Integrand size = 27, antiderivative size = 75 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}} \] Output:
1/2*b^(1/4)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(1/4)+1/2*b^(1/ 4)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(1/4)
Time = 0.03 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.80 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt [4]{b} \left (-\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right )}{\sqrt {2} \sqrt [4]{a}} \] Input:
Integrate[(Sqrt[a]*Sqrt[b] + b*x^2)/(a + b*x^4),x]
Output:
(b^(1/4)*(-ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + ArcTan[1 + (Sqrt[2]*b ^(1/4)*x)/a^(1/4)]))/(Sqrt[2]*a^(1/4))
Time = 0.34 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1476, 1082, 217}
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.
\(\displaystyle \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx+\frac {1}{2} \int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\sqrt [4]{b} \int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a}}-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a}}\) |
Input:
Int[(Sqrt[a]*Sqrt[b] + b*x^2)/(a + b*x^4),x]
Output:
-((b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(Sqrt[2]*a^(1/4))) + ( b^(1/4)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(Sqrt[2]*a^(1/4))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Leaf count of result is larger than twice the leaf count of optimal. \(203\) vs. \(2(51)=102\).
Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.72
method | result | size |
default | \(\frac {\sqrt {b}\, \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \sqrt {a}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(204\) |
Input:
int((a^(1/2)*b^(1/2)+b*x^2)/(b*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/8/a^(1/2)*b^(1/2)*(a/b)^(1/4)*2^(1/2)*(ln((x^2+(a/b)^(1/4)*x*2^(1/2)+(a/ b)^(1/2))/(x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^ (1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1))+1/8/(a/b)^(1/4)*2^(1/2)*(ln( (x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^( 1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1) )
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\left [\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \log \left (\frac {b x^{4} - 4 \, \sqrt {a} \sqrt {b} x^{2} + 4 \, \sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} - a x\right )} \sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} + a}{b x^{4} + a}\right ), \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\sqrt {\frac {1}{2}} x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\right ) + \sqrt {\frac {1}{2}} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}} \arctan \left (\frac {\sqrt {\frac {1}{2}} {\left (\sqrt {a} \sqrt {b} x^{3} + a x\right )} \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}{a}\right )\right ] \] Input:
integrate((a^(1/2)*b^(1/2)+b*x^2)/(b*x^4+a),x, algorithm="fricas")
Output:
[1/2*sqrt(1/2)*sqrt(-sqrt(b)/sqrt(a))*log((b*x^4 - 4*sqrt(a)*sqrt(b)*x^2 + 4*sqrt(1/2)*(sqrt(a)*sqrt(b)*x^3 - a*x)*sqrt(-sqrt(b)/sqrt(a)) + a)/(b*x^ 4 + a)), sqrt(1/2)*sqrt(sqrt(b)/sqrt(a))*arctan(sqrt(1/2)*x*sqrt(sqrt(b)/s qrt(a))) + sqrt(1/2)*sqrt(sqrt(b)/sqrt(a))*arctan(sqrt(1/2)*(sqrt(a)*sqrt( b)*x^3 + a*x)*sqrt(sqrt(b)/sqrt(a))/a)]
Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=- \frac {\sqrt {2} \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}} \log {\left (- \frac {\sqrt {2} \sqrt {a} x \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} - \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} + \frac {\sqrt {2} \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}} \log {\left (\frac {\sqrt {2} \sqrt {a} x \sqrt {- \frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {b}} - \frac {\sqrt {a}}{\sqrt {b}} + x^{2} \right )}}{4} \] Input:
integrate((a**(1/2)*b**(1/2)+b*x**2)/(b*x**4+a),x)
Output:
-sqrt(2)*sqrt(-sqrt(b)/sqrt(a))*log(-sqrt(2)*sqrt(a)*x*sqrt(-sqrt(b)/sqrt( a))/sqrt(b) - sqrt(a)/sqrt(b) + x**2)/4 + sqrt(2)*sqrt(-sqrt(b)/sqrt(a))*l og(sqrt(2)*sqrt(a)*x*sqrt(-sqrt(b)/sqrt(a))/sqrt(b) - sqrt(a)/sqrt(b) + x* *2)/4
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {\sqrt {a} \sqrt {b}}} \] Input:
integrate((a^(1/2)*b^(1/2)+b*x^2)/(b*x^4+a),x, algorithm="maxima")
Output:
1/2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1 /4))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + 1/2*sqrt(2)*sqrt(b)*ar ctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt (b)))/sqrt(sqrt(a)*sqrt(b))
Exception generated. \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^(1/2)*b^(1/2)+b*x^2)/(b*x^4+a),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Time = 17.67 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {\sqrt {2}\,b^{1/4}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/4}\,x}{2\,a^{1/4}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{3/4}\,x^3}{2\,a^{3/4}}+\frac {\sqrt {2}\,b^{1/4}\,x}{2\,a^{1/4}}\right )\right )}{4\,a^{1/4}} \] Input:
int((b*x^2 + a^(1/2)*b^(1/2))/(a + b*x^4),x)
Output:
(2^(1/2)*b^(1/4)*(2*atan((2^(1/2)*b^(1/4)*x)/(2*a^(1/4))) + 2*atan((2^(1/2 )*b^(3/4)*x^3)/(2*a^(3/4)) + (2^(1/2)*b^(1/4)*x)/(2*a^(1/4)))))/(4*a^(1/4) )
Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {a} \sqrt {b}+b x^2}{a+b x^4} \, dx=\frac {b^{\frac {1}{4}} \sqrt {2}\, \left (-\mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+\mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )\right )}{2 a^{\frac {1}{4}}} \] Input:
int((a^(1/2)*b^(1/2)+b*x^2)/(b*x^4+a),x)
Output:
(b**(1/4)*a**(3/4)*sqrt(2)*( - atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b) *x)/(b**(1/4)*a**(1/4)*sqrt(2))) + atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqr t(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))))/(2*a)