3.1056 \(\int \frac{x^2}{(2-b x^2)^{3/4} (4-b x^2)} \, dx\)

Optimal. Leaf size=119 \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-b x^2}}{\sqrt [4]{2} \sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-b x^2}+2}{\sqrt [4]{2} \sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

[Out]

ArcTan[(2 - Sqrt[2]*Sqrt[2 - b*x^2])/(2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))]/(2^(1/4)*b^(3/2)) - ArcTanh[(2 + S
qrt[2]*Sqrt[2 - b*x^2])/(2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))]/(2^(1/4)*b^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0317682, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {441} \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-b x^2}}{\sqrt [4]{2} \sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-b x^2}+2}{\sqrt [4]{2} \sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/((2 - b*x^2)^(3/4)*(4 - b*x^2)),x]

[Out]

ArcTan[(2 - Sqrt[2]*Sqrt[2 - b*x^2])/(2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))]/(2^(1/4)*b^(3/2)) - ArcTanh[(2 + S
qrt[2]*Sqrt[2 - b*x^2])/(2^(1/4)*Sqrt[b]*x*(2 - b*x^2)^(1/4))]/(2^(1/4)*b^(3/2))

Rule 441

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> -Simp[(b*ArcTan[(b + Rt[b^2/a, 4]
^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))])/(a*d*Rt[b^2/a, 4]^3), x] + Simp[(b*ArcTanh[(b - Rt[
b^2/a, 4]^2*Sqrt[a + b*x^2])/(Rt[b^2/a, 4]^3*x*(a + b*x^2)^(1/4))])/(a*d*Rt[b^2/a, 4]^3), x] /; FreeQ[{a, b, c
, d}, x] && EqQ[b*c - 2*a*d, 0] && PosQ[b^2/a]

Rubi steps

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

Mathematica [C]  time = 0.0485828, size = 39, normalized size = 0.33 \[ \frac{x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right )}{12\ 2^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3*AppellF1[3/2, 3/4, 1, 5/2, (b*x^2)/2, (b*x^2)/4])/(12*2^(3/4))

________________________________________________________________________________________

Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-b{x}^{2}+4} \left ( -b{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.72266, size = 1262, normalized size = 10.61 \begin{align*} \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \arctan \left (\frac{8 \, \sqrt{2} \sqrt{\frac{1}{2}} \left (\frac{1}{8}\right )^{\frac{3}{4}} b^{4} \sqrt{\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} - 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{x^{2}}} \frac{1}{b^{6}}^{\frac{3}{4}} x - 8 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{3}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{4} \frac{1}{b^{6}}^{\frac{3}{4}} + x}{x}\right ) + \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \arctan \left (\frac{8 \, \sqrt{2} \sqrt{\frac{1}{2}} \left (\frac{1}{8}\right )^{\frac{3}{4}} b^{4} x \sqrt{\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} + 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{x^{2}}} \frac{1}{b^{6}}^{\frac{3}{4}} - 8 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{3}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{4} \frac{1}{b^{6}}^{\frac{3}{4}} - x}{x}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \log \left (\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} + 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{2 \, x^{2}}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \log \left (\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} - 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{2 \, x^{2}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(1/8)^(1/4)*(b^(-6))^(1/4)*arctan((8*sqrt(2)*sqrt(1/2)*(1/8)^(3/4)*b^4*sqrt((sqrt(1/2)*b^4*sqrt(b^(-6)
)*x^2 - 2*sqrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)^(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))/x^2)*(b^(-6))^(3/4
)*x - 8*sqrt(2)*(1/8)^(3/4)*(-b*x^2 + 2)^(1/4)*b^4*(b^(-6))^(3/4) + x)/x) + sqrt(2)*(1/8)^(1/4)*(b^(-6))^(1/4)
*arctan((8*sqrt(2)*sqrt(1/2)*(1/8)^(3/4)*b^4*x*sqrt((sqrt(1/2)*b^4*sqrt(b^(-6))*x^2 + 2*sqrt(2)*(1/8)^(1/4)*(-
b*x^2 + 2)^(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))/x^2)*(b^(-6))^(3/4) - 8*sqrt(2)*(1/8)^(3/4)*(-b*x^
2 + 2)^(1/4)*b^4*(b^(-6))^(3/4) - x)/x) - 1/4*sqrt(2)*(1/8)^(1/4)*(b^(-6))^(1/4)*log(1/2*(sqrt(1/2)*b^4*sqrt(b
^(-6))*x^2 + 2*sqrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)^(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))/x^2) + 1/4*sq
rt(2)*(1/8)^(1/4)*(b^(-6))^(1/4)*log(1/2*(sqrt(1/2)*b^4*sqrt(b^(-6))*x^2 - 2*sqrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)^
(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))/x^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(x**2/(b*x**2*(-b*x**2 + 2)**(3/4) - 4*(-b*x**2 + 2)**(3/4)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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