3.787 \(\int \frac{a+b x}{x^4 \sqrt{c x^2}} \, dx\)

Optimal. Leaf size=35 \[ -\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}} \]

[Out]

-a/(4*x^3*Sqrt[c*x^2]) - b/(3*x^2*Sqrt[c*x^2])

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Rubi [A]  time = 0.0067296, 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} \[ -\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/(x^4*Sqrt[c*x^2]),x]

[Out]

-a/(4*x^3*Sqrt[c*x^2]) - b/(3*x^2*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^4 \sqrt{c x^2}} \, dx &=\frac{x \int \frac{a+b x}{x^5} \, dx}{\sqrt{c x^2}}\\ &=\frac{x \int \left (\frac{a}{x^5}+\frac{b}{x^4}\right ) \, dx}{\sqrt{c x^2}}\\ &=-\frac{a}{4 x^3 \sqrt{c x^2}}-\frac{b}{3 x^2 \sqrt{c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0049902, size = 24, normalized size = 0.69 \[ \frac{-3 a-4 b x}{12 x^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/(x^4*Sqrt[c*x^2]),x]

[Out]

(-3*a - 4*b*x)/(12*x^3*Sqrt[c*x^2])

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \begin{align*} -{\frac{4\,bx+3\,a}{12\,{x}^{3}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)/x^4/(c*x^2)^(1/2),x)

[Out]

-1/12*(4*b*x+3*a)/x^3/(c*x^2)^(1/2)

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Maxima [A]  time = 1.03886, size = 26, normalized size = 0.74 \begin{align*} -\frac{b}{3 \, \sqrt{c} x^{3}} - \frac{a}{4 \, \sqrt{c} x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.58097, size = 55, normalized size = 1.57 \begin{align*} -\frac{\sqrt{c x^{2}}{\left (4 \, b x + 3 \, a\right )}}{12 \, c x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x^4/(c*x^2)^(1/2),x, algorithm="fricas")

[Out]

-1/12*sqrt(c*x^2)*(4*b*x + 3*a)/(c*x^5)

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Sympy [A]  time = 0.729381, size = 37, normalized size = 1.06 \begin{align*} - \frac{a}{4 \sqrt{c} x^{3} \sqrt{x^{2}}} - \frac{b}{3 \sqrt{c} x^{2} \sqrt{x^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)/x**4/(c*x**2)**(1/2),x)

[Out]

-a/(4*sqrt(c)*x**3*sqrt(x**2)) - b/(3*sqrt(c)*x**2*sqrt(x**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)/x^4/(c*x^2)^(1/2),x, algorithm="giac")

[Out]

sage0*x