Integrand size = 28, antiderivative size = 335 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}+\frac {11 \sqrt {d x}}{48 a^2 d \left (a+b x^2\right )^2}+\frac {77 \sqrt {d x}}{192 a^3 d \left (a+b x^2\right )}-\frac {77 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}-\frac {77 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}}+\frac {77 \log \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {d} x+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}\right )}{256 \sqrt {2} a^{15/4} \sqrt [4]{b} \sqrt {d}} \]
-77/256*arctan(1-b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d^(1/2))/a^(15/4)/b^( 1/4)*2^(1/2)/d^(1/2)+77/256*arctan(1+b^(1/4)*2^(1/2)*(d*x)^(1/2)/a^(1/4)/d ^(1/2))/a^(15/4)/b^(1/4)*2^(1/2)/d^(1/2)-77/512*ln(a^(1/2)*d^(1/2)+x*b^(1/ 2)*d^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*(d*x)^(1/2))/a^(15/4)/b^(1/4)*2^(1/2)/d ^(1/2)+77/512*ln(a^(1/2)*d^(1/2)+x*b^(1/2)*d^(1/2)+a^(1/4)*b^(1/4)*2^(1/2) *(d*x)^(1/2))/a^(15/4)/b^(1/4)*2^(1/2)/d^(1/2)+1/6*(d*x)^(1/2)/a/d/(b*x^2+ a)^3+11/48*(d*x)^(1/2)/a^2/d/(b*x^2+a)^2+77/192*(d*x)^(1/2)/a^3/d/(b*x^2+a )
Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\sqrt {x} \left (\frac {4 a^{3/4} \sqrt {x} \left (153 a^2+198 a b x^2+77 b^2 x^4\right )}{\left (a+b x^2\right )^3}-\frac {231 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{b}}+\frac {231 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{b}}\right )}{768 a^{15/4} \sqrt {d x}} \]
(Sqrt[x]*((4*a^(3/4)*Sqrt[x]*(153*a^2 + 198*a*b*x^2 + 77*b^2*x^4))/(a + b* x^2)^3 - (231*Sqrt[2]*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4 )*Sqrt[x])])/b^(1/4) + (231*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[ x])/(Sqrt[a] + Sqrt[b]*x)])/b^(1/4)))/(768*a^(15/4)*Sqrt[d*x])
Time = 0.55 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.11, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {1380, 27, 253, 253, 253, 266, 755, 27, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^4 \int \frac {1}{b^4 \sqrt {d x} \left (b x^2+a\right )^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{\sqrt {d x} \left (a+b x^2\right )^4}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^3}dx}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \left (\frac {7 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )^2}dx}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{\sqrt {d x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \int \frac {1}{b x^2+a}d\sqrt {d x}}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {d^2 \left (\sqrt {a} d-\sqrt {b} d x\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}+\frac {\int \frac {d^2 \left (\sqrt {b} x d+\sqrt {a} d\right )}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a} d}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {11 \left (\frac {7 \left (\frac {3 \left (\frac {d \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}+\frac {d \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}\right )}{2 \sqrt {a}}\right )}{2 a d}+\frac {\sqrt {d x}}{2 a d \left (a+b x^2\right )}\right )}{8 a}+\frac {\sqrt {d x}}{4 a d \left (a+b x^2\right )^2}\right )}{12 a}+\frac {\sqrt {d x}}{6 a d \left (a+b x^2\right )^3}\) |
Sqrt[d*x]/(6*a*d*(a + b*x^2)^3) + (11*(Sqrt[d*x]/(4*a*d*(a + b*x^2)^2) + ( 7*(Sqrt[d*x]/(2*a*d*(a + b*x^2)) + (3*((d*(-(ArcTan[1 - (Sqrt[2]*b^(1/4)*S qrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqrt[d])]/(Sqrt[2]*a^(1/4)*b^(1/4) *Sqrt[d])))/(2*Sqrt[a]) + (d*(-1/2*Log[Sqrt[a]*d + Sqrt[b]*d*x - Sqrt[2]*a ^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]) + Log[ Sqrt[a]*d + Sqrt[b]*d*x + Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]*Sqrt[d*x]]/(2*Sq rt[2]*a^(1/4)*b^(1/4)*Sqrt[d])))/(2*Sqrt[a])))/(2*a*d)))/(8*a)))/(12*a)
3.8.6.3.1 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[((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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
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_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[(u_)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> S imp[1/c^p Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Time = 0.29 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.62
| method | result | size |
| derivativedivides | \(2 d^{7} \left (\frac {\frac {77 b^{2} \left (d x \right )^{\frac {9}{2}}}{384 a^{3} d^{6}}+\frac {33 b \left (d x \right )^{\frac {5}{2}}}{64 a^{2} d^{4}}+\frac {51 \sqrt {d x}}{128 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{4} d^{8}}\right )\) | \(209\) |
| default | \(2 d^{7} \left (\frac {\frac {77 b^{2} \left (d x \right )^{\frac {9}{2}}}{384 a^{3} d^{6}}+\frac {33 b \left (d x \right )^{\frac {5}{2}}}{64 a^{2} d^{4}}+\frac {51 \sqrt {d x}}{128 a \,d^{2}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {77 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{4} d^{8}}\right )\) | \(209\) |
| pseudoelliptic | \(\frac {77 \sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \ln \left (\frac {d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+154 \sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+154 \sqrt {2}\, \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+408 a \left (\frac {77}{153} b^{2} x^{4}+\frac {22}{17} a b \,x^{2}+a^{2}\right ) \sqrt {d x}}{512 d \,a^{4} \left (b \,x^{2}+a \right )^{3}}\) | \(255\) |
2*d^7*((77/384/a^3/d^6*b^2*(d*x)^(9/2)+33/64/a^2/d^4*b*(d*x)^(5/2)+51/128/ a/d^2*(d*x)^(1/2))/(b*d^2*x^2+a*d^2)^3+77/1024/a^4/d^8*(a*d^2/b)^(1/4)*2^( 1/2)*(ln((d*x+(a*d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2))/(d*x-(a *d^2/b)^(1/4)*(d*x)^(1/2)*2^(1/2)+(a*d^2/b)^(1/2)))+2*arctan(2^(1/2)/(a*d^ 2/b)^(1/4)*(d*x)^(1/2)+1)+2*arctan(2^(1/2)/(a*d^2/b)^(1/4)*(d*x)^(1/2)-1)) )
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {231 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \log \left (a^{4} d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 231 \, {\left (-i \, a^{3} b^{3} d x^{6} - 3 i \, a^{4} b^{2} d x^{4} - 3 i \, a^{5} b d x^{2} - i \, a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \log \left (i \, a^{4} d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 231 \, {\left (i \, a^{3} b^{3} d x^{6} + 3 i \, a^{4} b^{2} d x^{4} + 3 i \, a^{5} b d x^{2} + i \, a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \log \left (-i \, a^{4} d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) - 231 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )} \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} \log \left (-a^{4} d \left (-\frac {1}{a^{15} b d^{2}}\right )^{\frac {1}{4}} + \sqrt {d x}\right ) + 4 \, {\left (77 \, b^{2} x^{4} + 198 \, a b x^{2} + 153 \, a^{2}\right )} \sqrt {d x}}{768 \, {\left (a^{3} b^{3} d x^{6} + 3 \, a^{4} b^{2} d x^{4} + 3 \, a^{5} b d x^{2} + a^{6} d\right )}} \]
1/768*(231*(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/( a^15*b*d^2))^(1/4)*log(a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x)) - 231*(- I*a^3*b^3*d*x^6 - 3*I*a^4*b^2*d*x^4 - 3*I*a^5*b*d*x^2 - I*a^6*d)*(-1/(a^15 *b*d^2))^(1/4)*log(I*a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x)) - 231*(I*a ^3*b^3*d*x^6 + 3*I*a^4*b^2*d*x^4 + 3*I*a^5*b*d*x^2 + I*a^6*d)*(-1/(a^15*b* d^2))^(1/4)*log(-I*a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x)) - 231*(a^3*b ^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x^2 + a^6*d)*(-1/(a^15*b*d^2))^(1/4 )*log(-a^4*d*(-1/(a^15*b*d^2))^(1/4) + sqrt(d*x)) + 4*(77*b^2*x^4 + 198*a* b*x^2 + 153*a^2)*sqrt(d*x))/(a^3*b^3*d*x^6 + 3*a^4*b^2*d*x^4 + 3*a^5*b*d*x ^2 + a^6*d)
\[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {1}{\sqrt {d x} \left (a + b x^{2}\right )^{4}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 322, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {8 \, {\left (77 \, \left (d x\right )^{\frac {9}{2}} b^{2} d^{2} + 198 \, \left (d x\right )^{\frac {5}{2}} a b d^{4} + 153 \, \sqrt {d x} a^{2} d^{6}\right )}}{a^{3} b^{3} d^{6} x^{6} + 3 \, a^{4} b^{2} d^{6} x^{4} + 3 \, a^{5} b d^{6} x^{2} + a^{6} d^{6}} + \frac {231 \, {\left (\frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} d^{2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, \sqrt {2} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}} + \frac {2 \, \sqrt {2} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {a}}\right )}}{a^{3}}}{1536 \, d} \]
1/1536*(8*(77*(d*x)^(9/2)*b^2*d^2 + 198*(d*x)^(5/2)*a*b*d^4 + 153*sqrt(d*x )*a^2*d^6)/(a^3*b^3*d^6*x^6 + 3*a^4*b^2*d^6*x^4 + 3*a^5*b*d^6*x^2 + a^6*d^ 6) + 231*(sqrt(2)*d^2*log(sqrt(b)*d*x + sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^ (1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/4)) - sqrt(2)*d^2*log(sqrt(b)*d*x - sqrt(2)*(a*d^2)^(1/4)*sqrt(d*x)*b^(1/4) + sqrt(a)*d)/((a*d^2)^(3/4)*b^(1/ 4)) + 2*sqrt(2)*d*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) + 2*sq rt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a) ) + 2*sqrt(2)*d*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqr t(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*sqrt(a)) )/a^3)/d
Time = 0.29 (sec) , antiderivative size = 308, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {77 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{4} b d} + \frac {77 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{256 \, a^{4} b d} + \frac {77 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{4} b d} - \frac {77 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {1}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{512 \, a^{4} b d} + \frac {77 \, \sqrt {d x} b^{2} d^{5} x^{4} + 198 \, \sqrt {d x} a b d^{5} x^{2} + 153 \, \sqrt {d x} a^{2} d^{5}}{192 \, {\left (b d^{2} x^{2} + a d^{2}\right )}^{3} a^{3}} \]
77/256*sqrt(2)*(a*b^3*d^2)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/ 4) + 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a^4*b*d) + 77/256*sqrt(2)*(a*b^3*d^2)^ (1/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b )^(1/4))/(a^4*b*d) + 77/512*sqrt(2)*(a*b^3*d^2)^(1/4)*log(d*x + sqrt(2)*(a *d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a^4*b*d) - 77/512*sqrt(2)*(a*b^3 *d^2)^(1/4)*log(d*x - sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/( a^4*b*d) + 1/192*(77*sqrt(d*x)*b^2*d^5*x^4 + 198*sqrt(d*x)*a*b*d^5*x^2 + 1 53*sqrt(d*x)*a^2*d^5)/((b*d^2*x^2 + a*d^2)^3*a^3)
Time = 13.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.45 \[ \int \frac {1}{\sqrt {d x} \left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {51\,d^5\,\sqrt {d\,x}}{64\,a}+\frac {33\,b\,d^3\,{\left (d\,x\right )}^{5/2}}{32\,a^2}+\frac {77\,b^2\,d\,{\left (d\,x\right )}^{9/2}}{192\,a^3}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}+\frac {77\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{15/4}\,b^{1/4}\,\sqrt {d}}+\frac {77\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{15/4}\,b^{1/4}\,\sqrt {d}} \]
((51*d^5*(d*x)^(1/2))/(64*a) + (33*b*d^3*(d*x)^(5/2))/(32*a^2) + (77*b^2*d *(d*x)^(9/2))/(192*a^3))/(a^3*d^6 + b^3*d^6*x^6 + 3*a^2*b*d^6*x^2 + 3*a*b^ 2*d^6*x^4) + (77*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(128*(- a)^(15/4)*b^(1/4)*d^(1/2)) + (77*atanh((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d ^(1/2))))/(128*(-a)^(15/4)*b^(1/4)*d^(1/2))