^4}+\frac{b}{x^3}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a}{3 x^2 \sqrt{c x^2}}-\frac{b}{2 x \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0061041, size = 22, normalized size = 0.63 \[ \frac{c (-2 a-3 b x)}{6 \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*(-2*a - 3*b*x))/(6*(c*x^2)^(3/2))

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{3\,bx+2\,a}{6\,{x}^{2}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*(3*b*x+2*a)/x^2/(c*x^2)^(1/2)

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Maxima [A]  time = 1.02184, size = 26, normalized size = 0.74 \begin{align*} -\frac{b}{2 \, \sqrt{c} x^{2}} - \frac{a}{3 \, \sqrt{c} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54425, size = 54, normalized size = 1.54 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(c*x^2)*(3*b*x + 2*a)/(c*x^4)

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Sympy [A]  time = 0.592965, size = 36, normalized size = 1.03 \begin{align*} - \frac{a}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} - \frac{b}{2 \sqrt{c} x \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b x + a}{\sqrt{c x^{2}} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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