3.10.39 \(\int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx\) [939]

3.10.39.1 Optimal result

Integrand size = 20, antiderivative size = 296 \[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx=-\frac {2 \sqrt [4]{a-b x^2}}{c \sqrt {c x}}+\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} c^{3/2}}-\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt {c} \sqrt [4]{a-b x^2}}\right )}{\sqrt {2} c^{3/2}}-\frac {\sqrt [4]{b} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt {2} c^{3/2}}+\frac {\sqrt [4]{b} \log \left (\sqrt {c}+\frac {\sqrt {b} \sqrt {c} x}{\sqrt {a-b x^2}}+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {c x}}{\sqrt [4]{a-b x^2}}\right )}{2 \sqrt {2} c^{3/2}} \]

output

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

3.10.39.2 Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx=\frac {x \left (-4 \sqrt [4]{a-b x^2}+\sqrt {2} \sqrt [4]{b} \sqrt {x} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}{\sqrt {b} x-\sqrt {a-b x^2}}\right )+\sqrt {2} \sqrt [4]{b} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {b} x+\sqrt {a-b x^2}}{\sqrt {2} \sqrt [4]{b} \sqrt {x} \sqrt [4]{a-b x^2}}\right )\right )}{2 (c x)^{3/2}} \]

input

Integrate[(a - b*x^2)^(1/4)/(c*x)^(3/2),x]
 

output

(x*(-4*(a - b*x^2)^(1/4) + Sqrt[2]*b^(1/4)*Sqrt[x]*ArcTan[(Sqrt[2]*b^(1/4) 
*Sqrt[x]*(a - b*x^2)^(1/4))/(Sqrt[b]*x - Sqrt[a - b*x^2])] + Sqrt[2]*b^(1/ 
4)*Sqrt[x]*ArcTanh[(Sqrt[b]*x + Sqrt[a - b*x^2])/(Sqrt[2]*b^(1/4)*Sqrt[x]* 
(a - b*x^2)^(1/4))]))/(2*(c*x)^(3/2))
 

3.10.39.3 Rubi [A] (warning: unable to verify)

Time = 0.47 (sec) , antiderivative size = 293, normalized size of antiderivative = 0.99, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {247, 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.

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

\(\Big \downarrow \) 247

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

\(\Big \downarrow \) 266

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

\(\Big \downarrow \) 854

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

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

input

Int[(a - b*x^2)^(1/4)/(c*x)^(3/2),x]
 

output

(-2*(a - b*x^2)^(1/4))/(c*Sqrt[c*x]) - (2*b*((-(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])) + Arc 
Tan[1 + (Sqrt[2]*b^(1/4)*Sqrt[c*x])/(Sqrt[c]*(a - b*x^2)^(1/4))]/(Sqrt[2]* 
b^(1/4)*Sqrt[c]))/(2*Sqrt[b]) - (-1/2*Log[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])))/c
 

3.10.39.3.1 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 247

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^p/(c*(m + 1))), x] - Simp[2*b*(p/(c^2*(m + 1)))   Int[ 
(c*x)^(m + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] && GtQ[p, 
0] && LtQ[m, -1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomialQ[a, b, c, 2, 
m, p, x]
 

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]
 

3.10.39.4 Maple [F]

\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {1}{4}}}{\left (c x \right )^{\frac {3}{2}}}d x\]

input

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

output

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

3.10.39.5 Fricas [F(-1)]

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

input

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

output

Timed out
 

3.10.39.6 Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.17 \[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx=\frac {\sqrt [4]{a} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{2 c^{\frac {3}{2}} \sqrt {x} \Gamma \left (\frac {3}{4}\right )} \]

input

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

output

a**(1/4)*gamma(-1/4)*hyper((-1/4, -1/4), (3/4,), b*x**2*exp_polar(2*I*pi)/ 
a)/(2*c**(3/2)*sqrt(x)*gamma(3/4))
 

3.10.39.7 Maxima [F]

\[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {3}{2}}} \,d x } \]

input

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

output

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

3.10.39.8 Giac [F]

\[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {1}{4}}}{\left (c x\right )^{\frac {3}{2}}} \,d x } \]

input

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

output

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

3.10.39.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{a-b x^2}}{(c x)^{3/2}} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{1/4}}{{\left (c\,x\right )}^{3/2}} \,d x \]

input

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

output

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