3.1075 \(\int \frac{x^2}{(-2+b x^2) (-1+b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=72 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0242826, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {442} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 442

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

Rubi steps

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

Mathematica [C]  time = 0.0500228, size = 54, normalized size = 0.75 \[ -\frac{x^3 \left (1-b x^2\right )^{3/4} F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};b x^2,\frac{b x^2}{2}\right )}{6 \left (b x^2-1\right )^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(x^3*(1 - b*x^2)^(3/4)*AppellF1[3/2, 3/4, 1, 5/2, b*x^2, (b*x^2)/2])/(6*(-1 + b*x^2)^(3/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{b{x}^{2}-2} \left ( b{x}^{2}-1 \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)/(b*x^2-1)^(3/4),x)

[Out]

int(x^2/(b*x^2-2)/(b*x^2-1)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x^{2} - 1\right )}^{\frac{3}{4}}{\left (b x^{2} - 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)), x)

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Fricas [B]  time = 1.64654, size = 707, normalized size = 9.82 \begin{align*} \left [-\frac{2 \, \sqrt{2} \sqrt{b} \arctan \left (\frac{\sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}}}{\sqrt{b} x}\right ) - \sqrt{2} \sqrt{b} \log \left (-\frac{b^{2} x^{4} - 2 \, \sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}} b^{\frac{3}{2}} x^{3} + 4 \, \sqrt{b x^{2} - 1} b x^{2} + 4 \, b x^{2} - 4 \, \sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{3}{4}} \sqrt{b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}, \frac{2 \, \sqrt{2} \sqrt{-b} \arctan \left (\frac{\sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b}}{b x}\right ) - \sqrt{2} \sqrt{-b} \log \left (-\frac{b^{2} x^{4} - 2 \, \sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b} b x^{3} - 4 \, \sqrt{b x^{2} - 1} b x^{2} + 4 \, b x^{2} + 4 \, \sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{3}{4}} \sqrt{-b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (b x^{2} - 2\right ) \left (b x^{2} - 1\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)/(b*x**2-1)**(3/4),x)

[Out]

Integral(x**2/((b*x**2 - 2)*(b*x**2 - 1)**(3/4)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{{\left (b x^{2} - 1\right )}^{\frac{3}{4}}{\left (b x^{2} - 2\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)), x)