\(\int \frac {x}{\sqrt {x^3} \sqrt {a+b x^{3/2}}} \, dx\) [54]

Optimal result

Integrand size = 22, antiderivative size = 247 \[ \int \frac {x}{\sqrt {x^3} \sqrt {a+b x^{3/2}}} \, dx=\frac {4 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right ) x^{3/2} \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )^2}} \sqrt {x^3} \sqrt {a+b x^{3/2}}} \] Output:

4/3*(1/2*6^(1/2)+1/2*2^(1/2))*(a^(1/3)+b^(1/3)*x^(1/2))*x^(3/2)*((a^(2/3)- 
a^(1/3)*b^(1/3)*x^(1/2)+b^(2/3)*x)/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^(1/2))^2 
)^(1/2)*EllipticF(((1-3^(1/2))*a^(1/3)+b^(1/3)*x^(1/2))/((1+3^(1/2))*a^(1/ 
3)+b^(1/3)*x^(1/2)),I*3^(1/2)+2*I)*3^(3/4)/b^(1/3)/(a^(1/3)*(a^(1/3)+b^(1/ 
3)*x^(1/2))/((1+3^(1/2))*a^(1/3)+b^(1/3)*x^(1/2))^2)^(1/2)/(x^3)^(1/2)/(a+ 
b*x^(3/2))^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.04 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.25 \[ \int \frac {x}{\sqrt {x^3} \sqrt {a+b x^{3/2}}} \, dx=\frac {2 x^2 \sqrt {1+\frac {b x^{3/2}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^{3/2}}{a}\right )}{\sqrt {x^3} \sqrt {a+b x^{3/2}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {30, 864, 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.

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

\(\Big \downarrow \) 30

\(\displaystyle \frac {x^{3/2} \int \frac {1}{\sqrt {x} \sqrt {b x^{3/2}+a}}dx}{\sqrt {x^3}}\)

\(\Big \downarrow \) 864

\(\displaystyle \frac {2 x^{3/2} \int \frac {1}{\sqrt {b x^{3/2}+a}}d\sqrt {x}}{\sqrt {x^3}}\)

\(\Big \downarrow \) 759

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 30

Int[(u_.)*((a_.)*(x_))^(m_.)*((b_.)*(x_)^(i_.))^(p_), x_Symbol] :> Simp[b^I 
ntPart[p]*((b*x^i)^FracPart[p]/(a^(i*IntPart[p])*(a*x)^(i*FracPart[p]))) 
Int[u*(a*x)^(m + i*p), x], x] /; FreeQ[{a, b, i, m, p}, x] && IntegerQ[i] & 
&  !IntegerQ[p]
 

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 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]
 

Maple [A] (verified)

Time = 0.79 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.98

method	result	size
default	\(-\frac {2 i x^{\frac {3}{2}} \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}+2 b \sqrt {x}+\left (-b^{2} a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {b \sqrt {x}-\left (-b^{2} a \right )^{\frac {1}{3}}}{\left (-b^{2} a \right )^{\frac {1}{3}} \left (i \sqrt {3}-3\right )}}\, \sqrt {-\frac {i \left (i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}+2 b \sqrt {x}+\left (-b^{2} a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {\frac {i \left (-i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}+2 b \sqrt {x}+\left (-b^{2} a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{6}, \sqrt {2}\, \sqrt {\frac {i \sqrt {3}}{i \sqrt {3}-3}}\right )}{3 \sqrt {x^{3}}\, b \sqrt {a +b \,x^{\frac {3}{2}}}}\)	\(241\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(x^3)*sqrt(b*x^(3/2) + a)*(b*x^(3/2) - a)/(b^2*x^5 - a^2*x^2) 
, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x}{\sqrt {x^3} \sqrt {a+b x^{3/2}}} \, dx=-\left (\int \frac {\sqrt {\sqrt {x}\, b x +a}\, x}{-b^{2} x^{3}+a^{2}}d x \right ) b +\left (\int \frac {\sqrt {x}\, \sqrt {\sqrt {x}\, b x +a}}{-b^{2} x^{4}+a^{2} x}d x \right ) a \] Input:

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

Output:

 - int((sqrt(sqrt(x)*b*x + a)*x)/(a**2 - b**2*x**3),x)*b + int((sqrt(x)*sq 
rt(sqrt(x)*b*x + a))/(a**2*x - b**2*x**4),x)*a