\(\int \frac {\sqrt {c x}}{(a-b x^2)^{3/4}} \, dx\) [1006]

Optimal result

Integrand size = 20, antiderivative size = 191 \[ \int \frac {\sqrt {c x}}{\left (a-b x^2\right )^{3/4}} \, dx=-\frac {\sqrt {c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} b^{3/4}}+\frac {\sqrt {c} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} b^{3/4}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2} \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}\right )}\right )}{\sqrt {2} b^{3/4}} \] Output:

1/2*c^(1/2)*arctan(-1+2^(1/2)*b^(1/4)*(c*x)^(1/2)/c^(1/2)/(-b*x^2+a)^(1/4) 
)*2^(1/2)/b^(3/4)+1/2*c^(1/2)*arctan(1+2^(1/2)*b^(1/4)*(c*x)^(1/2)/c^(1/2) 
/(-b*x^2+a)^(1/4))*2^(1/2)/b^(3/4)-1/2*c^(1/2)*arctanh(2^(1/2)*b^(1/4)*(c* 
x)^(1/2)/(-b*x^2+a)^(1/4)/(c^(1/2)+b^(1/2)*c^(1/2)*x/(-b*x^2+a)^(1/2)))*2^ 
(1/2)/b^(3/4)
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {c x}}{\left (a-b x^2\right )^{3/4}} \, dx=\frac {\sqrt {c x} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}{-\sqrt {b} x+\sqrt {a-b x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {b} x+\sqrt {a-b x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}\right )\right )}{\sqrt {2} b^{3/4} \sqrt {x}} \] Input:

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

Output:

(Sqrt[c*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^(3/4)*Sqrt[x])
 

Rubi [A] (warning: unable to verify)

Time = 0.45 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.39, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {266, 854, 27, 826, 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.

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 854

\(\displaystyle \frac {2 \int \frac {c^3 x}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}}{c}\)

\(\Big \downarrow \) 27

\(\displaystyle 2 c \int \frac {c x}{b x^2 c^2+c^2}d\frac {\sqrt {c x}}{\sqrt [4]{a-b x^2}}\)

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle 2 c \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}+1\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {b} c x+c\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}-\frac {\log \left (-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c} \sqrt {c x}}{\sqrt [4]{a-b x^2}}+\sqrt {b} c x+c\right )}{2 \sqrt {2} \sqrt [4]{b} \sqrt {c}}}{2 \sqrt {b}}\right )\)

Input:

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

Output:

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])/(Sqr 
t[c]*(a - b*x^2)^(1/4))]/(Sqrt[2]*b^(1/4)*Sqrt[c]))/(2*Sqrt[b]) - (-1/2*Lo 
g[c + Sqrt[b]*c*x - (Sqrt[2]*b^(1/4)*Sqrt[c]*Sqrt[c*x])/(a - b*x^2)^(1/4)] 
/(Sqrt[2]*b^(1/4)*Sqrt[c]) + Log[c + Sqrt[b]*c*x + (Sqrt[2]*b^(1/4)*Sqrt[c 
]*Sqrt[c*x])/(a - b*x^2)^(1/4)]/(2*Sqrt[2]*b^(1/4)*Sqrt[c]))/(2*Sqrt[b]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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])
 

rule 266

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]
 

rule 826

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 
2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s)   Int[(r + s*x^2)/(a + b*x^ 
4), x], x] - Simp[1/(2*s)   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]]))
 

rule 854

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 
 1)/n)   Subst[Int[x^m/(1 - b*x^n)^(p + (m + 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)] && IntegersQ[m, p + (m + 1)/n]
 

rule 1082

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]
 

rule 1103

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]
 

rule 1476

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]
 

rule 1479

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]
 

Maple [F]

\[\int \frac {\sqrt {c x}}{\left (-b \,x^{2}+a \right )^{\frac {3}{4}}}d x\]

Input:

int((c*x)^(1/2)/(-b*x^2+a)^(3/4),x)
 

Output:

int((c*x)^(1/2)/(-b*x^2+a)^(3/4),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {\sqrt {c x}}{\left (a-b x^2\right )^{3/4}} \, dx=\text {Timed out} \] Input:

integrate((c*x)^(1/2)/(-b*x^2+a)^(3/4),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.70 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.24 \[ \int \frac {\sqrt {c x}}{\left (a-b x^2\right )^{3/4}} \, dx=\frac {\sqrt {c} x^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 a^{\frac {3}{4}} \Gamma \left (\frac {7}{4}\right )} \] Input:

integrate((c*x)**(1/2)/(-b*x**2+a)**(3/4),x)
 

Output:

sqrt(c)*x**(3/2)*gamma(3/4)*hyper((3/4, 3/4), (7/4,), b*x**2*exp_polar(2*I 
*pi)/a)/(2*a**(3/4)*gamma(7/4))
 

Maxima [F]

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

integrate((c*x)^(1/2)/(-b*x^2+a)^(3/4),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x)/(-b*x^2 + a)^(3/4), x)
 

Giac [F]

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

integrate((c*x)^(1/2)/(-b*x^2+a)^(3/4),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x)/(-b*x^2 + a)^(3/4), x)
 

Mupad [F(-1)]

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

int((c*x)^(1/2)/(a - b*x^2)^(3/4),x)
 

Output:

int((c*x)^(1/2)/(a - b*x^2)^(3/4), x)
 

Reduce [F]

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

int((c*x)^(1/2)/(-b*x^2+a)^(3/4),x)
 

Output:

sqrt(c)*int(sqrt(x)/(a - b*x**2)**(3/4),x)