3.3.5 \(\int \frac {1}{\sqrt {a x^2+b x^5}} \, dx\)

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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2008, 206} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*x^2 + b*x^5],x]

[Out]

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

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Mathematica [A]  time = 0.01, size = 54, normalized size = 1.69 \begin {gather*} -\frac {2 x \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 \sqrt {a} \sqrt {x^2 \left (a+b x^3\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*x^2 + b*x^5],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.05, size = 32, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^5}}\right )}{3 \sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/Sqrt[a*x^2 + b*x^5],x]

[Out]

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

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fricas [A]  time = 0.42, size = 75, normalized size = 2.34 \begin {gather*} \left [\frac {\log \left (\frac {b x^{4} + 2 \, a x - 2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {a}}{x^{4}}\right )}{3 \, \sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x^{5} + a x^{2}} \sqrt {-a}}{a x}\right )}{3 \, a}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.19, size = 47, normalized size = 1.47 \begin {gather*} -\frac {2 \, \arctan \left (\frac {\sqrt {a}}{\sqrt {-a}}\right ) \mathrm {sgn}\relax (x)}{3 \, \sqrt {-a}} + \frac {2 \, \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{3 \, \sqrt {-a} \mathrm {sgn}\relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*arctan(sqrt(a)/sqrt(-a))*sgn(x)/sqrt(-a) + 2/3*arctan(sqrt(b*x^3 + a)/sqrt(-a))/(sqrt(-a)*sgn(x))

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maple [A]  time = 0.05, size = 43, normalized size = 1.34 \begin {gather*} -\frac {2 \sqrt {b \,x^{3}+a}\, x \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right )}{3 \sqrt {b \,x^{5}+a \,x^{2}}\, \sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b x^{5} + a x^{2}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(b*x^5 + a*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {b\,x^5+a\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*x^2 + b*x^5)^(1/2),x)

[Out]

int(1/(a*x^2 + b*x^5)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x^{2} + b x^{5}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*x**2 + b*x**5), x)

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