\(\int \frac {x^2}{(a+b x^2)^{3/4} (2 a+b x^2)} \, dx\) [1497]

Optimal result

Integrand size = 26, antiderivative size = 115 \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {b^{3/2} x \sqrt [4]{a+b x^2}}{a^{3/4} \left (b+\frac {b \sqrt {a+b x^2}}{\sqrt {a}}\right )}\right )}{\sqrt [4]{a} b^{3/2}}+\frac {\text {arctanh}\left (\frac {a^{3/4} \left (1-\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {b} x \sqrt [4]{a+b x^2}}\right )}{\sqrt [4]{a} b^{3/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.85 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05 \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=\frac {\arctan \left (\frac {b x^2-2 \sqrt {a} \sqrt {a+b x^2}}{2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}\right )-\text {arctanh}\left (\frac {2 \sqrt [4]{a} \sqrt {b} x \sqrt [4]{a+b x^2}}{b x^2+2 \sqrt {a} \sqrt {a+b x^2}}\right )}{2 \sqrt [4]{a} b^{3/2}} \] Input:

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

Output:

(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*Sq 
rt[a]*Sqrt[a + b*x^2])])/(2*a^(1/4)*b^(3/2))
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {350}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 350

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 350

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))*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))], 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 
]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.72 \[ \int \frac {x^2}{\left (a+b x^2\right )^{3/4} \left (2 a+b x^2\right )} \, dx=-\frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (-\frac {\left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} - {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, \left (\frac {1}{4}\right )^{\frac {1}{4}} b^{2} x \left (-\frac {1}{a b^{6}}\right )^{\frac {1}{4}} + {\left (b x^{2} + a\right )}^{\frac {1}{4}}}{x}\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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