Integrand size = 19, antiderivative size = 235 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\frac {4 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}}{\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )^2}} \sqrt {a+b x^{3/2}}} \] Output:
4/3*(1/2*6^(1/2)+1/2*2^(1/2))*(a^(1/3)+b^(1/3)*x^(1/2))*((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)/(a+b*x^(3/2))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.24 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\frac {2 \sqrt {x} \sqrt {1+\frac {b x^{3/2}}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {4}{3},-\frac {b x^{3/2}}{a}\right )}{\sqrt {a+b x^{3/2}}} \] Input:
Integrate[1/(Sqrt[x]*Sqrt[a + b*x^(3/2)]),x]
Output:
(2*Sqrt[x]*Sqrt[1 + (b*x^(3/2))/a]*Hypergeometric2F1[1/3, 1/2, 4/3, -((b*x ^(3/2))/a)])/Sqrt[a + b*x^(3/2)]
Time = 0.44 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {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.
| \(\displaystyle \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx\) |
| \(\Big \downarrow \) 864 |
| \(\displaystyle 2 \int \frac {1}{\sqrt {b x^{3/2}+a}}d\sqrt {x}\) |
| \(\Big \downarrow \) 759 |
| \(\displaystyle \frac {4 \sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right ) \sqrt {\frac {a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \sqrt {x}+b^{2/3} x}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [3]{b} \sqrt {x}+\left (1-\sqrt {3}\right ) \sqrt [3]{a}}{\sqrt [3]{b} \sqrt {x}+\left (1+\sqrt {3}\right ) \sqrt [3]{a}}\right ),-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt {\frac {\sqrt [3]{a} \left (\sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )}{\left (\left (1+\sqrt {3}\right ) \sqrt [3]{a}+\sqrt [3]{b} \sqrt {x}\right )^2}} \sqrt {a+b x^{3/2}}}\) |
Input:
Int[1/(Sqrt[x]*Sqrt[a + b*x^(3/2)]),x]
Output:
(4*Sqrt[2 + Sqrt[3]]*(a^(1/3) + b^(1/3)*Sqrt[x])*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]*E llipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)*Sqrt[x])/((1 + Sqrt[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)*Sqrt[x])^2 ]*Sqrt[a + b*x^(3/2)])
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]
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]
Time = 0.82 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.24
| method | result | size |
| derivativedivides | \(-\frac {4 i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {x}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {\sqrt {x}-\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (\sqrt {x}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {x}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {a +b \,x^{\frac {3}{2}}}}\) | \(291\) |
| default | \(-\frac {4 i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}} \sqrt {\frac {i \left (\sqrt {x}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \sqrt {\frac {\sqrt {x}-\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{b}}{-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}}}\, \sqrt {-\frac {i \left (\sqrt {x}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (\sqrt {x}+\frac {\left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}-\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right ) \sqrt {3}\, b}{\left (-b^{2} a \right )^{\frac {1}{3}}}}}{3}, \sqrt {\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{b \left (-\frac {3 \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}+\frac {i \sqrt {3}\, \left (-b^{2} a \right )^{\frac {1}{3}}}{2 b}\right )}}\right )}{3 b \sqrt {a +b \,x^{\frac {3}{2}}}}\) | \(291\) |
Input:
int(1/x^(1/2)/(a+b*x^(3/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
-4/3*I*3^(1/2)/b*(-b^2*a)^(1/3)*(I*(x^(1/2)+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^( 1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a)^(1/3))^(1/2)*((x^(1/2)-1/b*(-b^2 *a)^(1/3))/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3)))^(1/2)*( -I*(x^(1/2)+1/2/b*(-b^2*a)^(1/3)+1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b /(-b^2*a)^(1/3))^(1/2)/(a+b*x^(3/2))^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x^(1/ 2)+1/2/b*(-b^2*a)^(1/3)-1/2*I*3^(1/2)/b*(-b^2*a)^(1/3))*3^(1/2)*b/(-b^2*a) ^(1/3))^(1/2),(I*3^(1/2)/b*(-b^2*a)^(1/3)/(-3/2/b*(-b^2*a)^(1/3)+1/2*I*3^( 1/2)/b*(-b^2*a)^(1/3)))^(1/2))
\[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\int { \frac {1}{\sqrt {b x^{\frac {3}{2}} + a} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(a+b*x^(3/2))^(1/2),x, algorithm="fricas")
Output:
integral((b*x^2 - a*sqrt(x))*sqrt(b*x^(3/2) + a)/(b^2*x^4 - a^2*x), x)
Time = 0.51 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.18 \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\frac {2 \sqrt {x} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{2} \\ \frac {4}{3} \end {matrix}\middle | {\frac {b x^{\frac {3}{2}} e^{i \pi }}{a}} \right )}}{3 \sqrt {a} \Gamma \left (\frac {4}{3}\right )} \] Input:
integrate(1/x**(1/2)/(a+b*x**(3/2))**(1/2),x)
Output:
2*sqrt(x)*gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*x**(3/2)*exp_polar(I*pi)/ a)/(3*sqrt(a)*gamma(4/3))
\[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\int { \frac {1}{\sqrt {b x^{\frac {3}{2}} + a} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(a+b*x^(3/2))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*x^(3/2) + a)*sqrt(x)), x)
\[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\int { \frac {1}{\sqrt {b x^{\frac {3}{2}} + a} \sqrt {x}} \,d x } \] Input:
integrate(1/x^(1/2)/(a+b*x^(3/2))^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b*x^(3/2) + a)*sqrt(x)), x)
Timed out. \[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=\int \frac {1}{\sqrt {x}\,\sqrt {a+b\,x^{3/2}}} \,d x \] Input:
int(1/(x^(1/2)*(a + b*x^(3/2))^(1/2)),x)
Output:
int(1/(x^(1/2)*(a + b*x^(3/2))^(1/2)), x)
\[ \int \frac {1}{\sqrt {x} \sqrt {a+b x^{3/2}}} \, dx=-\left (\int \frac {\sqrt {\sqrt {x}\, b x +a}\, x}{-b^{2} x^{3}+a^{2}}d x \right ) b +\left (\int \frac {\sqrt {x}\, \sqrt {\sqrt {x}\, b x +a}}{-b^{2} x^{4}+a^{2} x}d x \right ) a \] Input:
int(1/x^(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