Integrand size = 35, antiderivative size = 166 \[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx=x \sqrt [4]{b x^2+a x^4}-\frac {7 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}+\frac {3 \sqrt [4]{2} b \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{a^{3/4}}+\frac {7 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}-\frac {3 \sqrt [4]{2} b \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{a^{3/4}} \]
x*(a*x^4+b*x^2)^(1/4)-7/2*b*arctan(a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(3/4)+ 3*2^(1/4)*b*arctan(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(3/4)+7/2*b*ar ctanh(a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(3/4)-3*2^(1/4)*b*arctanh(2^(1/4)*a ^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(3/4)
Time = 0.64 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.13 \[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx=\frac {x^{3/2} \left (b+a x^2\right )^{3/4} \left (2 a^{3/4} x^{3/2} \sqrt [4]{b+a x^2}-7 b \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+6 \sqrt [4]{2} b \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+7 b \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-6 \sqrt [4]{2} b \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )}{2 a^{3/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
(x^(3/2)*(b + a*x^2)^(3/4)*(2*a^(3/4)*x^(3/2)*(b + a*x^2)^(1/4) - 7*b*ArcT an[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] + 6*2^(1/4)*b*ArcTan[(2^(1/4)*a^(1 /4)*Sqrt[x])/(b + a*x^2)^(1/4)] + 7*b*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2 )^(1/4)] - 6*2^(1/4)*b*ArcTanh[(2^(1/4)*a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4) ]))/(2*a^(3/4)*(x^2*(b + a*x^2))^(3/4))
Time = 0.58 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2467, 25, 443, 25, 27, 446, 2009}
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.
| \(\displaystyle \int \frac {\left (2 a x^2+b\right ) \sqrt [4]{a x^4+b x^2}}{a x^2-b} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^2} \int -\frac {\sqrt {x} \sqrt [4]{a x^2+b} \left (2 a x^2+b\right )}{b-a x^2}dx}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \int \frac {\sqrt {x} \sqrt [4]{a x^2+b} \left (2 a x^2+b\right )}{b-a x^2}dx}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 443 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (x^{3/2} \left (-\sqrt [4]{a x^2+b}\right )-\frac {\int -\frac {a b \sqrt {x} \left (7 a x^2+5 b\right )}{\left (b-a x^2\right ) \left (a x^2+b\right )^{3/4}}dx}{2 a}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (\frac {\int \frac {a b \sqrt {x} \left (7 a x^2+5 b\right )}{\left (b-a x^2\right ) \left (a x^2+b\right )^{3/4}}dx}{2 a}-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (\frac {1}{2} b \int \frac {\sqrt {x} \left (7 a x^2+5 b\right )}{\left (b-a x^2\right ) \left (a x^2+b\right )^{3/4}}dx-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 446 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (\frac {1}{2} b \int \left (\frac {12 b \sqrt {x}}{\left (b-a x^2\right ) \left (a x^2+b\right )^{3/4}}-\frac {7 \sqrt {x}}{\left (a x^2+b\right )^{3/4}}\right )dx-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (\frac {1}{2} b \left (\frac {7 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{a^{3/4}}-\frac {6 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{a^{3/4}}-\frac {7 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{a^{3/4}}+\frac {6 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{a^{3/4}}\right )-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
-(((b*x^2 + a*x^4)^(1/4)*(-(x^(3/2)*(b + a*x^2)^(1/4)) + (b*((7*ArcTan[(a^ (1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/a^(3/4) - (6*2^(1/4)*ArcTan[(2^(1/4)*a^ (1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/a^(3/4) - (7*ArcTanh[(a^(1/4)*Sqrt[x])/ (b + a*x^2)^(1/4)])/a^(3/4) + (6*2^(1/4)*ArcTanh[(2^(1/4)*a^(1/4)*Sqrt[x]) /(b + a*x^2)^(1/4)])/a^(3/4)))/2))/(Sqrt[x]*(b + a*x^2)^(1/4)))
3.23.27.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*g*(m + 2*(p + q + 1) + 1))), x] + Simp[1/(b*(m + 2* (p + q + 1) + 1)) Int[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(( b*e - a*f)*(m + 1) + b*e*2*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*2*q*(b *c - a*d) + b*e*d*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f , g, m, p}, x] && GtQ[q, 0] && !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^ 2])
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/( (c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^2)^ p*((e + f*x^2)/(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.45 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.10
| method | result | size |
| pseudoelliptic | \(\frac {4 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} x \,a^{\frac {3}{4}}-6 \,2^{\frac {1}{4}} \ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) b -12 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) b +7 b \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right )+14 b \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )}{4 a^{\frac {3}{4}}}\) | \(182\) |
1/4*(4*(x^2*(a*x^2+b))^(1/4)*x*a^(3/4)-6*2^(1/4)*ln((x*2^(1/4)*a^(1/4)+(x^ 2*(a*x^2+b))^(1/4))/(-x*2^(1/4)*a^(1/4)+(x^2*(a*x^2+b))^(1/4)))*b-12*2^(1/ 4)*arctan(1/2*(x^2*(a*x^2+b))^(1/4)/x*2^(3/4)/a^(1/4))*b+7*b*ln((a^(1/4)*x +(x^2*(a*x^2+b))^(1/4))/(-a^(1/4)*x+(x^2*(a*x^2+b))^(1/4)))+14*b*arctan(1/ a^(1/4)*(x^2*(a*x^2+b))^(1/4)/x))/a^(3/4)
Timed out. \[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx=\int \frac {\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (2 a x^{2} + b\right )}{a x^{2} - b}\, dx \]
\[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (2 \, a x^{2} + b\right )}}{a x^{2} - b} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (128) = 256\).
Time = 0.31 (sec) , antiderivative size = 394, normalized size of antiderivative = 2.37 \[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx={\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} x^{2} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{2 \, a} - \frac {3 \cdot 2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{2 \, a} + \frac {3 \cdot 2^{\frac {3}{4}} b \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{4 \, \left (-a\right )^{\frac {3}{4}}} + \frac {3 \cdot 2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{4 \, a} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{4 \, a} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{4 \, a} + \frac {7 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{8 \, a} + \frac {7 \, \sqrt {2} b \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{8 \, \left (-a\right )^{\frac {3}{4}}} \]
(a + b/x^2)^(1/4)*x^2 - 3/2*2^(3/4)*(-a)^(1/4)*b*arctan(1/2*2^(1/4)*(2^(3/ 4)*(-a)^(1/4) + 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/a - 3/2*2^(3/4)*(-a)^(1/4 )*b*arctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1 /4))/a + 3/4*2^(3/4)*b*log(2^(3/4)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(2)* sqrt(-a) + sqrt(a + b/x^2))/(-a)^(3/4) + 3/4*2^(3/4)*(-a)^(1/4)*b*log(-2^( 3/4)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^2))/a + 7/4*sqrt(2)*(-a)^(1/4)*b*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/a + 7/4*sqrt(2)*(-a)^(1/4)*b*arctan(-1/2*sqrt(2 )*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/a + 7/8*sqrt(2)*( -a)^(1/4)*b*log(sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(-a) + sqrt(a + b/x^2))/a + 7/8*sqrt(2)*b*log(-sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqr t(-a) + sqrt(a + b/x^2))/(-a)^(3/4)
Timed out. \[ \int \frac {\left (b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{-b+a x^2} \, dx=\int -\frac {\left (2\,a\,x^2+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{b-a\,x^2} \,d x \]