\(\int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx\) [54]

Optimal result

Integrand size = 24, antiderivative size = 347 \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\frac {b x^{3/2} \left (a+b x^2+c x^4\right )}{3 \sqrt {c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {a x+b x^3+c x^5}}+\frac {1}{3} \sqrt {x} \sqrt {a x+b x^3+c x^5}-\frac {\sqrt [4]{a} b \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 c^{3/4} \sqrt {a x+b x^3+c x^5}}+\frac {\sqrt [4]{a} \left (b+2 \sqrt {a} \sqrt {c}\right ) \sqrt {x} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 c^{3/4} \sqrt {a x+b x^3+c x^5}} \] Output:

1/3*b*x^(3/2)*(c*x^4+b*x^2+a)/c^(1/2)/(a^(1/2)+c^(1/2)*x^2)/(c*x^5+b*x^3+a 
*x)^(1/2)+1/3*x^(1/2)*(c*x^5+b*x^3+a*x)^(1/2)-1/3*a^(1/4)*b*x^(1/2)*(a^(1/ 
2)+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE( 
sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(3/4)/ 
(c*x^5+b*x^3+a*x)^(1/2)+1/6*a^(1/4)*(b+2*a^(1/2)*c^(1/2))*x^(1/2)*(a^(1/2) 
+c^(1/2)*x^2)*((c*x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacob 
iAM(2*arctan(c^(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/c^(3/4)/( 
c*x^5+b*x^3+a*x)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 6.36 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\frac {\sqrt {x} \left (4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (a+b x^2+c x^4\right )+i b \left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )-i \left (-b^2+4 a c+b \sqrt {b^2-4 a c}\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{12 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {x \left (a+b x^2+c x^4\right )}} \] Input:

Integrate[Sqrt[a*x + b*x^3 + c*x^5]/Sqrt[x],x]
 

Output:

(Sqrt[x]*(4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(a + b*x^2 + c*x^4) + I*b* 
(-b + Sqrt[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[ 
b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 
4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], ( 
b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - I*(-b^2 + 4*a*c + b*Sqrt 
[b^2 - 4*a*c])*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a* 
c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*E 
llipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b 
^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])]))/(12*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c 
])]*Sqrt[x*(a + b*x^2 + c*x^4)])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1968, 2000, 1511, 27, 1416, 1509}

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.

\(\displaystyle \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx\)

\(\Big \downarrow \) 1968

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

\(\Big \downarrow \) 2000

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

Int[Sqrt[a*x + b*x^3 + c*x^5]/Sqrt[x],x]
 

Output:

(Sqrt[x]*Sqrt[a*x + b*x^3 + c*x^5])/3 + (Sqrt[x]*Sqrt[a + b*x^2 + c*x^4]*( 
-((b*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + Sqrt[c]*x^2)) + (a^(1/4)*(S 
qrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]* 
EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(c^ 
(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a^(1/4)*(2*Sqrt[a] + b/Sqrt[c 
])*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2 
)^2]*EllipticF[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4] 
)/(2*c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/(3*Sqrt[a*x + b*x^3 + c*x^5])
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

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

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 1968

Int[(x_)^(m_.)*((b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(p_ 
), x_Symbol] :> Simp[x^(m + 1)*((a*x^q + b*x^n + c*x^(2*n - q))^p/(m + p*(2 
*n - q) + 1)), x] + Simp[(n - q)*(p/(m + p*(2*n - q) + 1))   Int[x^(m + q)* 
(2*a + b*x^(n - q))*(a*x^q + b*x^n + c*x^(2*n - q))^(p - 1), x], x] /; Free 
Q[{a, b, c}, x] && EqQ[r, 2*n - q] && PosQ[n - q] &&  !IntegerQ[p] && NeQ[b 
^2 - 4*a*c, 0] && IGtQ[n, 0] && GtQ[p, 0] && RationalQ[m, q] && GtQ[m + p*q 
 + 1, -(n - q)] && NeQ[m + p*(2*n - q) + 1, 0]
 

rule 2000

Int[((x_)^(m_.)*((A_) + (B_.)*(x_)^(j_.)))/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x 
_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Simp[x^(q/2)*(Sqrt[a + b*x^(n - 
q) + c*x^(2*(n - q))]/Sqrt[a*x^q + b*x^n + c*x^(2*n - q)])   Int[x^(m - q/2 
)*((A + B*x^(n - q))/Sqrt[a + b*x^(n - q) + c*x^(2*(n - q))]), x], x] /; Fr 
eeQ[{a, b, c, A, B, m, n, q}, x] && EqQ[j, n - q] && EqQ[r, 2*n - q] && Pos 
Q[n - q] && (EqQ[m, 1/2] || EqQ[m, -2^(-1)]) && EqQ[n, 3] && EqQ[q, 1]
 

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.24

method	result	size
risch	\(\frac {x^{\frac {3}{2}} \left (c \,x^{4}+b \,x^{2}+a \right )}{3 \sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}+\frac {\left (\frac {a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {b a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{6 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\right ) \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {x}}{\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}}\)	\(430\)
default	\(\frac {\sqrt {x \left (c \,x^{4}+b \,x^{2}+a \right )}\, \left (\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, c \,x^{5}+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b c \,x^{5}+\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b \,x^{3}+\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, b^{2} x^{3}+a \sqrt {-\frac {2 \left (\sqrt {-4 a c +b^{2}}\, x^{2}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, x^{2}+b \,x^{2}+2 a}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right ) \sqrt {-4 a c +b^{2}}+b a \sqrt {-\frac {2 \left (\sqrt {-4 a c +b^{2}}\, x^{2}-b \,x^{2}-2 a \right )}{a}}\, \sqrt {\frac {\sqrt {-4 a c +b^{2}}\, x^{2}+b \,x^{2}+2 a}{a}}\, \operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {2}\, \sqrt {\frac {b \sqrt {-4 a c +b^{2}}-2 a c +b^{2}}{a c}}}{2}\right )+\sqrt {-4 a c +b^{2}}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a x +\sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, a b x \right )}{3 \sqrt {x}\, \left (c \,x^{4}+b \,x^{2}+a \right ) \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(508\)

Input:

int((c*x^5+b*x^3+a*x)^(1/2)/x^(1/2),x,method=_RETURNVERBOSE)
 

Output:

1/3*x^(3/2)*(c*x^4+b*x^2+a)/(x*(c*x^4+b*x^2+a))^(1/2)+(1/6*a*2^(1/2)/((-b+ 
(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+ 
2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2* 
x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1 
/2))/a/c)^(1/2))-1/6*b*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2*(- 
b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2 
)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2*x*2^(1/2)*(( 
-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1 
/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b 
*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))))*(c*x^4+b*x^2+a)^(1/2)*x^(1/2)/(x*(c* 
x^4+b*x^2+a))^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 314, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (b c x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b^{2} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} E(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) - \sqrt {\frac {1}{2}} {\left ({\left (b c - 2 \, c^{2}\right )} x^{2} \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - {\left (b^{2} + 2 \, b c\right )} x^{2}\right )} \sqrt {c} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}} F(\arcsin \left (\frac {\sqrt {\frac {1}{2}} \sqrt {\frac {c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} - b}{c}}}{x}\right )\,|\,\frac {b c \sqrt {\frac {b^{2} - 4 \, a c}{c^{2}}} + b^{2} - 2 \, a c}{2 \, a c}) + 2 \, \sqrt {c x^{5} + b x^{3} + a x} {\left (c^{2} x^{2} + b c\right )} \sqrt {x}}{6 \, c^{2} x^{2}} \] Input:

integrate((c*x^5+b*x^3+a*x)^(1/2)/x^(1/2),x, algorithm="fricas")
 

Output:

1/6*(sqrt(1/2)*(b*c*x^2*sqrt((b^2 - 4*a*c)/c^2) - b^2*x^2)*sqrt(c)*sqrt((c 
*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1/2)*sqrt((c*sqrt( 
(b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c)/c^2) + b^2 - 2* 
a*c)/(a*c)) - sqrt(1/2)*((b*c - 2*c^2)*x^2*sqrt((b^2 - 4*a*c)/c^2) - (b^2 
+ 2*b*c)*x^2)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_f(a 
rcsin(sqrt(1/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt( 
(b^2 - 4*a*c)/c^2) + b^2 - 2*a*c)/(a*c)) + 2*sqrt(c*x^5 + b*x^3 + a*x)*(c^ 
2*x^2 + b*c)*sqrt(x))/(c^2*x^2)
 

Sympy [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\int \frac {\sqrt {x \left (a + b x^{2} + c x^{4}\right )}}{\sqrt {x}}\, dx \] Input:

integrate((c*x**5+b*x**3+a*x)**(1/2)/x**(1/2),x)
 

Output:

Integral(sqrt(x*(a + b*x**2 + c*x**4))/sqrt(x), x)
 

Maxima [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\int { \frac {\sqrt {c x^{5} + b x^{3} + a x}}{\sqrt {x}} \,d x } \] Input:

integrate((c*x^5+b*x^3+a*x)^(1/2)/x^(1/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(c*x^5 + b*x^3 + a*x)/sqrt(x), x)
 

Giac [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\int { \frac {\sqrt {c x^{5} + b x^{3} + a x}}{\sqrt {x}} \,d x } \] Input:

integrate((c*x^5+b*x^3+a*x)^(1/2)/x^(1/2),x, algorithm="giac")
 

Output:

integrate(sqrt(c*x^5 + b*x^3 + a*x)/sqrt(x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\int \frac {\sqrt {c\,x^5+b\,x^3+a\,x}}{\sqrt {x}} \,d x \] Input:

int((a*x + b*x^3 + c*x^5)^(1/2)/x^(1/2),x)
 

Output:

int((a*x + b*x^3 + c*x^5)^(1/2)/x^(1/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {a x+b x^3+c x^5}}{\sqrt {x}} \, dx=\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x}{3}+\frac {2 \left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c \,x^{4}+b \,x^{2}+a}d x \right ) a}{3}+\frac {\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c \,x^{4}+b \,x^{2}+a}d x \right ) b}{3} \] Input:

int((c*x^5+b*x^3+a*x)^(1/2)/x^(1/2),x)
 

Output:

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