3.1053 \(\int \frac {x^2}{(2+3 x^2)^{3/4} (4+3 x^2)} \, dx\)

Optimal. Leaf size=129 \[ \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]

[Out]

-1/18*arctan(1/6*(2*2^(3/4)+2*2^(1/4)*(3*x^2+2)^(1/2))/x/(3*x^2+2)^(1/4)*3^(1/2))*2^(3/4)*3^(1/2)+1/18*arctanh
(1/6*(2*2^(3/4)-2*2^(1/4)*(3*x^2+2)^(1/2))/x/(3*x^2+2)^(1/4)*3^(1/2))*2^(3/4)*3^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 129, 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 = {441} \[ \frac {\tanh ^{-1}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 0.04, size = 37, normalized size = 0.29 \[ \frac {x^3 F_1\left (\frac {3}{2};\frac {3}{4},1;\frac {5}{2};-\frac {3 x^2}{2},-\frac {3 x^2}{4}\right )}{12\ 2^{3/4}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [B]  time = 1.16, size = 282, normalized size = 2.19 \[ \frac {1}{216} \cdot 72^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {72^{\frac {1}{4}} \sqrt {6} \sqrt {2} x \sqrt {\frac {72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 18 \, \sqrt {2} x^{2} + 24 \, \sqrt {3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} - 36 \, x}{36 \, x}\right ) + \frac {1}{216} \cdot 72^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {72^{\frac {1}{4}} \sqrt {6} \sqrt {2} x \sqrt {-\frac {72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x - 18 \, \sqrt {2} x^{2} - 24 \, \sqrt {3 \, x^{2} + 2}}{x^{2}}} - 12 \cdot 72^{\frac {1}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} + 36 \, x}{36 \, x}\right ) - \frac {1}{864} \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {96 \, {\left (72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x + 18 \, \sqrt {2} x^{2} + 24 \, \sqrt {3 \, x^{2} + 2}\right )}}{x^{2}}\right ) + \frac {1}{864} \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (-\frac {96 \, {\left (72^{\frac {3}{4}} \sqrt {2} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}} x - 18 \, \sqrt {2} x^{2} - 24 \, \sqrt {3 \, x^{2} + 2}\right )}}{x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/216*72^(3/4)*sqrt(2)*arctan(1/36*(72^(1/4)*sqrt(6)*sqrt(2)*x*sqrt((72^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 18
*sqrt(2)*x^2 + 24*sqrt(3*x^2 + 2))/x^2) - 12*72^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4) - 36*x)/x) + 1/216*72^(3/4)*sq
rt(2)*arctan(1/36*(72^(1/4)*sqrt(6)*sqrt(2)*x*sqrt(-(72^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x - 18*sqrt(2)*x^2 - 2
4*sqrt(3*x^2 + 2))/x^2) - 12*72^(1/4)*sqrt(2)*(3*x^2 + 2)^(1/4) + 36*x)/x) - 1/864*72^(3/4)*sqrt(2)*log(96*(72
^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x + 18*sqrt(2)*x^2 + 24*sqrt(3*x^2 + 2))/x^2) + 1/864*72^(3/4)*sqrt(2)*log(-9
6*(72^(3/4)*sqrt(2)*(3*x^2 + 2)^(1/4)*x - 18*sqrt(2)*x^2 - 24*sqrt(3*x^2 + 2))/x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 7.44, size = 186, normalized size = 1.44 \[ -\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (\frac {3 x \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+\left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+9 \sqrt {3 x^{2}+2}\, x -6 \left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}+4}\right )}{18}-\frac {\RootOf \left (\textit {\_Z}^{4}+18\right ) \ln \left (-\frac {3 x \RootOf \left (\textit {\_Z}^{4}+18\right )^{2}+\left (3 x^{2}+2\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )^{3}-9 \sqrt {3 x^{2}+2}\, x +6 \left (3 x^{2}+2\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+18\right )}{3 x^{2}+4}\right )}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*RootOf(_Z^2+RootOf(_Z^4+18)^2)*ln((RootOf(_Z^4+18)^2*RootOf(_Z^2+RootOf(_Z^4+18)^2)*(3*x^2+2)^(3/4)+3*Ro
otOf(_Z^4+18)^2*x+9*(3*x^2+2)^(1/2)*x-6*RootOf(_Z^2+RootOf(_Z^4+18)^2)*(3*x^2+2)^(1/4))/(3*x^2+4))-1/18*RootOf
(_Z^4+18)*ln(-((3*x^2+2)^(3/4)*RootOf(_Z^4+18)^3+3*RootOf(_Z^4+18)^2*x-9*(3*x^2+2)^(1/2)*x+6*(3*x^2+2)^(1/4)*R
ootOf(_Z^4+18))/(3*x^2+4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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