\(\int \sqrt {a+b \sqrt {c x^3}} \, dx\) [66]

Optimal result

Integrand size = 17, antiderivative size = 770 \[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\frac {4}{7} x \sqrt {a+b \sqrt {c x^3}}+\frac {12 a \sqrt {a+b \sqrt {c x^3}}}{7 b^{2/3} \sqrt [3]{c} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}-\frac {6 \sqrt [4]{3} \sqrt {2-\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} E\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right )|-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}}+\frac {4 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right ),-7-4 \sqrt {3}\right )}{7 b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}} \] Output:

4/7*x*(a+b*(c*x^3)^(1/2))^(1/2)+12/7*a*(a+b*(c*x^3)^(1/2))^(1/2)/b^(2/3)/c 
^(1/3)/((1+3^(1/2))*a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))-6/7*3^(1/4) 
*(1/2*6^(1/2)-1/2*2^(1/2))*a^(4/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1 
/2))*((a^(2/3)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2) 
)/((1+3^(1/2))*a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))^2)^(1/2)*Ellipti 
cE(((1-3^(1/2))*a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/((1+3^(1/2))*a^ 
(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2)),I*3^(1/2)+2*I)/b^(2/3)/c^(1/3)/(a 
^(1/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/((1+3^(1/2))*a^(1/3)+b^ 
(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))^2)^(1/2)/(a+b*(c*x^3)^(1/2))^(1/2)+4/7*2^ 
(1/2)*3^(3/4)*a^(4/3)*(a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))*((a^(2/3 
)+b^(2/3)*c^(1/3)*x-a^(1/3)*b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/((1+3^(1/2) 
)*a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))^2)^(1/2)*EllipticF(((1-3^(1/2 
))*a^(1/3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/((1+3^(1/2))*a^(1/3)+b^(1/3) 
*c^(2/3)*x^2/(c*x^3)^(1/2)),I*3^(1/2)+2*I)/b^(2/3)/c^(1/3)/(a^(1/3)*(a^(1/ 
3)+b^(1/3)*c^(2/3)*x^2/(c*x^3)^(1/2))/((1+3^(1/2))*a^(1/3)+b^(1/3)*c^(2/3) 
*x^2/(c*x^3)^(1/2))^2)^(1/2)/(a+b*(c*x^3)^(1/2))^(1/2)
 

Mathematica [F]

\[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int \sqrt {a+b \sqrt {c x^3}} \, dx \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.07 (sec) , antiderivative size = 822, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {787, 774, 811, 832, 759, 2416}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 787

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

\(\Big \downarrow \) 774

\(\displaystyle 2 \int \frac {\sqrt {c x^3} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}{\sqrt {c} x}d\frac {\sqrt {c x^3}}{\sqrt {c} x}\)

\(\Big \downarrow \) 811

\(\displaystyle 2 \left (\frac {3}{7} a \int \frac {\sqrt {c x^3}}{\sqrt {c} x \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}\right )\)

\(\Big \downarrow \) 832

\(\displaystyle 2 \left (\frac {3}{7} a \left (\frac {\int \frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt [3]{b} \sqrt [6]{c}}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt [3]{b} \sqrt [6]{c}}\right )+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}\right )\)

\(\Big \downarrow \) 759

\(\displaystyle 2 \left (\frac {3}{7} a \left (\frac {\int \frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}}{\sqrt [3]{b} \sqrt [6]{c}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right ) \sqrt {\frac {a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+b^{2/3} \sqrt [3]{c} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}\right )^2}} \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}\right )+\frac {2}{7} x \sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}\right )\)

\(\Big \downarrow \) 2416

\(\displaystyle 2 \left (\frac {2}{7} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a} x+\frac {3}{7} a \left (\frac {\frac {2 \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}{\sqrt [3]{b} \sqrt [6]{c} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )}-\frac {\sqrt [4]{3} \sqrt {2-\sqrt {3}} \sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} \sqrt [3]{c} x+a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} E\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right )|-7-4 \sqrt {3}\right )}{\sqrt [3]{b} \sqrt [6]{c} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}}{\sqrt [3]{b} \sqrt [6]{c}}-\frac {2 \left (1-\sqrt {3}\right ) \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right ) \sqrt {\frac {b^{2/3} \sqrt [3]{c} x+a^{2/3}-\frac {\sqrt [3]{a} \sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} b^{2/3} \sqrt [3]{c} \sqrt {\frac {\sqrt [3]{a} \left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\sqrt [3]{a}\right )}{\left (\frac {\sqrt [3]{b} \sqrt {c x^3}}{\sqrt [3]{c} x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}\right )^2}} \sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}\right )\right )\)

Input:

Int[Sqrt[a + b*Sqrt[c*x^3]],x]
 

Output:

2*((2*x*Sqrt[a + (b*(c*x^3)^(3/2))/(c*x^3)])/7 + (3*a*(((2*Sqrt[a + (b*(c* 
x^3)^(3/2))/(c*x^3)])/(b^(1/3)*c^(1/6)*((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*S 
qrt[c*x^3])/(c^(1/3)*x))) - (3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*(a^(1/3) + 
(b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^ 
(1/3)*b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)* 
Sqrt[c*x^3])/(c^(1/3)*x))^2]*EllipticE[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^ 
(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x 
^3])/(c^(1/3)*x))], -7 - 4*Sqrt[3]])/(b^(1/3)*c^(1/6)*Sqrt[(a^(1/3)*(a^(1/ 
3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x)))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3) 
*Sqrt[c*x^3])/(c^(1/3)*x))^2]*Sqrt[a + (b*(c*x^3)^(3/2))/(c*x^3)]))/(b^(1/ 
3)*c^(1/6)) - (2*(1 - Sqrt[3])*Sqrt[2 + Sqrt[3]]*a^(1/3)*(a^(1/3) + (b^(1/ 
3)*Sqrt[c*x^3])/(c^(1/3)*x))*Sqrt[(a^(2/3) + b^(2/3)*c^(1/3)*x - (a^(1/3)* 
b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c 
*x^3])/(c^(1/3)*x))^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + (b^(1/3)* 
Sqrt[c*x^3])/(c^(1/3)*x))/((1 + Sqrt[3])*a^(1/3) + (b^(1/3)*Sqrt[c*x^3])/( 
c^(1/3)*x))], -7 - 4*Sqrt[3]])/(3^(1/4)*b^(2/3)*c^(1/3)*Sqrt[(a^(1/3)*(a^( 
1/3) + (b^(1/3)*Sqrt[c*x^3])/(c^(1/3)*x)))/((1 + Sqrt[3])*a^(1/3) + (b^(1/ 
3)*Sqrt[c*x^3])/(c^(1/3)*x))^2]*Sqrt[a + (b*(c*x^3)^(3/2))/(c*x^3)])))/7)
 

Defintions of rubi rules used

rule 759

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

rule 774

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

rule 787

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_), x_Symbol] :> With[{k = Den 
ominator[n]}, Subst[Int[(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, p, q}, x] && FractionQ[n]
 

rule 811

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] + Simp[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] && I 
GtQ[n, 0] && GtQ[p, 0] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b, c, n, m 
, p, x]
 

rule 832

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

rule 2416

Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N 
umer[Simplify[(1 - Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 - Sqrt[3])*(d/c) 
]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 + Sqrt[3])*s + r*x))), x] - S 
imp[3^(1/4)*Sqrt[2 - Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( 
(1 + Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[s*((s + r*x)/((1 + Sqrt 
[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 - Sqrt[3])*s + r*x)/((1 + Sqrt[3]) 
*s + r*x)], -7 - 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && PosQ[a] && Eq 
Q[b*c^3 - 2*(5 - 3*Sqrt[3])*a*d^3, 0]
 

Maple [A] (verified)

Time = 1.14 (sec) , antiderivative size = 854, normalized size of antiderivative = 1.11

method	result	size
default	\(\frac {3 i \sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) \left (-a \,b^{2} c \right )^{\frac {2}{3}} a -2 i \sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) \left (-a \,b^{2} c \right )^{\frac {2}{3}} a +3 \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \operatorname {EllipticE}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) \left (-a \,b^{2} c \right )^{\frac {2}{3}} a +4 \sqrt {c \,x^{3}}\, b^{3} c x +4 a \,b^{2} c x}{7 c \,b^{2} \sqrt {a +b \sqrt {c \,x^{3}}}}\)	\(854\)

Input:

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

Output:

1/7/c*(3*I*3^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^( 
1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2)*((b*(c*x^3)^(1/ 
2)-(-a*b^2*c)^(1/3)*x)/x/(-a*b^2*c)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1 
/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b 
^2*c)^(1/3)/x)^(1/2)*EllipticE(1/6*3^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^ 
2*c)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/ 
x)^(1/2),2^(1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-a*b^2*c)^(2/3)*a-2*I*3 
^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*b^2 
*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2)*((b*(c*x^3)^(1/2)-(-a*b^2*c 
)^(1/3)*x)/x/(-a*b^2*c)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^ 
2*c)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/ 
x)^(1/2)*EllipticF(1/6*3^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2 
*b*(c*x^3)^(1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2),2^( 
1/2)*(I*3^(1/2)/(I*3^(1/2)-3))^(1/2))*(-a*b^2*c)^(2/3)*a+3*2^(1/2)*(I*(-I* 
3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/( 
-a*b^2*c)^(1/3)/x)^(1/2)*((b*(c*x^3)^(1/2)-(-a*b^2*c)^(1/3)*x)/x/(-a*b^2*c 
)^(1/3)/(I*3^(1/2)-3))^(1/2)*(-I*(I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3) 
^(1/2)+(-a*b^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2)*EllipticE(1/6 
*3^(1/2)*2^(1/2)*(I*(-I*3^(1/2)*x*(-a*b^2*c)^(1/3)+2*b*(c*x^3)^(1/2)+(-a*b 
^2*c)^(1/3)*x)*3^(1/2)/(-a*b^2*c)^(1/3)/x)^(1/2),2^(1/2)*(I*3^(1/2)/(I*...
 

Fricas [F]

\[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int { \sqrt {\sqrt {c x^{3}} b + a} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int \sqrt {a + b \sqrt {c x^{3}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int { \sqrt {\sqrt {c x^{3}} b + a} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int { \sqrt {\sqrt {c x^{3}} b + a} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int \sqrt {a+b\,\sqrt {c\,x^3}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \sqrt {c x^3}} \, dx=\int \sqrt {a +b \sqrt {c \,x^{3}}}d x \] Input:

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

Output:

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