\(\int \frac {x^2}{(2+3 x^2)^{3/4} (4+3 x^2)} \, dx\) [1053]

Optimal result

Integrand size = 24, antiderivative size = 129 \[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {2\ 2^{3/4}+2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {2+3 x^2}}{2 \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]

[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)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {452} \[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt {3 x^2+2}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}}-\frac {\arctan \left (\frac {2 \sqrt [4]{2} \sqrt {3 x^2+2}+2\ 2^{3/4}}{2 \sqrt {3} x \sqrt [4]{3 x^2+2}}\right )}{3 \sqrt [4]{2} \sqrt {3}} \]

[In]

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

[Out]

-1/3*ArcTan[(2*2^(3/4) + 2*2^(1/4)*Sqrt[2 + 3*x^2])/(2*Sqrt[3]*x*(2 + 3*x^2)^(1/4))]/(2^(1/4)*Sqrt[3]) + ArcTa
nh[(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 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*} \text {integral}& = -\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*}

Mathematica [A] (verified)

Time = 1.65 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.84 \[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=-\frac {\arctan \left (\frac {-3 \sqrt {2} x^2+4 \sqrt {2+3 x^2}}{2\ 2^{3/4} \sqrt {3} x \sqrt [4]{2+3 x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt {3} x \sqrt [4]{4+6 x^2}}{3 x^2+2 \sqrt {4+6 x^2}}\right )}{6 \sqrt [4]{2} \sqrt {3}} \]

[In]

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

[Out]

-1/6*(ArcTan[(-3*Sqrt[2]*x^2 + 4*Sqrt[2 + 3*x^2])/(2*2^(3/4)*Sqrt[3]*x*(2 + 3*x^2)^(1/4))] + ArcTanh[(2*Sqrt[3
]*x*(4 + 6*x^2)^(1/4))/(3*x^2 + 2*Sqrt[4 + 6*x^2])])/(2^(1/4)*Sqrt[3])

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 5.70 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.45

method	result	size
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right ) \ln \left (-\frac {\left (3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{3}-9 \sqrt {3 x^{2}+2}\, x +3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}+6 \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right ) \left (3 x^{2}+2\right )^{\frac {1}{4}}}{3 x^{2}+4}\right )}{18}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right ) \ln \left (-\frac {\left (3 x^{2}+2\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right )-9 \sqrt {3 x^{2}+2}\, x -3 x \operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}-6 \left (3 x^{2}+2\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+18\right )^{2}\right )}{3 x^{2}+4}\right )}{18}\)	\(187\)

[In]

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

[Out]

-1/18*RootOf(_Z^4+18)*ln(-((3*x^2+2)^(3/4)*RootOf(_Z^4+18)^3-9*(3*x^2+2)^(1/2)*x+3*x*RootOf(_Z^4+18)^2+6*RootO
f(_Z^4+18)*(3*x^2+2)^(1/4))/(3*x^2+4))+1/18*RootOf(_Z^2+RootOf(_Z^4+18)^2)*ln(-((3*x^2+2)^(3/4)*RootOf(_Z^4+18
)^2*RootOf(_Z^2+RootOf(_Z^4+18)^2)-9*(3*x^2+2)^(1/2)*x-3*x*RootOf(_Z^4+18)^2-6*(3*x^2+2)^(1/4)*RootOf(_Z^2+Roo
tOf(_Z^4+18)^2))/(3*x^2+4))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.25 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=-\left (\frac {1}{864} i + \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {\left (i + 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{864} i - \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{864} i - \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {\left (i - 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{864} i + \frac {1}{864}\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \cdot 72^{\frac {3}{4}} \sqrt {2} x + 48 \, {\left (3 \, x^{2} + 2\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

-(1/864*I + 1/864)*72^(3/4)*sqrt(2)*log(((I + 1)*72^(3/4)*sqrt(2)*x + 48*(3*x^2 + 2)^(1/4))/x) + (1/864*I - 1/
864)*72^(3/4)*sqrt(2)*log((-(I - 1)*72^(3/4)*sqrt(2)*x + 48*(3*x^2 + 2)^(1/4))/x) - (1/864*I - 1/864)*72^(3/4)
*sqrt(2)*log(((I - 1)*72^(3/4)*sqrt(2)*x + 48*(3*x^2 + 2)^(1/4))/x) + (1/864*I + 1/864)*72^(3/4)*sqrt(2)*log((
-(I + 1)*72^(3/4)*sqrt(2)*x + 48*(3*x^2 + 2)^(1/4))/x)

Sympy [F]

\[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=\int \frac {x^{2}}{\left (3 x^{2} + 2\right )^{\frac {3}{4}} \cdot \left (3 x^{2} + 4\right )}\, dx \]

[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)

Maxima [F]

\[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} + 4\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=\int { \frac {x^{2}}{{\left (3 \, x^{2} + 4\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {3}{4}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\left (2+3 x^2\right )^{3/4} \left (4+3 x^2\right )} \, dx=\int \frac {x^2}{{\left (3\,x^2+2\right )}^{3/4}\,\left (3\,x^2+4\right )} \,d x \]

[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)