3.136 \(\int \frac {\sqrt {x}}{\sqrt {x^3 (a+b x^2+c x^4)}} \, dx\)

Optimal. Leaf size=53 \[ -\frac {\tanh ^{-1}\left (\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^3+b x^5+c x^7}}\right )}{2 \sqrt {a}} \]

[Out]

-1/2*arctanh(1/2*x^(3/2)*(b*x^2+2*a)/a^(1/2)/(c*x^7+b*x^5+a*x^3)^(1/2))/a^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.08, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1997, 1913, 206} \[ -\frac {\tanh ^{-1}\left (\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^3+b x^5+c x^7}}\right )}{2 \sqrt {a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[(x^(3/2)*(2*a + b*x^2))/(2*Sqrt[a]*Sqrt[a*x^3 + b*x^5 + c*x^7])]/(2*Sqrt[a])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1913

Int[(x_)^(m_.)/Sqrt[(b_.)*(x_)^(n_.) + (a_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - q), Sub
st[Int[1/(4*a - x^2), x], x, (x^(m + 1)*(2*a + b*x^(n - q)))/Sqrt[a*x^q + b*x^n + c*x^r]], x] /; FreeQ[{a, b,
c, m, n, q, r}, x] && EqQ[r, 2*n - q] && PosQ[n - q] && NeQ[b^2 - 4*a*c, 0] && EqQ[m, q/2 - 1]

Rule 1997

Int[(u_)^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Int[(d*x)^m*ExpandToSum[u, x]^p, x] /; FreeQ[{d, m, p}, x] &&
GeneralizedTrinomialQ[u, x] &&  !GeneralizedTrinomialMatchQ[u, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {x}}{\sqrt {x^3 \left (a+b x^2+c x^4\right )}} \, dx &=\int \frac {\sqrt {x}}{\sqrt {a x^3+b x^5+c x^7}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{4 a-x^2} \, dx,x,\frac {x^{3/2} \left (2 a+b x^2\right )}{\sqrt {a x^3+b x^5+c x^7}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x^{3/2} \left (2 a+b x^2\right )}{2 \sqrt {a} \sqrt {a x^3+b x^5+c x^7}}\right )}{2 \sqrt {a}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 85, normalized size = 1.60 \[ -\frac {x^{3/2} \sqrt {a+b x^2+c x^4} \tanh ^{-1}\left (\frac {2 a+b x^2}{2 \sqrt {a} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {a} \sqrt {x^3 \left (a+b x^2+c x^4\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 145, normalized size = 2.74 \[ \left [\frac {\log \left (-\frac {{\left (b^{2} + 4 \, a c\right )} x^{6} + 8 \, a b x^{4} + 8 \, a^{2} x^{2} - 4 \, \sqrt {c x^{7} + b x^{5} + a x^{3}} {\left (b x^{2} + 2 \, a\right )} \sqrt {a} \sqrt {x}}{x^{6}}\right )}{4 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {c x^{7} + b x^{5} + a x^{3}} {\left (b x^{2} + 2 \, a\right )} \sqrt {-a} \sqrt {x}}{2 \, {\left (a c x^{6} + a b x^{4} + a^{2} x^{2}\right )}}\right )}{2 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*log(-((b^2 + 4*a*c)*x^6 + 8*a*b*x^4 + 8*a^2*x^2 - 4*sqrt(c*x^7 + b*x^5 + a*x^3)*(b*x^2 + 2*a)*sqrt(a)*sqr
t(x))/x^6)/sqrt(a), 1/2*sqrt(-a)*arctan(1/2*sqrt(c*x^7 + b*x^5 + a*x^3)*(b*x^2 + 2*a)*sqrt(-a)*sqrt(x)/(a*c*x^
6 + a*b*x^4 + a^2*x^2))/a]

________________________________________________________________________________________

giac [A]  time = 0.55, size = 56, normalized size = 1.06 \[ \frac {\arctan \left (-\frac {\sqrt {c} x^{2} - \sqrt {c x^{4} + b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right )}{\sqrt {-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 74, normalized size = 1.40 \[ -\frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{\frac {3}{2}} \ln \left (\frac {b \,x^{2}+2 a +2 \sqrt {c \,x^{4}+b \,x^{2}+a}\, \sqrt {a}}{x^{2}}\right )}{2 \sqrt {\left (c \,x^{4}+b \,x^{2}+a \right ) x^{3}}\, \sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x}}{\sqrt {{\left (c x^{4} + b x^{2} + a\right )} x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {x}}{\sqrt {x^3\,\left (c\,x^4+b\,x^2+a\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________