Integrand size = 19, antiderivative size = 126 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\frac {2 \sqrt {c x} \sqrt {a+b x^2}}{3 c}+\frac {2 a^{3/4} \left (\sqrt {a}+\sqrt {b} x\right ) \sqrt {\frac {a+b x^2}{\left (\sqrt {a}+\sqrt {b} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} \sqrt {c} \sqrt {a+b x^2}} \] Output:
2/3*(c*x)^(1/2)*(b*x^2+a)^(1/2)/c+2/3*a^(3/4)*(a^(1/2)+b^(1/2)*x)*((b*x^2+ a)/(a^(1/2)+b^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*arctan(b^(1/4)*(c*x)^(1/ 2)/a^(1/4)/c^(1/2)),1/2*2^(1/2))/b^(1/4)/c^(1/2)/(b*x^2+a)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\frac {2 x \sqrt {a+b x^2} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-\frac {b x^2}{a}\right )}{\sqrt {c x} \sqrt {1+\frac {b x^2}{a}}} \] Input:
Integrate[Sqrt[a + b*x^2]/Sqrt[c*x],x]
Output:
(2*x*Sqrt[a + b*x^2]*Hypergeometric2F1[-1/2, 1/4, 5/4, -((b*x^2)/a)])/(Sqr t[c*x]*Sqrt[1 + (b*x^2)/a])
Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.10, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {248, 266, 761}
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.
\(\displaystyle \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx\) |
\(\Big \downarrow \) 248 |
\(\displaystyle \frac {2}{3} a \int \frac {1}{\sqrt {c x} \sqrt {b x^2+a}}dx+\frac {2 \sqrt {c x} \sqrt {a+b x^2}}{3 c}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {4 a \int \frac {1}{\sqrt {b x^2+a}}d\sqrt {c x}}{3 c}+\frac {2 \sqrt {c x} \sqrt {a+b x^2}}{3 c}\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {2 a^{3/4} \left (\sqrt {a} c+\sqrt {b} c x\right ) \sqrt {\frac {a c^2+b c^2 x^2}{\left (\sqrt {a} c+\sqrt {b} c x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a} \sqrt {c}}\right ),\frac {1}{2}\right )}{3 \sqrt [4]{b} c^{3/2} \sqrt {a+b x^2}}+\frac {2 \sqrt {c x} \sqrt {a+b x^2}}{3 c}\) |
Input:
Int[Sqrt[a + b*x^2]/Sqrt[c*x],x]
Output:
(2*Sqrt[c*x]*Sqrt[a + b*x^2])/(3*c) + (2*a^(3/4)*(Sqrt[a]*c + Sqrt[b]*c*x) *Sqrt[(a*c^2 + b*c^2*x^2)/(Sqrt[a]*c + Sqrt[b]*c*x)^2]*EllipticF[2*ArcTan[ (b^(1/4)*Sqrt[c*x])/(a^(1/4)*Sqrt[c])], 1/2])/(3*b^(1/4)*c^(3/2)*Sqrt[a + b*x^2])
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^p/(c*(m + 2*p + 1))), x] + Simp[2*a*(p/(m + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, m}, x] && GtQ[ p, 0] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Time = 0.39 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\frac {2 \sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {2}\, \sqrt {\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, a}{3}+\frac {2 b^{2} x^{3}}{3}+\frac {2 a b x}{3}}{\sqrt {b \,x^{2}+a}\, \sqrt {c x}\, b}\) | \(119\) |
risch | \(\frac {2 x \sqrt {b \,x^{2}+a}}{3 \sqrt {c x}}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {c x \left (b \,x^{2}+a \right )}}{3 b \sqrt {b c \,x^{3}+a c x}\, \sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(156\) |
elliptic | \(\frac {\sqrt {c x \left (b \,x^{2}+a \right )}\, \left (\frac {2 \sqrt {b c \,x^{3}+a c x}}{3 c}+\frac {2 a \sqrt {-a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}\, \sqrt {-\frac {b x}{\sqrt {-a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-a b}}{b}\right ) b}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{3 b \sqrt {b c \,x^{3}+a c x}}\right )}{\sqrt {c x}\, \sqrt {b \,x^{2}+a}}\) | \(158\) |
Input:
int((b*x^2+a)^(1/2)/(c*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/3/(b*x^2+a)^(1/2)*(((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*2^(1/2)*((-b* x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2)*(-b/(-a*b)^(1/2)*x)^(1/2)*EllipticF((( b*x+(-a*b)^(1/2))/(-a*b)^(1/2))^(1/2),1/2*2^(1/2))*(-a*b)^(1/2)*a+b^2*x^3+ a*b*x)/(c*x)^(1/2)/b
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.33 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\frac {2 \, {\left (2 \, \sqrt {b c} a {\rm weierstrassPInverse}\left (-\frac {4 \, a}{b}, 0, x\right ) + \sqrt {b x^{2} + a} \sqrt {c x} b\right )}}{3 \, b c} \] Input:
integrate((b*x^2+a)^(1/2)/(c*x)^(1/2),x, algorithm="fricas")
Output:
2/3*(2*sqrt(b*c)*a*weierstrassPInverse(-4*a/b, 0, x) + sqrt(b*x^2 + a)*sqr t(c*x)*b)/(b*c)
Result contains complex when optimal does not.
Time = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.37 \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\frac {\sqrt {a} \sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt {c} \Gamma \left (\frac {5}{4}\right )} \] Input:
integrate((b*x**2+a)**(1/2)/(c*x)**(1/2),x)
Output:
sqrt(a)*sqrt(x)*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), b*x**2*exp_polar(I*p i)/a)/(2*sqrt(c)*gamma(5/4))
\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {c x}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(c*x)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^2 + a)/sqrt(c*x), x)
\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\int { \frac {\sqrt {b x^{2} + a}}{\sqrt {c x}} \,d x } \] Input:
integrate((b*x^2+a)^(1/2)/(c*x)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^2 + a)/sqrt(c*x), x)
Timed out. \[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\int \frac {\sqrt {b\,x^2+a}}{\sqrt {c\,x}} \,d x \] Input:
int((a + b*x^2)^(1/2)/(c*x)^(1/2),x)
Output:
int((a + b*x^2)^(1/2)/(c*x)^(1/2), x)
\[ \int \frac {\sqrt {a+b x^2}}{\sqrt {c x}} \, dx=\frac {2 \sqrt {c}\, \left (\sqrt {x}\, \sqrt {b \,x^{2}+a}+\left (\int \frac {\sqrt {x}\, \sqrt {b \,x^{2}+a}}{b \,x^{3}+a x}d x \right ) a \right )}{3 c} \] Input:
int((b*x^2+a)^(1/2)/(c*x)^(1/2),x)
Output:
(2*sqrt(c)*(sqrt(x)*sqrt(a + b*x**2) + int((sqrt(x)*sqrt(a + b*x**2))/(a*x + b*x**3),x)*a))/(3*c)