3.11.55 \(\int \frac {x^2}{(2+b x^2)^{3/4} (4+b x^2)} \, dx\) [1055]

Optimal. Leaf size=124 \[ -\frac {\tan ^{-1}\left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 124, 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 = {452} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {b x^2+2}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac {\text {ArcTan}\left (\frac {2 \sqrt [4]{2} \sqrt {b x^2+2}+2\ 2^{3/4}}{2 \sqrt {b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 452

Int[(x_)^2/(((a_) + (b_.)*(x_)^2)^(3/4)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Simp[(-b/(a*d*Rt[b^2/a, 4]^3))*Ar
cTan[(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))], x] + Simp[(b/(a*d*Rt[b^2/a, 4
]^3))*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))], 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\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}+\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+b x^2}}{2 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 2.10, size = 112, normalized size = 0.90 \begin {gather*} -\frac {\tan ^{-1}\left (\frac {-2^{3/4} b x^2+4 \sqrt [4]{2} \sqrt {2+b x^2}}{4 \sqrt {b} x \sqrt [4]{2+b x^2}}\right )+\tanh ^{-1}\left (\frac {2 \sqrt {b} x \sqrt [4]{4+2 b x^2}}{b x^2+2 \sqrt {4+2 b x^2}}\right )}{2 \sqrt [4]{2} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (b \,x^{2}+2\right )^{\frac {3}{4}} \left (b \,x^{2}+4\right )}\, dx\]

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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 393 vs. \(2 (86) = 172\).
time = 0.91, size = 393, normalized size = 3.17 \begin {gather*} \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 {gather*}

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)*ar
ctan((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*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)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\left (b x^{2} + 2\right )^{\frac {3}{4}} \left (b x^{2} + 4\right )}\, dx \end {gather*}

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 + 2)**(3/4)*(b*x**2 + 4)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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

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)

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