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

Optimal. Leaf size=33 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}-b x^2}}\right )}{5 \sqrt {b}} \]

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Rubi [A]  time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1979, 2008, 203} \begin {gather*} \frac {2 \tan ^{-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*ArcTan[(Sqrt[b]*x)/Sqrt[a/x^3 - b*x^2]])/(5*Sqrt[b])

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[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 \tan ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {\frac {a}{x^3}-b x^2}}\right )}{5 \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 66, normalized size = 2.00 \begin {gather*} \frac {2 \sqrt {a-b x^5} \tan ^{-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]*ArcTan[(Sqrt[b]*x^(5/2))/Sqrt[a - b*x^5]])/(5*Sqrt[b]*x^(3/2)*Sqrt[(a - b*x^5)/x^3])

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IntegrateAlgebraic [C]  time = 74.85, size = 72, normalized size = 2.18 \begin {gather*} -\frac {2 i x^{3/2} \sqrt {\frac {a-b x^5}{x^3}} \log \left (\sqrt {a-b x^5}+i \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*I)/5)*x^(3/2)*Sqrt[(a - b*x^5)/x^3]*Log[I*Sqrt[b]*x^(5/2) + Sqrt[a - b*x^5]])/(Sqrt[b]*Sqrt[a - b*x^5])

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fricas [A]  time = 0.93, size = 111, normalized size = 3.36 \begin {gather*} \left [-\frac {\sqrt {-b} \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 \, b}, -\frac {\arctan \left (\frac {2 \, \sqrt {b} x^{4} \sqrt {-\frac {b x^{5} - a}{x^{3}}}}{2 \, b x^{5} - a}\right )}{5 \, \sqrt {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*sqrt(-b)*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))/b, -1
/5*arctan(2*sqrt(b)*x^4*sqrt(-(b*x^5 - a)/x^3)/(2*b*x^5 - a))/sqrt(b)]

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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)]Undef/Unsigned Inf encountered in limitLimit: Max order reached or unable to mak
e series expansion Error: Bad Argument Value

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maple [F]  time = 0.16, 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)

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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)

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

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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

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