Integrand size = 32, antiderivative size = 72 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {1}{4} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}^3+\log \left (\sqrt [4]{b x^2+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-a+\text {$\#$1}^4}\&\right ] \]
Time = 0.31 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.49 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=-\frac {\sqrt {x} \sqrt [4]{b+a x^2} \text {RootSum}\left [a^2+a b-2 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right ) \text {$\#$1}^3+\log \left (\sqrt [4]{b+a x^2}-\sqrt {x} \text {$\#$1}\right ) \text {$\#$1}^3}{-a+\text {$\#$1}^4}\&\right ]}{4 \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
-1/4*(Sqrt[x]*(b + a*x^2)^(1/4)*RootSum[a^2 + a*b - 2*a*#1^4 + #1^8 & , (- (Log[Sqrt[x]]*#1^3) + Log[(b + a*x^2)^(1/4) - Sqrt[x]*#1]*#1^3)/(-a + #1^4 ) & ])/(x^2*(b + a*x^2))^(1/4)
Leaf count is larger than twice the leaf count of optimal. \(531\) vs. \(2(72)=144\).
Time = 0.88 (sec) , antiderivative size = 531, normalized size of antiderivative = 7.38, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2467, 1593, 1759, 916, 770, 756, 216, 219, 902, 756, 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.
| \(\displaystyle \int \frac {a x^2+b}{\left (a x^4+b\right ) \sqrt [4]{a x^4+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \frac {\left (a x^2+b\right )^{3/4}}{\sqrt {x} \left (a x^4+b\right )}dx}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 1593 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \int \frac {\left (a x^2+b\right )^{3/4}}{a x^4+b}d\sqrt {x}}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 1759 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \int \frac {\left (a x^2+b\right )^{3/4}}{\sqrt {-a} \sqrt {b}-a x^2}d\sqrt {x}}{2 \sqrt {b}}+\frac {\sqrt {-a} \int \frac {\left (a x^2+b\right )^{3/4}}{a x^2+\sqrt {-a} \sqrt {b}}d\sqrt {x}}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 916 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\int \frac {1}{\sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\int \frac {1}{\sqrt [4]{a x^2+b}}d\sqrt {x}-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \int \frac {1}{\left (a x^2+\sqrt {-a} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\int \frac {1}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\int \frac {1}{1-a x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \int \frac {1}{\left (a x^2+\sqrt {-a} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (-\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\frac {1}{2} \int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {1}{2} \int \frac {1}{\sqrt {a} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \int \frac {1}{\left (a x^2+\sqrt {-a} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (-\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {a} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \int \frac {1}{\left (a x^2+\sqrt {-a} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\left (\sqrt {-a} \sqrt {b}-a x^2\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \int \frac {1}{\left (a x^2+\sqrt {-a} \sqrt {b}\right ) \sqrt [4]{a x^2+b}}d\sqrt {x}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 902 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \int \frac {1}{\sqrt {-a} \sqrt {b}-\left (\sqrt {-a} a \sqrt {b}-a b\right ) x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \int \frac {1}{\sqrt {-a} \sqrt {b}-\left (b a+\sqrt {-a} \sqrt {b} a\right ) x^2}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \left (\frac {\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {-a} \sqrt {b}}+\frac {\int \frac {1}{\sqrt {a-\sqrt {-a} \sqrt {b}} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {-a} \sqrt {b}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \left (\frac {\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {-a} \sqrt {b}}+\frac {\int \frac {1}{\sqrt {a+\sqrt {-a} \sqrt {b}} x+1}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {-a} \sqrt {b}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \left (\frac {\int \frac {1}{1-\sqrt {a-\sqrt {-a} \sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {-a} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \left (\frac {\int \frac {1}{1-\sqrt {a+\sqrt {-a} \sqrt {b}} x}d\frac {\sqrt {x}}{\sqrt [4]{a x^2+b}}}{2 \sqrt {-a} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {\sqrt {-a} \left (\sqrt {b} \left (\sqrt {-a}+\sqrt {b}\right ) \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{a-\sqrt {-a} \sqrt {b}}}\right )-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}+\frac {\sqrt {-a} \left (-\sqrt {b} \left (\sqrt {-a}-\sqrt {b}\right ) \left (\frac {\arctan \left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}+\frac {\text {arctanh}\left (\frac {\sqrt {x} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt {-a} \sqrt {b} \sqrt [4]{\sqrt {-a} \sqrt {b}+a}}\right )+\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{2 \sqrt [4]{a}}\right )}{2 \sqrt {b}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
(2*Sqrt[x]*(b + a*x^2)^(1/4)*((Sqrt[-a]*(-1/2*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/a^(1/4) - ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2* a^(1/4)) + (Sqrt[-a] + Sqrt[b])*Sqrt[b]*(ArcTan[((a - Sqrt[-a]*Sqrt[b])^(1 /4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2*Sqrt[-a]*(a - Sqrt[-a]*Sqrt[b])^(1/4)*S qrt[b]) + ArcTanh[((a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4) ]/(2*Sqrt[-a]*(a - Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]))))/(2*Sqrt[b]) + (Sqrt [-a]*(ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2*a^(1/4)) + ArcTanh[(a ^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2*a^(1/4)) - (Sqrt[-a] - Sqrt[b])*Sqrt [b]*(ArcTan[((a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2*S qrt[-a]*(a + Sqrt[-a]*Sqrt[b])^(1/4)*Sqrt[b]) + ArcTanh[((a + Sqrt[-a]*Sqr t[b])^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2*Sqrt[-a]*(a + Sqrt[-a]*Sqrt[b]) ^(1/4)*Sqrt[b]))))/(2*Sqrt[b])))/(b*x^2 + a*x^4)^(1/4)
3.10.51.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 1 /n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[b/d Int[(a + b*x^n)^(p - 1), x], x] - Simp[(b*c - a*d)/d Int[(a + b* x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]
Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_ ), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/f Subst[Int[x^(k*(m + 1 ) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)], x] ] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> W ith[{r = Rt[(-a)*c, 2]}, Simp[-c/(2*r) Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Simp[c/(2*r) Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] && !IntegerQ[q]
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 = 1.41 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.82
| method | result | size |
| pseudoelliptic | \(-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-2 \textit {\_Z}^{4} a +a^{2}+a b \right )}{\sum }\frac {\textit {\_R}^{3} \ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{4}-a}\right )}{4}\) | \(59\) |
-1/4*sum(_R^3*ln((-_R*x+(x^2*(a*x^2+b))^(1/4))/x)/(_R^4-a),_R=RootOf(_Z^8- 2*_Z^4*a+a^2+a*b))
Timed out. \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
Not integrable
Time = 5.53 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.38 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a x^{2} + b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{4} + b\right )}\, dx \]
Not integrable
Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}} \,d x } \]
Not integrable
Time = 0.86 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {a x^{2} + b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{4} + b\right )}} \,d x } \]
Not integrable
Time = 8.39 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.44 \[ \int \frac {b+a x^2}{\left (b+a x^4\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {a\,x^2+b}{\left (a\,x^4+b\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \]