3.594 \(\int \frac{\sqrt{a+b x^2}}{(c x)^{3/2}} \, dx\)

Optimal. Leaf size=263 \[ \frac{2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} \text{EllipticF}\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right ),\frac{1}{2}\right )}{c^{3/2} \sqrt{a+b x^2}}+\frac{4 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{c^2 \left (\sqrt{a}+\sqrt{b} x\right )}-\frac{4 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{c^{3/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{a+b x^2}}{c \sqrt{c x}} \]

[Out]

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

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Rubi [A]  time = 0.194235, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {277, 329, 305, 220, 1196} \[ \frac{4 \sqrt{b} \sqrt{c x} \sqrt{a+b x^2}}{c^2 \left (\sqrt{a}+\sqrt{b} x\right )}+\frac{2 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{c^{3/2} \sqrt{a+b x^2}}-\frac{4 \sqrt [4]{a} \sqrt [4]{b} \left (\sqrt{a}+\sqrt{b} x\right ) \sqrt{\frac{a+b x^2}{\left (\sqrt{a}+\sqrt{b} x\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} \sqrt{c x}}{\sqrt [4]{a} \sqrt{c}}\right )|\frac{1}{2}\right )}{c^{3/2} \sqrt{a+b x^2}}-\frac{2 \sqrt{a+b x^2}}{c \sqrt{c x}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*x^2]/(c*x)^(3/2),x]

[Out]

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

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 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 305

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, Int[1/Sqrt[a + b*x^4], x],
x] - Dist[1/q, Int[(1 - q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rule 1196

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, -Simp[(d*x*Sqrt[a + c
*x^4])/(a*(1 + q^2*x^2)), x] + Simp[(d*(1 + q^2*x^2)*Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]*EllipticE[2*ArcTan[
q*x], 1/2])/(q*Sqrt[a + c*x^4]), x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

Rubi steps

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

Mathematica [C]  time = 0.0137922, size = 54, normalized size = 0.21 \[ -\frac{2 x \sqrt{a+b x^2} \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};-\frac{b x^2}{a}\right )}{(c x)^{3/2} \sqrt{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.026, size = 194, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{\sqrt{b{x}^{2}+a}c\sqrt{cx}} \left ( 2\,\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticE} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) a-\sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{2}\sqrt{{\frac{-bx+\sqrt{-ab}}{\sqrt{-ab}}}}\sqrt{-{\frac{bx}{\sqrt{-ab}}}}{\it EllipticF} \left ( \sqrt{{\frac{bx+\sqrt{-ab}}{\sqrt{-ab}}}},1/2\,\sqrt{2} \right ) a-b{x}^{2}-a \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/
2))^(1/2)*EllipticE(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a-((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(
1/2)*2^(1/2)*((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-x*b/(-a*b)^(1/2))^(1/2)*EllipticF(((b*x+(-a*b)^(1/2))/
(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*a-b*x^2-a)/(b*x^2+a)^(1/2)/c/(c*x)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 1.37052, size = 49, normalized size = 0.19 \begin{align*} \frac{\sqrt{a} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{3}{2}} \sqrt{x} \Gamma \left (\frac{3}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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