3.226 \(\int \frac{\sqrt{b x^2+c x^4}}{x^5} \, dx\)

Optimal. Leaf size=25 \[ -\frac{\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(3*b*x^6)

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Rubi [A]  time = 0.0393856, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2014} \[ -\frac{\left (b x^2+c x^4\right )^{3/2}}{3 b x^6} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[b*x^2 + c*x^4]/x^5,x]

[Out]

-(b*x^2 + c*x^4)^(3/2)/(3*b*x^6)

Rule 2014

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && N
eQ[n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{b x^2+c x^4}}{x^5} \, dx &=-\frac{\left (b x^2+c x^4\right )^{3/2}}{3 b x^6}\\ \end{align*}

Mathematica [A]  time = 0.0100465, size = 25, normalized size = 1. \[ -\frac{\left (x^2 \left (b+c x^2\right )\right )^{3/2}}{3 b x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[b*x^2 + c*x^4]/x^5,x]

[Out]

-(x^2*(b + c*x^2))^(3/2)/(3*b*x^6)

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Maple [A]  time = 0.045, size = 29, normalized size = 1.2 \begin{align*} -{\frac{c{x}^{2}+b}{3\,b{x}^{4}}\sqrt{c{x}^{4}+b{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2)^(1/2)/x^5,x)

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.5287, size = 62, normalized size = 2.48 \begin{align*} -\frac{\sqrt{c x^{4} + b x^{2}}{\left (c x^{2} + b\right )}}{3 \, b x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^5,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2)**(1/2)/x**5,x)

[Out]

Integral(sqrt(x**2*(b + c*x**2))/x**5, x)

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Giac [B]  time = 1.34027, size = 85, normalized size = 3.4 \begin{align*} \frac{2 \,{\left (3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{4} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right ) + b^{2} c^{\frac{3}{2}} \mathrm{sgn}\left (x\right )\right )}}{3 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2)^(1/2)/x^5,x, algorithm="giac")

[Out]

2/3*(3*(sqrt(c)*x - sqrt(c*x^2 + b))^4*c^(3/2)*sgn(x) + b^2*c^(3/2)*sgn(x))/((sqrt(c)*x - sqrt(c*x^2 + b))^2 -
b)^3