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

Optimal. Leaf size=77 \[ \frac{b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2} \]

[Out]

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

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Rubi [A]  time = 0.0540936, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1927, 1904, 206} \[ \frac{b \tanh ^{-1}\left (\frac{x (2 a+b x)}{2 \sqrt{a} \sqrt{a x^2+b x^3+c x^4}}\right )}{2 a^{3/2}}-\frac{\sqrt{a x^2+b x^3+c x^4}}{a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[a*x^2 + b*x^3 + c*x^4]),x]

[Out]

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

Rule 1927

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

Rule 1904

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

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])

Rubi steps

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

Mathematica [A]  time = 0.0555978, size = 89, normalized size = 1.16 \[ \frac{b x \sqrt{a+x (b+c x)} \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+x (b+c x)}}\right )-2 \sqrt{a} (a+x (b+c x))}{2 a^{3/2} \sqrt{x^2 (a+x (b+c x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[a*x^2 + b*x^3 + c*x^4]),x]

[Out]

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

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Maple [A]  time = 0.006, size = 88, normalized size = 1.1 \begin{align*} -{\frac{1}{2}\sqrt{c{x}^{2}+bx+a} \left ( 2\,\sqrt{c{x}^{2}+bx+a}{a}^{3/2}-b\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ) ax \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{3}+a{x}^{2}}}}{a}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(c*x^2+b*x+a)^(1/2)*(2*(c*x^2+b*x+a)^(1/2)*a^(3/2)-b*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)*a*x)/(
c*x^4+b*x^3+a*x^2)^(1/2)/a^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c x^{4} + b x^{3} + a x^{2}} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73191, size = 444, normalized size = 5.77 \begin{align*} \left [\frac{\sqrt{a} b x^{2} \log \left (-\frac{8 \, a b x^{2} +{\left (b^{2} + 4 \, a c\right )} x^{3} + 8 \, a^{2} x + 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{a}}{x^{3}}\right ) - 4 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}{4 \, a^{2} x^{2}}, -\frac{\sqrt{-a} b x^{2} \arctan \left (\frac{\sqrt{c x^{4} + b x^{3} + a x^{2}}{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \,{\left (a c x^{3} + a b x^{2} + a^{2} x\right )}}\right ) + 2 \, \sqrt{c x^{4} + b x^{3} + a x^{2}} a}{2 \, a^{2} x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{x^{2} \left (a + b x + c x^{2}\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(x**2*(a + b*x + c*x**2))), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError