3.303 \(\int \frac{1}{x^{3/2} \sqrt{a x^2+b x^5}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 \sqrt{a x^2+b x^5}}{3 a x^{5/2}} \]

[Out]

(-2*Sqrt[a*x^2 + b*x^5])/(3*a*x^(5/2))

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Rubi [A]  time = 0.0400189, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {2014} \[ -\frac{2 \sqrt{a x^2+b x^5}}{3 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*Sqrt[a*x^2 + b*x^5])/(3*a*x^(5/2))

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{1}{x^{3/2} \sqrt{a x^2+b x^5}} \, dx &=-\frac{2 \sqrt{a x^2+b x^5}}{3 a x^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0108333, size = 27, normalized size = 1. \[ -\frac{2 \sqrt{x^2 \left (a+b x^3\right )}}{3 a x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*Sqrt[a*x^2 + b*x^5]),x]

[Out]

(-2*Sqrt[x^2*(a + b*x^3)])/(3*a*x^(5/2))

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Maple [A]  time = 0.003, size = 29, normalized size = 1.1 \begin{align*} -{\frac{2\,b{x}^{3}+2\,a}{3\,a}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{b{x}^{5}+a{x}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(3/2)/(b*x^5+a*x^2)^(1/2),x)

[Out]

-2/3*(b*x^3+a)/x^(1/2)/a/(b*x^5+a*x^2)^(1/2)

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Maxima [A]  time = 1.11984, size = 35, normalized size = 1.3 \begin{align*} -\frac{2 \,{\left (b x^{4} + a x\right )}}{3 \, \sqrt{b x^{3} + a} a x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.920576, size = 51, normalized size = 1.89 \begin{align*} -\frac{2 \, \sqrt{b x^{5} + a x^{2}}}{3 \, a x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="fricas")

[Out]

-2/3*sqrt(b*x^5 + a*x^2)/(a*x^(5/2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{\frac{3}{2}} \sqrt{x^{2} \left (a + b x^{3}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**(3/2)/(b*x**5+a*x**2)**(1/2),x)

[Out]

Integral(1/(x**(3/2)*sqrt(x**2*(a + b*x**3))), x)

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Giac [A]  time = 1.4494, size = 31, normalized size = 1.15 \begin{align*} -\frac{2 \, \sqrt{b + \frac{a}{x^{3}}}}{3 \, a} + \frac{2 \, \sqrt{b}}{3 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(3/2)/(b*x^5+a*x^2)^(1/2),x, algorithm="giac")

[Out]

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