3.4.12 \(\int \frac {1}{\sqrt {\frac {a+b x^5}{x^3}}} \, dx\)

Optimal. Leaf size=32 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}+b x^2}}\right )}{5 \sqrt {b}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1979, 2008, 206} \begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}+b x^2}}\right )}{5 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(a + b*x^5)/x^3],x]

[Out]

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

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 1979

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedBinomialQ[u, x] &&  !Gene
ralizedBinomialMatchQ[u, x]

Rule 2008

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.)], x_Symbol] :> Dist[2/(2 - n), Subst[Int[1/(1 - a*x^2), x], x, x/Sq
rt[a*x^2 + b*x^n]], x] /; FreeQ[{a, b, n}, x] && NeQ[n, 2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 63, normalized size = 1.97 \begin {gather*} \frac {2 \sqrt {a+b x^5} \tanh ^{-1}\left (\frac {\sqrt {b} x^{5/2}}{\sqrt {a+b x^5}}\right )}{5 \sqrt {b} x^{3/2} \sqrt {\frac {a+b x^5}{x^3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(a + b*x^5)/x^3],x]

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 75.15, size = 64, normalized size = 2.00 \begin {gather*} \frac {2 x^{3/2} \sqrt {\frac {a+b x^5}{x^3}} \log \left (\sqrt {a+b x^5}+\sqrt {b} x^{5/2}\right )}{5 \sqrt {b} \sqrt {a+b x^5}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[(a + b*x^5)/x^3],x]

[Out]

(2*x^(3/2)*Sqrt[(a + b*x^5)/x^3]*Log[Sqrt[b]*x^(5/2) + Sqrt[a + b*x^5]])/(5*Sqrt[b]*Sqrt[a + b*x^5])

________________________________________________________________________________________

fricas [A]  time = 0.93, size = 102, normalized size = 3.19 \begin {gather*} \left [\frac {\log \left (-8 \, b^{2} x^{10} - 8 \, a b x^{5} - a^{2} - 4 \, {\left (2 \, b x^{9} + a x^{4}\right )} \sqrt {b} \sqrt {\frac {b x^{5} + a}{x^{3}}}\right )}{10 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (\frac {2 \, \sqrt {-b} x^{4} \sqrt {\frac {b x^{5} + a}{x^{3}}}}{2 \, b x^{5} + a}\right )}{5 \, b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/10*log(-8*b^2*x^10 - 8*a*b*x^5 - a^2 - 4*(2*b*x^9 + a*x^4)*sqrt(b)*sqrt((b*x^5 + a)/x^3))/sqrt(b), -1/5*sqr
t(-b)*arctan(2*sqrt(-b)*x^4*sqrt((b*x^5 + a)/x^3)/(2*b*x^5 + a))/b]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(t_nostep)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the ar
gument is real):Check [abs(x)]Limit: Max order reached or unable to make series expansion Error: Bad Argument
Value

________________________________________________________________________________________

maple [F]  time = 0.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\frac {b \,x^{5}+a}{x^{3}}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {\frac {b x^{5} + a}{x^{3}}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt((b*x^5 + a)/x^3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {\frac {b\,x^5+a}{x^3}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*x^5)/x^3)^(1/2),x)

[Out]

int(1/((a + b*x^5)/x^3)^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________