Integrand size = 35, antiderivative size = 84 \[ \int \frac {\left (-b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^2} \, dx=x \sqrt [4]{b x^2+a x^4}+\frac {5 b \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}}-\frac {5 b \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{2 a^{3/4}} \]
x*(a*x^4+b*x^2)^(1/4)+5/2*b*arctan(a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(3/4)- 5/2*b*arctanh(a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(3/4)
Time = 0.44 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.46 \[ \int \frac {\left (-b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^2} \, dx=\frac {x^{3/2} \left (2 a^{3/4} x^{3/2} \left (b+a x^2\right )+5 b \left (b+a x^2\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-5 b \left (b+a x^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )}{2 a^{3/4} \left (x^2 \left (b+a x^2\right )\right )^{3/4}} \]
(x^(3/2)*(2*a^(3/4)*x^(3/2)*(b + a*x^2) + 5*b*(b + a*x^2)^(3/4)*ArcTan[(a^ (1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] - 5*b*(b + a*x^2)^(3/4)*ArcTanh[(a^(1/4) *Sqrt[x])/(b + a*x^2)^(1/4)]))/(2*a^(3/4)*(x^2*(b + a*x^2))^(3/4))
Time = 0.36 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.43, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {2467, 25, 363, 266, 854, 827, 216, 219}
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 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^2} \int -\frac {\sqrt {x} \left (b-2 a x^2\right )}{\left (a x^2+b\right )^{3/4}}dx}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \int \frac {\sqrt {x} \left (b-2 a x^2\right )}{\left (a x^2+b\right )^{3/4}}dx}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (\frac {5}{2} b \int \frac {\sqrt {x}}{\left (a x^2+b\right )^{3/4}}dx-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (5 b \int \frac {x}{\left (a x^2+b\right )^{3/4}}d\sqrt {x}-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (5 b \int \frac {x}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (5 b \left (\frac {\int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a}}\right )-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (5 b \left (\frac {\int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4}}\right )-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^2} \left (5 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 a^{3/4}}\right )-x^{3/2} \sqrt [4]{a x^2+b}\right )}{\sqrt {x} \sqrt [4]{a x^2+b}}\) |
-(((b*x^2 + a*x^4)^(1/4)*(-(x^(3/2)*(b + a*x^2)^(1/4)) + 5*b*(-1/2*ArcTan[ (a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/a^(3/4) + ArcTanh[(a^(1/4)*Sqrt[x])/( b + a*x^2)^(1/4)]/(2*a^(3/4)))))/(Sqrt[x]*(b + a*x^2)^(1/4)))
3.12.27.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
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 = 2.37 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(-\frac {5 \left (-\frac {4 \left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} x \,a^{\frac {3}{4}}}{5}+\ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right ) b +2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b \right )}{4 a^{\frac {3}{4}}}\) | \(95\) |
-5/4*(-4/5*(x^2*(a*x^2+b))^(1/4)*x*a^(3/4)+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)))*b+2*arctan(1/a^(1/4)/x*(x^2*(a*x ^2+b))^(1/4))*b)/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. 209 vs. \(2 (66) = 132\).
Time = 0.30 (sec) , antiderivative size = 209, normalized size of antiderivative = 2.49 \[ \int \frac {\left (-b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^2} \, dx=\frac {8 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} b x^{2} + \frac {10 \, \sqrt {2} b^{2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {10 \, \sqrt {2} b^{2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {5 \, \sqrt {2} b^{2} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{\left (-a\right )^{\frac {3}{4}}} + \frac {5 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{2} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{a}}{8 \, b} \]
1/8*(8*(a + b/x^2)^(1/4)*b*x^2 + 10*sqrt(2)*b^2*arctan(1/2*sqrt(2)*(sqrt(2 )*(-a)^(1/4) + 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/(-a)^(3/4) + 10*sqrt(2)*b^ 2*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1/4 ))/(-a)^(3/4) + 5*sqrt(2)*b^2*log(sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + s qrt(-a) + sqrt(a + b/x^2))/(-a)^(3/4) + 5*sqrt(2)*(-a)^(1/4)*b^2*log(-sqrt (2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(-a) + sqrt(a + b/x^2))/a)/b
Timed out. \[ \int \frac {\left (-b+2 a x^2\right ) \sqrt [4]{b x^2+a x^4}}{b+a x^2} \, dx=\int -\frac {\left (b-2\,a\,x^2\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}}{a\,x^2+b} \,d x \]