3.283 \(\int \frac{x^2}{(b x^2+c x^4)^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0618186, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2023, 2008, 206} \[ \frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(b*x^2 + c*x^4)^(3/2),x]

[Out]

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

Rule 2023

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(c^(j - 1)*(c*x)^(m - j
+ 1)*(a*x^j + b*x^n)^(p + 1))/(a*(n - j)*(p + 1)), x] + Dist[(c^j*(m + n*p + n - j + 1))/(a*(n - j)*(p + 1)),
Int[(c*x)^(m - j)*(a*x^j + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] &
& (IntegersQ[j, n] || GtQ[c, 0]) && LtQ[p, -1]

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]

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

Mathematica [C]  time = 0.0091179, size = 38, normalized size = 0.75 \[ \frac{x \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c x^2}{b}+1\right )}{b \sqrt{x^2 \left (b+c x^2\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (c*x^2)/b])/(b*Sqrt[x^2*(b + c*x^2)])

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Maple [A]  time = 0.046, size = 67, normalized size = 1.3 \begin{align*} -{{x}^{3} \left ( c{x}^{2}+b \right ) \left ( \ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) b\sqrt{c{x}^{2}+b}-{b}^{{\frac{3}{2}}} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.32026, size = 348, normalized size = 6.82 \begin{align*} \left [\frac{{\left (c x^{3} + b x\right )} \sqrt{b} \log \left (-\frac{c x^{3} + 2 \, b x - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{b}}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} b}{2 \,{\left (b^{2} c x^{3} + b^{3} x\right )}}, \frac{{\left (c x^{3} + b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{c x^{4} + b x^{2}} \sqrt{-b}}{c x^{3} + b x}\right ) + \sqrt{c x^{4} + b x^{2}} b}{b^{2} c x^{3} + b^{3} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: NotImplementedError