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

Optimal. Leaf size=41 \[ -\frac{a}{3 c x^2 \sqrt{c x^2}}-\frac{b}{2 c x \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0077756, antiderivative size = 41, 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} \[ -\frac{a}{3 c x^2 \sqrt{c x^2}}-\frac{b}{2 c x \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x*(c*x^2)^(3/2)),x]

[Out]

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

Mathematica [A]  time = 0.0087747, size = 25, normalized size = 0.61 \[ \frac{c x^2 (-2 a-3 b x)}{6 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(c*x^2*(-2*a - 3*b*x))/(6*(c*x^2)^(5/2))

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Maple [A]  time = 0.003, size = 18, normalized size = 0.4 \begin{align*} -{\frac{3\,bx+2\,a}{6} \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)/x/(c*x^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.12331, size = 26, normalized size = 0.63 \begin{align*} -\frac{b}{2 \, c^{\frac{3}{2}} x^{2}} - \frac{a}{3 \, c^{\frac{3}{2}} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45478, size = 57, normalized size = 1.39 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (3 \, b x + 2 \, a\right )}}{6 \, c^{2} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.637379, size = 32, normalized size = 0.78 \begin{align*} - \frac{a}{3 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)/x/(c*x**2)**(3/2),x)

[Out]

-a/(3*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*} \int \frac{b x + a}{\left (c x^{2}\right )^{\frac{3}{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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