3.2965 \(\int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^2} \, dx\)

Optimal. Leaf size=355 \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}\right ),-7-4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{x} \]

[Out]

-(Sqrt[a + b*Sqrt[c*x^3]]/x) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/3)*c^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3
) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt
[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(Sqrt[(a^(1/3)*(a^(1/3
) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[a
+ b*Sqrt[c*x^3]])

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Rubi [A]  time = 0.148777, antiderivative size = 355, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {369, 341, 277, 218} \[ \frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1-\sqrt{3}\right ) \sqrt [3]{a}}{\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}+\left (1+\sqrt{3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}-\frac{\sqrt{a+b \sqrt{c x^3}}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c*x^3]]/x^2,x]

[Out]

-(Sqrt[a + b*Sqrt[c*x^3]]/x) + (3^(3/4)*Sqrt[2 + Sqrt[3]]*b^(2/3)*c^(1/3)*(a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqr
t[c*x^3])*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)*b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])/((1 + Sqrt[3])*a^(1/3
) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt
[c*x^3])/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])], -7 - 4*Sqrt[3]])/(Sqrt[(a^(1/3)*(a^(1/3
) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3]))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*c^(2/3)*x^2)/Sqrt[c*x^3])^2]*Sqrt[a
+ b*Sqrt[c*x^3]])

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rule 341

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

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 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^2} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x^{3/2}}}{x^2} \, dx,\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{\sqrt{a+b \sqrt{c} x^3}}{x^3} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{x}+\operatorname{Subst}\left (\frac{1}{2} \left (3 b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b \sqrt{c} x^3}} \, dx,x,\sqrt{x}\right ),\sqrt{x},\frac{\sqrt{c x^3}}{\sqrt{c} x}\right )\\ &=-\frac{\sqrt{a+b \sqrt{c x^3}}}{x}+\frac{3^{3/4} \sqrt{2+\sqrt{3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right ) \sqrt{\frac{a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac{\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}}\right )|-7-4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b} c^{2/3} x^2}{\sqrt{c x^3}}\right )^2}} \sqrt{a+b \sqrt{c x^3}}}\\ \end{align*}

Mathematica [F]  time = 0.0355507, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b \sqrt{c x^3}}}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.188, size = 304, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,x} \left ( i\sqrt{3}\sqrt [3]{-ac{b}^{2}}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}\sqrt{-2\,{\frac{-b\sqrt{c{x}^{3}}+\sqrt [3]{-ac{b}^{2}}x}{\sqrt [3]{-ac{b}^{2}}x \left ( i\sqrt{3}-3 \right ) }}}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}+2\,b\sqrt{c{x}^{3}}+\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}{\it EllipticF} \left ({\frac{\sqrt{2}\sqrt{3}}{6}\sqrt{{\frac{-i\sqrt{3}}{x} \left ( i\sqrt{3}x\sqrt [3]{-ac{b}^{2}}-2\,b\sqrt{c{x}^{3}}-\sqrt [3]{-ac{b}^{2}}x \right ){\frac{1}{\sqrt [3]{-ac{b}^{2}}}}}}},\sqrt{2}\sqrt{{\frac{i\sqrt{3}}{i\sqrt{3}-3}}} \right ) x+2\,a+2\,b\sqrt{c{x}^{3}} \right ){\frac{1}{\sqrt{a+b\sqrt{c{x}^{3}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(I*3^(1/2)*(-a*c*b^2)^(1/3)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/
2)/(-a*c*b^2)^(1/3)/x)^(1/2)*(-2*(-b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)/x/(-a*c*b^2)^(1/3)/(I*3^(1/2)-3))^(1/2)
*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a*c*b^2)^(1/3)/x)^(1/2)*Ell
ipticF(1/6*3^(1/2)*2^(1/2)*(-I*(I*3^(1/2)*x*(-a*c*b^2)^(1/3)-2*b*(c*x^3)^(1/2)-(-a*c*b^2)^(1/3)*x)*3^(1/2)/(-a
*c*b^2)^(1/3)/x)^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*x+2*a+2*b*(c*x^3)^(1/2))/x/(a+b*(c*x^3)^(1/2))
^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \sqrt{c x^{3}}}}{x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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