\(\int \frac {1}{\sqrt [4]{a+b x^2} (2 a+b x^2)} \, dx\) [310]

Optimal result

Integrand size = 23, antiderivative size = 120 \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {a^{3/4} \left (1+\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {406} \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=-\frac {\arctan \left (\frac {a^{3/4} \left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}+1\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt {b}} \]

[In]

Int[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]

[Out]

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

Rule 406

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

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {b x^2-2 \sqrt {a} \sqrt {a+b x^2}}{2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}\right )+\text {arctanh}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}{b x^2+2 \sqrt {a} \sqrt {a+b x^2}}\right )}{4 a^{3/4} \sqrt {b}} \]

[In]

Integrate[1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x]

[Out]

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

Maple [F]

\[\int \frac {1}{\left (b \,x^{2}+a \right )^{\frac {1}{4}} \left (b \,x^{2}+2 a \right )}d x\]

[In]

int(1/(b*x^2+a)^(1/4)/(b*x^2+2*a),x)

[Out]

int(1/(b*x^2+a)^(1/4)/(b*x^2+2*a),x)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 24.40 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.76 \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=-\frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) + \frac {1}{4} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (-\frac {2 \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} - {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) + \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} + i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) - \frac {1}{4} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} \log \left (\frac {-2 i \, \left (\frac {1}{4}\right )^{\frac {3}{4}} \sqrt {b x^{2} + a} a^{2} b^{2} x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {3}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}} a^{2} b \sqrt {-\frac {1}{a^{3} b^{2}}} - i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} a b x \left (-\frac {1}{a^{3} b^{2}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {3}{4}}}{b x^{2} + 2 \, a}\right ) \]

[In]

integrate(1/(b*x^2+a)^(1/4)/(b*x^2+2*a),x, algorithm="fricas")

[Out]

-1/4*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log((2*(1/4)^(3/4)*sqrt(b*x^2 + a)*a^2*b^2*x*(-1/(a^3*b^2))^(3/4) + (b*x
^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) - (1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) + (b*x^2 + a)^(3/4))/(b*x^2 +
2*a)) + 1/4*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log(-(2*(1/4)^(3/4)*sqrt(b*x^2 + a)*a^2*b^2*x*(-1/(a^3*b^2))^(3/4
) - (b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) - (1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) - (b*x^2 + a)^(3/4))/(
b*x^2 + 2*a)) + 1/4*I*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log((2*I*(1/4)^(3/4)*sqrt(b*x^2 + a)*a^2*b^2*x*(-1/(a^3
*b^2))^(3/4) - (b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) + I*(1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(1/4) + (b*x^2
+ a)^(3/4))/(b*x^2 + 2*a)) - 1/4*I*(1/4)^(1/4)*(-1/(a^3*b^2))^(1/4)*log((-2*I*(1/4)^(3/4)*sqrt(b*x^2 + a)*a^2*
b^2*x*(-1/(a^3*b^2))^(3/4) - (b*x^2 + a)^(1/4)*a^2*b*sqrt(-1/(a^3*b^2)) - I*(1/4)^(1/4)*a*b*x*(-1/(a^3*b^2))^(
1/4) + (b*x^2 + a)^(3/4))/(b*x^2 + 2*a))

Sympy [F]

\[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=\int \frac {1}{\sqrt [4]{a + b x^{2}} \cdot \left (2 a + b x^{2}\right )}\, dx \]

[In]

integrate(1/(b*x**2+a)**(1/4)/(b*x**2+2*a),x)

[Out]

Integral(1/((a + b*x**2)**(1/4)*(2*a + b*x**2)), x)

Maxima [F]

\[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(b*x^2+a)^(1/4)/(b*x^2+2*a),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + 2*a)*(b*x^2 + a)^(1/4)), x)

Giac [F]

\[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=\int { \frac {1}{{\left (b x^{2} + 2 \, a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{4}}} \,d x } \]

[In]

integrate(1/(b*x^2+a)^(1/4)/(b*x^2+2*a),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + 2*a)*(b*x^2 + a)^(1/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right )} \, dx=\int \frac {1}{{\left (b\,x^2+a\right )}^{1/4}\,\left (b\,x^2+2\,a\right )} \,d x \]

[In]

int(1/((a + b*x^2)^(1/4)*(2*a + b*x^2)),x)

[Out]

int(1/((a + b*x^2)^(1/4)*(2*a + b*x^2)), x)