3.961 \(\int \frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}} \, dx\)

Optimal. Leaf size=90 \[ \frac{\sqrt{a} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^2}}-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}} \]

[Out]

-((c*(a - b*x^2)^(3/4))/(b*Sqrt[c*x])) + (Sqrt[a]*(1 - a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCsc[(Sqrt[b]*x)
/Sqrt[a]]/2, 2])/(Sqrt[b]*(a - b*x^2)^(1/4))

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Rubi [A]  time = 0.0357489, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {315, 317, 335, 228} \[ \frac{\sqrt{a} \sqrt{c x} \sqrt [4]{1-\frac{a}{b x^2}} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^2}}-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c*x]/(a - b*x^2)^(1/4),x]

[Out]

-((c*(a - b*x^2)^(3/4))/(b*Sqrt[c*x])) + (Sqrt[a]*(1 - a/(b*x^2))^(1/4)*Sqrt[c*x]*EllipticE[ArcCsc[(Sqrt[b]*x)
/Sqrt[a]]/2, 2])/(Sqrt[b]*(a - b*x^2)^(1/4))

Rule 315

Int[Sqrt[(c_)*(x_)]/((a_) + (b_.)*(x_)^2)^(1/4), x_Symbol] :> Simp[(c*(a + b*x^2)^(3/4))/(b*Sqrt[c*x]), x] + D
ist[(a*c^2)/(2*b), Int[1/((c*x)^(3/2)*(a + b*x^2)^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]

Rule 317

Int[1/(((c_.)*(x_))^(3/2)*((a_) + (b_.)*(x_)^2)^(1/4)), x_Symbol] :> Dist[(Sqrt[c*x]*(1 + a/(b*x^2))^(1/4))/(c
^2*(a + b*x^2)^(1/4)), Int[1/(x^2*(1 + a/(b*x^2))^(1/4)), x], x] /; FreeQ[{a, b, c}, x] && NegQ[b/a]

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 228

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \frac{\sqrt{c x}}{\sqrt [4]{a-b x^2}} \, dx &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}-\frac{\left (a c^2\right ) \int \frac{1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx}{2 b}\\ &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}-\frac{\left (a \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \int \frac{1}{\sqrt [4]{1-\frac{a}{b x^2}} x^2} \, dx}{2 b \sqrt [4]{a-b x^2}}\\ &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}+\frac{\left (a \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [4]{1-\frac{a x^2}{b}}} \, dx,x,\frac{1}{x}\right )}{2 b \sqrt [4]{a-b x^2}}\\ &=-\frac{c \left (a-b x^2\right )^{3/4}}{b \sqrt{c x}}+\frac{\sqrt{a} \sqrt [4]{1-\frac{a}{b x^2}} \sqrt{c x} E\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{b} \sqrt [4]{a-b x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0121084, size = 57, normalized size = 0.63 \[ \frac{2 x \sqrt{c x} \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{b x^2}{a}\right )}{3 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*Sqrt[c*x]*(1 - (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, (b*x^2)/a])/(3*(a - b*x^2)^(1/4))

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Maple [F]  time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{cx}{\frac{1}{\sqrt [4]{-b{x}^{2}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{b x^{2} - a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.01386, size = 46, normalized size = 0.51 \begin{align*} \frac{\sqrt{c} x^{\frac{3}{2}} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \Gamma \left (\frac{7}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(c)*x**(3/2)*gamma(3/4)*hyper((1/4, 3/4), (7/4,), b*x**2*exp_polar(2*I*pi)/a)/(2*a**(1/4)*gamma(7/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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