\(\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx\) [63]

Optimal result

Integrand size = 21, antiderivative size = 355 \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=-\frac {\sqrt {a+b \sqrt {c x^3}}}{x}+\frac {3^{3/4} \sqrt {2+\sqrt {3}} b^{2/3} \sqrt [3]{c} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right ) \sqrt {\frac {a^{2/3}+b^{2/3} \sqrt [3]{c} x-\frac {\sqrt [3]{a} \sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\frac {\sqrt [3]{b} c^{2/3} x^2}{\sqrt {c x^3}}\right )^2}} \sqrt {a+b \sqrt {c x^3}}} \] Output:

-(a+b*(c*x^3)^(1/2))^(1/2)/x+3^(3/4)*(1/2*6^(1/2)+1/2*2^(1/2))*b^(2/3)*c^( 
1/3)*(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)/(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]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 0.55 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {893, 864, 809, 759}

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.

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

\(\Big \downarrow \) 893

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

\(\Big \downarrow \) 864

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

\(\Big \downarrow \) 809

\(\displaystyle 2 \left (\frac {3}{4} b \sqrt {c} \int \frac {1}{\sqrt {\frac {b \left (c x^3\right )^{3/2}}{c x^3}+a}}d\frac {\sqrt {c x^3}}{\sqrt {c} x}-\frac {\sqrt {a+\frac {b \left (c x^3\right )^{3/2}}{c x^3}}}{2 x}\right )\)

\(\Big \downarrow \) 759

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

Input:

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

Output:

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

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 809

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] - Simp[b*n*(p/(c^n*(m + 1)))   I 
nt[(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] && IntB 
inomialQ[a, b, c, n, m, p, x]
 

rule 864

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denomi 
nator[n]}, Simp[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 893

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbo 
l] :> With[{k = Denominator[n]}, Subst[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]
 

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.86

method	result	size
default	\(-\frac {i \sqrt {3}\, \left (-a \,b^{2} c \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \sqrt {\frac {b \sqrt {c \,x^{3}}-\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}{x \left (-a \,b^{2} c \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, x \left (-a \,b^{2} c \right )^{\frac {1}{3}}+2 b \sqrt {c \,x^{3}}+\left (-a \,b^{2} c \right )^{\frac {1}{3}} x \right ) \sqrt {3}}{\left (-a \,b^{2} c \right )^{\frac {1}{3}} x}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right ) x +2 a +2 b \sqrt {c \,x^{3}}}{2 x \sqrt {a +b \sqrt {c \,x^{3}}}}\)	\(304\)

Input:

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

Output:

-1/2*(I*3^(1/2)*(-a*b^2*c)^(1/3)*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)*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))*x+2*a+2*b* 
(c*x^3)^(1/2))/x/(a+b*(c*x^3)^(1/2))^(1/2)
 

Fricas [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a + b \sqrt {c x^{3}}}}{x^{2}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int { \frac {\sqrt {\sqrt {c x^{3}} b + a}}{x^{2}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a+b\,\sqrt {c\,x^3}}}{x^2} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {a+b \sqrt {c x^3}}}{x^2} \, dx=\int \frac {\sqrt {a +b \sqrt {c \,x^{3}}}}{x^{2}}d x \] Input:

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

Output:

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