3.962 \(\int \frac {1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx\)

Optimal. Leaf size=68 \[ -\frac {2 \sqrt {b} \sqrt {c x} \sqrt [4]{1-\frac {a}{b x^2}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a-b x^2}} \]

[Out]

-2*(1-a/b/x^2)^(1/4)*(cos(1/2*arccsc(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arccsc(x*b^(1/2)/a^(1/2)))*EllipticE
(sin(1/2*arccsc(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2)*(c*x)^(1/2)/c^2/(-b*x^2+a)^(1/4)/a^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {317, 335, 228} \[ -\frac {2 \sqrt {b} \sqrt {c x} \sqrt [4]{1-\frac {a}{b x^2}} E\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} c^2 \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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]

Rubi steps

\begin {align*} \int \frac {1}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \, dx &=\frac {\left (\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}{c^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (\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 )}{c^2 \sqrt [4]{a-b x^2}}\\ &=-\frac {2 \sqrt {b} \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 {a} c^2 \sqrt [4]{a-b x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 55, normalized size = 0.81 \[ -\frac {2 x \sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {3}{4};\frac {b x^2}{a}\right )}{(c x)^{3/2} \sqrt [4]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x}}{b c^{2} x^{4} - a c^{2} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(3/2)/(-b*x^2+a)^(1/4),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {3}{2}} \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(3/2)/(-b*x^2+a)^(1/4),x, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (c\,x\right )}^{3/2}\,{\left (a-b\,x^2\right )}^{1/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((c*x)^(3/2)*(a - b*x^2)^(1/4)),x)

[Out]

int(1/((c*x)^(3/2)*(a - b*x^2)^(1/4)), x)

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sympy [C]  time = 2.06, size = 32, normalized size = 0.47 \[ \frac {i e^{\frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {a}{b x^{2}}} \right )}}{\sqrt [4]{b} c^{\frac {3}{2}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(3/2)/(-b*x**2+a)**(1/4),x)

[Out]

I*exp(I*pi/4)*hyper((1/4, 1/2), (3/2,), a/(b*x**2))/(b**(1/4)*c**(3/2)*x)

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