\(\int \frac {x^2}{(b+a x^2)^{3/4} (2 b+a x^2)} \, dx\) [1940]

Optimal result

Integrand size = 26, antiderivative size = 136 \[ \int \frac {x^2}{\left (b+a x^2\right )^{3/4} \left (2 b+a x^2\right )} \, dx=-\frac {\arctan \left (\frac {-\frac {\sqrt {a} x^2}{2 \sqrt [4]{b}}+\frac {\sqrt [4]{b} \sqrt {b+a x^2}}{\sqrt {a}}}{x \sqrt [4]{b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}}-\frac {\text {arctanh}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{b+a x^2}}{a x^2+2 \sqrt {b} \sqrt {b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.85, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {452} \[ \int \frac {x^2}{\left (b+a x^2\right )^{3/4} \left (2 b+a x^2\right )} \, dx=\frac {\text {arctanh}\left (\frac {b^{3/4} \left (1-\frac {\sqrt {a x^2+b}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{a x^2+b}}\right )}{a^{3/2} \sqrt [4]{b}}-\frac {\arctan \left (\frac {b^{3/4} \left (\frac {\sqrt {a x^2+b}}{\sqrt {b}}+1\right )}{\sqrt {a} x \sqrt [4]{a x^2+b}}\right )}{a^{3/2} \sqrt [4]{b}} \]

[In]

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

[Out]

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

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 {\arctan \left (\frac {b^{3/4} \left (1+\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{b+a x^2}}\right )}{a^{3/2} \sqrt [4]{b}}+\frac {\text {arctanh}\left (\frac {b^{3/4} \left (1-\frac {\sqrt {b+a x^2}}{\sqrt {b}}\right )}{\sqrt {a} x \sqrt [4]{b+a x^2}}\right )}{a^{3/2} \sqrt [4]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.92 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89 \[ \int \frac {x^2}{\left (b+a x^2\right )^{3/4} \left (2 b+a x^2\right )} \, dx=\frac {\arctan \left (\frac {a x^2-2 \sqrt {b} \sqrt {b+a x^2}}{2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{b+a x^2}}\right )-\text {arctanh}\left (\frac {2 \sqrt {a} \sqrt [4]{b} x \sqrt [4]{b+a x^2}}{a x^2+2 \sqrt {b} \sqrt {b+a x^2}}\right )}{2 a^{3/2} \sqrt [4]{b}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x^{2}}{\left (a \,x^{2}+b \right )^{\frac {3}{4}} \left (a \,x^{2}+2 b \right )}d x\]

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.46 \[ \int \frac {x^2}{\left (b+a x^2\right )^{3/4} \left (2 b+a x^2\right )} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} + b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} - {\left (a x^{2} + b\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} + b\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a^{2} x \left (-\frac {1}{a^{6} b}\right )^{\frac {1}{4}} + {\left (a x^{2} + b\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

-1/2*(1/4)^(1/4)*(-1/(a^6*b))^(1/4)*log(((1/4)^(1/4)*a^2*x*(-1/(a^6*b))^(1/4) + (a*x^2 + b)^(1/4))/x) + 1/2*(1
/4)^(1/4)*(-1/(a^6*b))^(1/4)*log(-((1/4)^(1/4)*a^2*x*(-1/(a^6*b))^(1/4) - (a*x^2 + b)^(1/4))/x) - 1/2*I*(1/4)^
(1/4)*(-1/(a^6*b))^(1/4)*log((I*(1/4)^(1/4)*a^2*x*(-1/(a^6*b))^(1/4) + (a*x^2 + b)^(1/4))/x) + 1/2*I*(1/4)^(1/
4)*(-1/(a^6*b))^(1/4)*log((-I*(1/4)^(1/4)*a^2*x*(-1/(a^6*b))^(1/4) + (a*x^2 + b)^(1/4))/x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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