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

Optimal. Leaf size=92 \[ \frac {\sqrt {c x} \sqrt [4]{a-b x^2}}{c}-\frac {\sqrt {a} \sqrt {b} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),2\right )}{c^2 \left (a-b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {279, 329, 237, 335, 275, 232} \[ \frac {\sqrt {c x} \sqrt [4]{a-b x^2}}{c}-\frac {\sqrt {a} \sqrt {b} (c x)^{3/2} \left (1-\frac {a}{b x^2}\right )^{3/4} F\left (\left .\frac {1}{2} \csc ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{c^2 \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(1/4)/Sqrt[c*x],x]

[Out]

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

Rule 232

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(3/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 237

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

Rule 275

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

Rule 279

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

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

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

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Mathematica [C]  time = 0.01, size = 55, normalized size = 0.60 \[ \frac {2 x \sqrt [4]{a-b x^2} \, _2F_1\left (-\frac {1}{4},\frac {1}{4};\frac {5}{4};\frac {b x^2}{a}\right )}{\sqrt {c x} \sqrt [4]{1-\frac {b x^2}{a}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(a - b*x^2)^(1/4)*Hypergeometric2F1[-1/4, 1/4, 5/4, (b*x^2)/a])/(Sqrt[c*x]*(1 - (b*x^2)/a)^(1/4))

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fricas [F]  time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{c x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{\sqrt {c x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a - b*x^2)^(1/4)/(c*x)^(1/2),x)

[Out]

int((a - b*x^2)^(1/4)/(c*x)^(1/2), x)

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sympy [C]  time = 1.34, size = 39, normalized size = 0.42 \[ - \frac {i \sqrt [4]{b} x e^{\frac {3 i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {a}{b x^{2}}} \right )}}{\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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