\(\int \frac {1}{\sqrt {x} (b x^2+c x^4)^{3/2}} \, dx\) [276]

Optimal result

Integrand size = 21, antiderivative size = 320 \[ \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}-\frac {21 c^{3/2} x^{3/2} \left (b+c x^2\right )}{5 b^3 \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {b x^2+c x^4}}-\frac {7 \sqrt {b x^2+c x^4}}{5 b^2 x^{7/2}}+\frac {21 c \sqrt {b x^2+c x^4}}{5 b^3 x^{3/2}}+\frac {21 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 b^{11/4} \sqrt {b x^2+c x^4}}-\frac {21 c^{5/4} x \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{10 b^{11/4} \sqrt {b x^2+c x^4}} \] Output:

1/b/x^(3/2)/(c*x^4+b*x^2)^(1/2)-21/5*c^(3/2)*x^(3/2)*(c*x^2+b)/b^3/(b^(1/2 
)+c^(1/2)*x)/(c*x^4+b*x^2)^(1/2)-7/5*(c*x^4+b*x^2)^(1/2)/b^2/x^(7/2)+21/5* 
c*(c*x^4+b*x^2)^(1/2)/b^3/x^(3/2)+21/5*c^(5/4)*x*(b^(1/2)+c^(1/2)*x)*((c*x 
^2+b)/(b^(1/2)+c^(1/2)*x)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4)*x^(1/2)/ 
b^(1/4))),1/2*2^(1/2))/b^(11/4)/(c*x^4+b*x^2)^(1/2)-21/10*c^(5/4)*x*(b^(1/ 
2)+c^(1/2)*x)*((c*x^2+b)/(b^(1/2)+c^(1/2)*x)^2)^(1/2)*InverseJacobiAM(2*ar 
ctan(c^(1/4)*x^(1/2)/b^(1/4)),1/2*2^(1/2))/b^(11/4)/(c*x^4+b*x^2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.02 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.19 \[ \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {c x^2}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {3}{2},-\frac {1}{4},-\frac {c x^2}{b}\right )}{5 b x^{3/2} \sqrt {x^2 \left (b+c x^2\right )}} \] Input:

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

Output:

(-2*Sqrt[1 + (c*x^2)/b]*Hypergeometric2F1[-5/4, 3/2, -1/4, -((c*x^2)/b)])/ 
(5*b*x^(3/2)*Sqrt[x^2*(b + c*x^2)])
 

Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1428, 1430, 1430, 1431, 266, 834, 27, 761, 1510}

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.

\(\displaystyle \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1428

\(\displaystyle \frac {7 \int \frac {1}{x^{5/2} \sqrt {c x^4+b x^2}}dx}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 1430

\(\displaystyle \frac {7 \left (-\frac {3 c \int \frac {1}{\sqrt {x} \sqrt {c x^4+b x^2}}dx}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 1430

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {c \int \frac {x^{3/2}}{\sqrt {c x^4+b x^2}}dx}{b}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 1431

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {c x \sqrt {b+c x^2} \int \frac {\sqrt {x}}{\sqrt {c x^2+b}}dx}{b \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {2 c x \sqrt {b+c x^2} \int \frac {x}{\sqrt {c x^2+b}}d\sqrt {x}}{b \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 834

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {2 c x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\sqrt {b} \int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {b} \sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{b \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {2 c x \sqrt {b+c x^2} \left (\frac {\sqrt {b} \int \frac {1}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{b \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 761

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {2 c x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\int \frac {\sqrt {b}-\sqrt {c} x}{\sqrt {c x^2+b}}d\sqrt {x}}{\sqrt {c}}\right )}{b \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

\(\Big \downarrow \) 1510

\(\displaystyle \frac {7 \left (-\frac {3 c \left (\frac {2 c x \sqrt {b+c x^2} \left (\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 c^{3/4} \sqrt {b+c x^2}}-\frac {\frac {\sqrt [4]{b} \left (\sqrt {b}+\sqrt {c} x\right ) \sqrt {\frac {b+c x^2}{\left (\sqrt {b}+\sqrt {c} x\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{\sqrt [4]{c} \sqrt {b+c x^2}}-\frac {\sqrt {x} \sqrt {b+c x^2}}{\sqrt {b}+\sqrt {c} x}}{\sqrt {c}}\right )}{b \sqrt {b x^2+c x^4}}-\frac {2 \sqrt {b x^2+c x^4}}{b x^{3/2}}\right )}{5 b}-\frac {2 \sqrt {b x^2+c x^4}}{5 b x^{7/2}}\right )}{2 b}+\frac {1}{b x^{3/2} \sqrt {b x^2+c x^4}}\)

Input:

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

Output:

1/(b*x^(3/2)*Sqrt[b*x^2 + c*x^4]) + (7*((-2*Sqrt[b*x^2 + c*x^4])/(5*b*x^(7 
/2)) - (3*c*((-2*Sqrt[b*x^2 + c*x^4])/(b*x^(3/2)) + (2*c*x*Sqrt[b + c*x^2] 
*(-((-((Sqrt[x]*Sqrt[b + c*x^2])/(Sqrt[b] + Sqrt[c]*x)) + (b^(1/4)*(Sqrt[b 
] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*EllipticE[2*ArcTa 
n[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(c^(1/4)*Sqrt[b + c*x^2]))/Sqrt[c]) + 
(b^(1/4)*(Sqrt[b] + Sqrt[c]*x)*Sqrt[(b + c*x^2)/(Sqrt[b] + Sqrt[c]*x)^2]*E 
llipticF[2*ArcTan[(c^(1/4)*Sqrt[x])/b^(1/4)], 1/2])/(2*c^(3/4)*Sqrt[b + c* 
x^2])))/(b*Sqrt[b*x^2 + c*x^4])))/(5*b)))/(2*b)
 

Defintions of rubi rules used

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 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 761

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]
 

rule 834

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, S 
imp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[a 
 + b*x^4], x], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
 

rule 1428

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(-d)*(d*x)^(m - 1)*((b*x^2 + c*x^4)^(p + 1)/(2*b*(p + 1))), x] + Simp[d^2* 
((m + 4*p + 3)/(2*b*(p + 1)))   Int[(d*x)^(m - 2)*(b*x^2 + c*x^4)^(p + 1), 
x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && LtQ[p, -1]
 

rule 1430

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[d*(d*x)^(m - 1)*((b*x^2 + c*x^4)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[c*( 
(m + 4*p + 3)/(b*d^2*(m + 2*p + 1)))   Int[(d*x)^(m + 2)*(b*x^2 + c*x^4)^p, 
 x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p] && LtQ[m + 2*p + 1, 0 
]
 

rule 1431

Int[((d_.)*(x_))^(m_)*((b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp 
[(b*x^2 + c*x^4)^p/((d*x)^(2*p)*(b + c*x^2)^p)   Int[(d*x)^(m + 2*p)*(b + c 
*x^2)^p, x], x] /; FreeQ[{b, c, d, m, p}, x] &&  !IntegerQ[p]
 

rule 1510

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d* 
(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4]))*E 
llipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e 
}, x] && PosQ[c/a]
 

Maple [A] (verified)

Time = 2.30 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {\sqrt {x}\, \left (c \,x^{2}+b \right ) \left (42 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticE}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b c \,x^{2}-21 \sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {2}\, \sqrt {\frac {-c x +\sqrt {-b c}}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {c x +\sqrt {-b c}}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right ) b c \,x^{2}-42 c^{2} x^{4}-28 b c \,x^{2}+4 b^{2}\right )}{10 \left (c \,x^{4}+b \,x^{2}\right )^{\frac {3}{2}} b^{3}}\)	\(222\)
risch	\(-\frac {2 \left (c \,x^{2}+b \right ) \left (-8 c \,x^{2}+b \right )}{5 b^{3} x^{\frac {3}{2}} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}-\frac {c^{2} \left (\frac {8 \sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{c \sqrt {c \,x^{3}+b x}}-5 b \left (\frac {x^{2}}{b \sqrt {\left (x^{2}+\frac {b}{c}\right ) c x}}-\frac {\sqrt {-b c}\, \sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}\, \sqrt {-\frac {c x}{\sqrt {-b c}}}\, \left (-\frac {2 \sqrt {-b c}\, \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}+\frac {\sqrt {-b c}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {-b c}}{c}\right ) c}{\sqrt {-b c}}}, \frac {\sqrt {2}}{2}\right )}{c}\right )}{2 b c \sqrt {c \,x^{3}+b x}}\right )\right ) \sqrt {x}\, \sqrt {x \left (c \,x^{2}+b \right )}}{5 b^{3} \sqrt {x^{2} \left (c \,x^{2}+b \right )}}\)	\(418\)

Input:

int(1/x^(1/2)/(c*x^4+b*x^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/10/(c*x^4+b*x^2)^(3/2)*x^(1/2)*(c*x^2+b)*(42*((c*x+(-b*c)^(1/2))/(-b*c) 
^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-c/(-b*c)^ 
(1/2)*x)^(1/2)*EllipticE(((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/ 
2))*b*c*x^2-21*((c*x+(-b*c)^(1/2))/(-b*c)^(1/2))^(1/2)*2^(1/2)*((-c*x+(-b* 
c)^(1/2))/(-b*c)^(1/2))^(1/2)*(-c/(-b*c)^(1/2)*x)^(1/2)*EllipticF(((c*x+(- 
b*c)^(1/2))/(-b*c)^(1/2))^(1/2),1/2*2^(1/2))*b*c*x^2-42*c^2*x^4-28*b*c*x^2 
+4*b^2)/b^3
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx=\frac {21 \, {\left (c^{2} x^{6} + b c x^{4}\right )} \sqrt {c} {\rm weierstrassZeta}\left (-\frac {4 \, b}{c}, 0, {\rm weierstrassPInverse}\left (-\frac {4 \, b}{c}, 0, x\right )\right ) + {\left (21 \, c^{2} x^{4} + 14 \, b c x^{2} - 2 \, b^{2}\right )} \sqrt {c x^{4} + b x^{2}} \sqrt {x}}{5 \, {\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} \] Input:

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

Output:

1/5*(21*(c^2*x^6 + b*c*x^4)*sqrt(c)*weierstrassZeta(-4*b/c, 0, weierstrass 
PInverse(-4*b/c, 0, x)) + (21*c^2*x^4 + 14*b*c*x^2 - 2*b^2)*sqrt(c*x^4 + b 
*x^2)*sqrt(x))/(b^3*c*x^6 + b^4*x^4)
 

Sympy [F]

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

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

Output:

Integral(1/(sqrt(x)*(x**2*(b + c*x**2))**(3/2)), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\sqrt {x} \left (b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {x}\, \sqrt {c \,x^{2}+b}}{c^{2} x^{8}+2 b c \,x^{6}+b^{2} x^{4}}d x \] Input:

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

Output:

int((sqrt(x)*sqrt(b + c*x**2))/(b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)