3.987 \(\int \frac{1}{\sqrt{c x} (a+b x^2)^{5/4}} \, dx\)

Optimal. Leaf size=26 \[ \frac{2 \sqrt{c x}}{a c \sqrt [4]{a+b x^2}} \]

[Out]

(2*Sqrt[c*x])/(a*c*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.0055901, antiderivative size = 26, 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 = {264} \[ \frac{2 \sqrt{c x}}{a c \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[c*x]*(a + b*x^2)^(5/4)),x]

[Out]

(2*Sqrt[c*x])/(a*c*(a + b*x^2)^(1/4))

Rule 264

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

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{c x} \left (a+b x^2\right )^{5/4}} \, dx &=\frac{2 \sqrt{c x}}{a c \sqrt [4]{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0060322, size = 24, normalized size = 0.92 \[ \frac{2 x}{a \sqrt{c x} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c*x]*(a + b*x^2)^(5/4)),x]

[Out]

(2*x)/(a*Sqrt[c*x]*(a + b*x^2)^(1/4))

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Maple [A]  time = 0.005, size = 21, normalized size = 0.8 \begin{align*} 2\,{\frac{x}{\sqrt [4]{b{x}^{2}+a}a\sqrt{cx}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \sqrt{c x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.50464, size = 69, normalized size = 2.65 \begin{align*} \frac{2 \,{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{a b c x^{2} + a^{2} c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 7.04361, size = 34, normalized size = 1.31 \begin{align*} \frac{\Gamma \left (\frac{1}{4}\right )}{2 a \sqrt [4]{b} \sqrt{c} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(1/4)/(2*a*b**(1/4)*sqrt(c)*(a/(b*x**2) + 1)**(1/4)*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \sqrt{c x}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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