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]

1/2*arctan(1/2*x*b^(1/2)/(b*x^2-1)^(1/4)*2^(1/2))/b^(3/2)*2^(1/2)-1/2*arctanh(1/2*x*b^(1/2)/(b*x^2-1)^(1/4)*2^
(1/2))/b^(3/2)*2^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, 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*}

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Mathematica [C]  time = 0.05, 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]

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

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fricas [B]  time = 0.90, size = 275, normalized size = 3.82 \[ \left [-\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}}}{\sqrt {b} x}\right ) - \sqrt {2} \sqrt {b} \log \left (-\frac {b^{2} x^{4} - 2 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} b^{\frac {3}{2}} x^{3} + 4 \, \sqrt {b x^{2} - 1} b x^{2} + 4 \, b x^{2} - 4 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}, \frac {2 \, \sqrt {2} \sqrt {-b} \arctan \left (\frac {\sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b}}{b x}\right ) - \sqrt {2} \sqrt {-b} \log \left (-\frac {b^{2} x^{4} - 2 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {1}{4}} \sqrt {-b} b x^{3} - 4 \, \sqrt {b x^{2} - 1} b x^{2} + 4 \, b x^{2} + 4 \, \sqrt {2} {\left (b x^{2} - 1\right )}^{\frac {3}{4}} \sqrt {-b} x - 4}{b^{2} x^{4} - 4 \, b x^{2} + 4}\right )}{4 \, b^{2}}\right ] \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b x^{2} - 1\right )}^{\frac {3}{4}} {\left (b x^{2} - 2\right )}}\,{d x} \]

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)

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maple [F]  time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b \,x^{2}-2\right ) \left (b \,x^{2}-1\right )^{\frac {3}{4}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (b x^{2} - 1\right )}^{\frac {3}{4}} {\left (b x^{2} - 2\right )}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2}{{\left (b\,x^2-1\right )}^{3/4}\,\left (b\,x^2-2\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\left (b x^{2} - 2\right ) \left (b x^{2} - 1\right )^{\frac {3}{4}}}\, dx \]

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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