Integrand size = 20, antiderivative size = 272 \[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}} \]
1/2*arctan(-1+b^(1/4)*2^(1/2)*(c*x)^(1/2)/(-b*x^2+a)^(1/4)/c^(1/2))/b^(1/4 )*2^(1/2)/c^(1/2)+1/2*arctan(1+b^(1/4)*2^(1/2)*(c*x)^(1/2)/(-b*x^2+a)^(1/4 )/c^(1/2))/b^(1/4)*2^(1/2)/c^(1/2)-1/4*ln(c^(1/2)-b^(1/4)*2^(1/2)*(c*x)^(1 /2)/(-b*x^2+a)^(1/4)+x*b^(1/2)*c^(1/2)/(-b*x^2+a)^(1/2))/b^(1/4)*2^(1/2)/c ^(1/2)+1/4*ln(c^(1/2)+b^(1/4)*2^(1/2)*(c*x)^(1/2)/(-b*x^2+a)^(1/4)+x*b^(1/ 2)*c^(1/2)/(-b*x^2+a)^(1/2))/b^(1/4)*2^(1/2)/c^(1/2)
Time = 0.34 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=\frac {\sqrt {x} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}{-\sqrt {b} x+\sqrt {a-b x^2}}\right )+\text {arctanh}\left (\frac {\sqrt {b} x+\sqrt {a-b x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}\right )\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c x}} \]
(Sqrt[x]*(ArcTan[(Sqrt[2]*b^(1/4)*Sqrt[x]*(a - b*x^2)^(1/4))/(-(Sqrt[b]*x) + Sqrt[a - b*x^2])] + ArcTanh[(Sqrt[b]*x + Sqrt[a - b*x^2])/(Sqrt[2]*b^(1 /4)*Sqrt[x]*(a - b*x^2)^(1/4))]))/(Sqrt[2]*b^(1/4)*Sqrt[c*x])
Time = 0.43 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {266, 770, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 \int \frac {1}{\sqrt [4]{a-b x^2}}d\sqrt {c x}}{c}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {2 \int \frac {1}{b x^2+1}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{c}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {c^2 \left (c-\sqrt {b} c x\right )}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 c}+\frac {\int \frac {c^2 \left (\sqrt {b} x c+c\right )}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 c}\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \int \frac {c-\sqrt {b} c x}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}+\frac {1}{2} c \int \frac {\sqrt {b} x c+c}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{c}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \int \frac {c-\sqrt {b} c x}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}+\frac {1}{2} c \left (\frac {\int \frac {1}{x c+\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {b}}+\frac {\int \frac {1}{x c+\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {b}}\right )\right )}{c}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \int \frac {c-\sqrt {b} c x}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}+\frac {1}{2} c \left (\frac {\int \frac {1}{-c x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\int \frac {1}{-c x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )\right )}{c}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \int \frac {c-\sqrt {b} c x}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}+\frac {1}{2} c \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )\right )}{c}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \left (-\frac {\int -\frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}}{\sqrt [4]{b} \left (x c+\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}\right )}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {c}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{b} \left (x c+\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}\right )}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )+\frac {1}{2} c \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )\right )}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \left (\frac {\int \frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}}{\sqrt [4]{b} \left (x c+\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}\right )}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {c}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{\sqrt [4]{b} \left (x c+\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}\right )}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )+\frac {1}{2} c \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )\right )}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \left (\frac {\int \frac {\sqrt {2} \sqrt {c}-\frac {2 \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}}{x c+\frac {c}{\sqrt {b}}-\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {2} \sqrt {b} \sqrt {c}}+\frac {\int \frac {\sqrt {c}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}}{x c+\frac {c}{\sqrt {b}}+\frac {\sqrt {2} \sqrt {c x} \sqrt {c}}{\sqrt [4]{b} \sqrt [4]{a-b x^2}}}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{2 \sqrt {b} \sqrt {c}}\right )+\frac {1}{2} c \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )\right )}{c}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} c \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )+\frac {1}{2} c \left (\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {b} c x+c\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {b} c x+c\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}\right )\right )}{c}\) |
(2*((c*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4 ))]/(Sqrt[2]*b^(1/4)*Sqrt[c])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(S qrt[c]*(a - b*x^2)^(1/4))]/(Sqrt[2]*b^(1/4)*Sqrt[c])))/2 + (c*(-1/2*Log[c + Sqrt[b]*c*x - (Sqrt[2]*b^(1/4)*Sqrt[c]*Sqrt[c*x])/(a - b*x^2)^(1/4)]/(Sq rt[2]*b^(1/4)*Sqrt[c]) + Log[c + Sqrt[b]*c*x + (Sqrt[2]*b^(1/4)*Sqrt[c]*Sq rt[c*x])/(a - b*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4)*Sqrt[c])))/2))/c
3.10.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 1/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
\[\int \frac {1}{\sqrt {c x}\, \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}d x\]
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=-\frac {1}{2} \, \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} + {\left (b c x^{2} - a c\right )} \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}}}{b x^{2} - a}\right ) + \frac {1}{2} \, \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} - {\left (b c x^{2} - a c\right )} \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}}}{b x^{2} - a}\right ) + \frac {1}{2} i \, \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} - {\left (i \, b c x^{2} - i \, a c\right )} \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}}}{b x^{2} - a}\right ) - \frac {1}{2} i \, \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}} \log \left (\frac {{\left (-b x^{2} + a\right )}^{\frac {3}{4}} \sqrt {c x} - {\left (-i \, b c x^{2} + i \, a c\right )} \left (-\frac {1}{b c^{2}}\right )^{\frac {1}{4}}}{b x^{2} - a}\right ) \]
-1/2*(-1/(b*c^2))^(1/4)*log(((-b*x^2 + a)^(3/4)*sqrt(c*x) + (b*c*x^2 - a*c )*(-1/(b*c^2))^(1/4))/(b*x^2 - a)) + 1/2*(-1/(b*c^2))^(1/4)*log(((-b*x^2 + a)^(3/4)*sqrt(c*x) - (b*c*x^2 - a*c)*(-1/(b*c^2))^(1/4))/(b*x^2 - a)) + 1 /2*I*(-1/(b*c^2))^(1/4)*log(((-b*x^2 + a)^(3/4)*sqrt(c*x) - (I*b*c*x^2 - I *a*c)*(-1/(b*c^2))^(1/4))/(b*x^2 - a)) - 1/2*I*(-1/(b*c^2))^(1/4)*log(((-b *x^2 + a)^(3/4)*sqrt(c*x) - (-I*b*c*x^2 + I*a*c)*(-1/(b*c^2))^(1/4))/(b*x^ 2 - a))
Result contains complex when optimal does not.
Time = 0.83 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=\frac {\sqrt {x} \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \sqrt {c} \Gamma \left (\frac {5}{4}\right )} \]
sqrt(x)*gamma(1/4)*hyper((1/4, 1/4), (5/4,), b*x**2*exp_polar(2*I*pi)/a)/( 2*a**(1/4)*sqrt(c)*gamma(5/4))
\[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}} \,d x } \]
\[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=\int { \frac {1}{{\left (-b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {c x} \sqrt [4]{a-b x^2}} \, dx=\int \frac {1}{\sqrt {c\,x}\,{\left (a-b\,x^2\right )}^{1/4}} \,d x \]