Integrand size = 28, antiderivative size = 256 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}-\frac {11 d^3 (d x)^{7/2}}{48 b^2 \left (a+b x^2\right )^2}-\frac {77 d^5 (d x)^{3/2}}{192 b^3 \left (a+b x^2\right )}-\frac {77 d^{13/2} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}+\frac {77 d^{13/2} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}}-\frac {77 d^{13/2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{128 \sqrt {2} \sqrt [4]{a} b^{15/4}} \] Output:
-1/6*d*(d*x)^(11/2)/b/(b*x^2+a)^3-11/48*d^3*(d*x)^(7/2)/b^2/(b*x^2+a)^2-77 /192*d^5*(d*x)^(3/2)/b^3/(b*x^2+a)-77/256*d^(13/2)*arctan(1-2^(1/2)*b^(1/4 )*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(1/4)/b^(15/4)+77/256*d^(13/2)*ar ctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(1/4)/b^(15/ 4)-77/256*d^(13/2)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*(d*x)^(1/2)/d^(1/2)/(a^ (1/2)+b^(1/2)*x))*2^(1/2)/a^(1/4)/b^(15/4)
Time = 0.52 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.64 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {d^6 \sqrt {d x} \left (-\frac {4 b^{3/4} x^{3/2} \left (77 a^2+198 a b x^2+153 b^2 x^4\right )}{\left (a+b x^2\right )^3}-\frac {231 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )}{\sqrt [4]{a}}-\frac {231 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{\sqrt [4]{a}}\right )}{768 b^{15/4} \sqrt {x}} \] Input:
Integrate[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
(d^6*Sqrt[d*x]*((-4*b^(3/4)*x^(3/2)*(77*a^2 + 198*a*b*x^2 + 153*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])])/a^(1/4) - (231*Sqrt[2]*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4 )*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/a^(1/4)))/(768*b^(15/4)*Sqrt[x])
Time = 0.95 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.45, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {1380, 27, 252, 252, 252, 266, 27, 826, 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.
| \(\displaystyle \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^4 \int \frac {(d x)^{13/2}}{b^4 \left (b x^2+a\right )^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(d x)^{13/2}}{\left (a+b x^2\right )^4}dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {11 d^2 \int \frac {(d x)^{9/2}}{\left (b x^2+a\right )^3}dx}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \int \frac {(d x)^{5/2}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^2 \int \frac {\sqrt {d x}}{b x^2+a}dx}{4 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {11 d^2 \left (\frac {7 d^2 \left (\frac {3 d^3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{2 b}-\frac {d (d x)^{3/2}}{2 b \left (a+b x^2\right )}\right )}{8 b}-\frac {d (d x)^{7/2}}{4 b \left (a+b x^2\right )^2}\right )}{12 b}-\frac {d (d x)^{11/2}}{6 b \left (a+b x^2\right )^3}\) |
Input:
Int[(d*x)^(13/2)/(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]
Output:
-1/6*(d*(d*x)^(11/2))/(b*(a + b*x^2)^3) + (11*d^2*(-1/4*(d*(d*x)^(7/2))/(b *(a + b*x^2)^2) + (7*d^2*(-1/2*(d*(d*x)^(3/2))/(b*(a + b*x^2)) + (3*d^3*(( -(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])) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[d*x])/(a^(1/4)*Sqr t[d])]/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b]) - (-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*Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[d]))/(2*Sqrt[b])))/( 2*b)))/(8*b)))/(12*b)
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*(c*x )^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* (p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c }, x] && LtQ[p, -1] && GtQ[m, 1] && !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi alQ[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[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) 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 = 2.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(2 d^{7} \left (\frac {-\frac {51 \left (d x \right )^{\frac {11}{2}}}{128 b}-\frac {33 d^{2} a \left (d x \right )^{\frac {7}{2}}}{64 b^{2}}-\frac {77 d^{4} a^{2} \left (d x \right )^{\frac {3}{2}}}{384 b^{3}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {77 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(203\) |
| default | \(2 d^{7} \left (\frac {-\frac {51 \left (d x \right )^{\frac {11}{2}}}{128 b}-\frac {33 d^{2} a \left (d x \right )^{\frac {7}{2}}}{64 b^{2}}-\frac {77 d^{4} a^{2} \left (d x \right )^{\frac {3}{2}}}{384 b^{3}}}{\left (b \,d^{2} x^{2}+a \,d^{2}\right )^{3}}+\frac {77 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 b^{4} \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\) | \(203\) |
| pseudoelliptic | \(\frac {77 d^{6} \left (-8 \left (\frac {153}{77} b^{2} x^{4}+\frac {18}{7} a b \,x^{2}+a^{2}\right ) \sqrt {d x}\, b x \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}+3 d \sqrt {2}\, \left (b \,x^{2}+a \right )^{3} \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right )\right )}{1536 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{3} b^{4}}\) | \(224\) |
Input:
int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x,method=_RETURNVERBOSE)
Output:
2*d^7*((-51/128/b*(d*x)^(11/2)-33/64/b^2*d^2*a*(d*x)^(7/2)-77/384/b^3*d^4* a^2*(d*x)^(3/2))/(b*d^2*x^2+a*d^2)^3+77/1024/b^4/(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 = 0.14 (sec) , antiderivative size = 406, normalized size of antiderivative = 1.59 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {231 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} + 456533 \, \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {3}{4}} a b^{11}\right ) - 231 \, {\left (i \, b^{6} x^{6} + 3 i \, a b^{5} x^{4} + 3 i \, a^{2} b^{4} x^{2} + i \, a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} + 456533 i \, \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {3}{4}} a b^{11}\right ) - 231 \, {\left (-i \, b^{6} x^{6} - 3 i \, a b^{5} x^{4} - 3 i \, a^{2} b^{4} x^{2} - i \, a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} - 456533 i \, \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {3}{4}} a b^{11}\right ) - 231 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )} \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {1}{4}} \log \left (456533 \, \sqrt {d x} d^{19} - 456533 \, \left (-\frac {d^{26}}{a b^{15}}\right )^{\frac {3}{4}} a b^{11}\right ) - 4 \, {\left (153 \, b^{2} d^{6} x^{5} + 198 \, a b d^{6} x^{3} + 77 \, a^{2} d^{6} x\right )} \sqrt {d x}}{768 \, {\left (b^{6} x^{6} + 3 \, a b^{5} x^{4} + 3 \, a^{2} b^{4} x^{2} + a^{3} b^{3}\right )}} \] Input:
integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="fricas")
Output:
1/768*(231*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)*(-d^26/(a*b^1 5))^(1/4)*log(456533*sqrt(d*x)*d^19 + 456533*(-d^26/(a*b^15))^(3/4)*a*b^11 ) - 231*(I*b^6*x^6 + 3*I*a*b^5*x^4 + 3*I*a^2*b^4*x^2 + I*a^3*b^3)*(-d^26/( a*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 + 456533*I*(-d^26/(a*b^15))^(3/4) *a*b^11) - 231*(-I*b^6*x^6 - 3*I*a*b^5*x^4 - 3*I*a^2*b^4*x^2 - I*a^3*b^3)* (-d^26/(a*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 - 456533*I*(-d^26/(a*b^15 ))^(3/4)*a*b^11) - 231*(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)*( -d^26/(a*b^15))^(1/4)*log(456533*sqrt(d*x)*d^19 - 456533*(-d^26/(a*b^15))^ (3/4)*a*b^11) - 4*(153*b^2*d^6*x^5 + 198*a*b*d^6*x^3 + 77*a^2*d^6*x)*sqrt( d*x))/(b^6*x^6 + 3*a*b^5*x^4 + 3*a^2*b^4*x^2 + a^3*b^3)
\[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\int \frac {\left (d x\right )^{\frac {13}{2}}}{\left (a + b x^{2}\right )^{4}}\, dx \] Input:
integrate((d*x)**(13/2)/(b**2*x**4+2*a*b*x**2+a**2)**2,x)
Output:
Integral((d*x)**(13/2)/(a + b*x**2)**4, x)
Time = 0.12 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.24 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {\frac {231 \, d^{8} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{b^{3}} - \frac {8 \, {\left (153 \, \left (d x\right )^{\frac {11}{2}} b^{2} d^{8} + 198 \, \left (d x\right )^{\frac {7}{2}} a b d^{10} + 77 \, \left (d x\right )^{\frac {3}{2}} a^{2} d^{12}\right )}}{b^{6} d^{6} x^{6} + 3 \, a b^{5} d^{6} x^{4} + 3 \, a^{2} b^{4} d^{6} x^{2} + a^{3} b^{3} d^{6}}}{1536 \, d} \] Input:
integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="maxima")
Output:
1/1536*(231*d^8*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/ 4) + 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d )*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a*d^2)^(1/4)*b^(1/4) - 2*sqrt(d*x)*sqrt(b))/sqrt(sqrt(a)*sqrt(b)*d))/(sqrt(sqrt(a)*sqrt(b)*d)*s qrt(b)) - sqrt(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)^(1/4)*b^(3/4)) + sqrt(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)^(1/4)*b^(3/4)))/b^3 - 8*(153*(d*x)^(11/2)*b^2*d^8 + 198*(d*x)^(7/2)*a*b*d^10 + 77*(d*x)^(3/2) *a^2*d^12)/(b^6*d^6*x^6 + 3*a*b^5*d^6*x^4 + 3*a^2*b^4*d^6*x^2 + a^3*b^3*d^ 6))/d
Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.23 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=-\frac {1}{1536} \, d^{6} {\left (\frac {8 \, {\left (153 \, \sqrt {d x} b^{2} d^{6} x^{5} + 198 \, \sqrt {d x} a b d^{6} x^{3} + 77 \, \sqrt {d x} a^{2} d^{6} x\right )}}{{\left (b d^{2} x^{2} + a d^{2}\right )}^{3} b^{3}} - \frac {462 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{6} d} - \frac {462 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a b^{6} d} + \frac {231 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{6} d} - \frac {231 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a b^{6} d}\right )} \] Input:
integrate((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x, algorithm="giac")
Output:
-1/1536*d^6*(8*(153*sqrt(d*x)*b^2*d^6*x^5 + 198*sqrt(d*x)*a*b*d^6*x^3 + 77 *sqrt(d*x)*a^2*d^6*x)/((b*d^2*x^2 + a*d^2)^3*b^3) - 462*sqrt(2)*(a*b^3*d^2 )^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a*d^2/b)^(1/4) + 2*sqrt(d*x))/(a*d^2/ b)^(1/4))/(a*b^6*d) - 462*sqrt(2)*(a*b^3*d^2)^(3/4)*arctan(-1/2*sqrt(2)*(s qrt(2)*(a*d^2/b)^(1/4) - 2*sqrt(d*x))/(a*d^2/b)^(1/4))/(a*b^6*d) + 231*sqr t(2)*(a*b^3*d^2)^(3/4)*log(d*x + sqrt(2)*(a*d^2/b)^(1/4)*sqrt(d*x) + sqrt( a*d^2/b))/(a*b^6*d) - 231*sqrt(2)*(a*b^3*d^2)^(3/4)*log(d*x - sqrt(2)*(a*d ^2/b)^(1/4)*sqrt(d*x) + sqrt(a*d^2/b))/(a*b^6*d))
Time = 0.11 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.60 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx=\frac {77\,d^{13/2}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{1/4}\,b^{15/4}}-\frac {\frac {51\,d^7\,{\left (d\,x\right )}^{11/2}}{64\,b}+\frac {77\,a^2\,d^{11}\,{\left (d\,x\right )}^{3/2}}{192\,b^3}+\frac {33\,a\,d^9\,{\left (d\,x\right )}^{7/2}}{32\,b^2}}{a^3\,d^6+3\,a^2\,b\,d^6\,x^2+3\,a\,b^2\,d^6\,x^4+b^3\,d^6\,x^6}-\frac {77\,d^{13/2}\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {d\,x}}{{\left (-a\right )}^{1/4}\,\sqrt {d}}\right )}{128\,{\left (-a\right )}^{1/4}\,b^{15/4}} \] Input:
int((d*x)^(13/2)/(a^2 + b^2*x^4 + 2*a*b*x^2)^2,x)
Output:
(77*d^(13/2)*atan((b^(1/4)*(d*x)^(1/2))/((-a)^(1/4)*d^(1/2))))/(128*(-a)^( 1/4)*b^(15/4)) - ((51*d^7*(d*x)^(11/2))/(64*b) + (77*a^2*d^11*(d*x)^(3/2)) /(192*b^3) + (33*a*d^9*(d*x)^(7/2))/(32*b^2))/(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*d^(13/2)*atanh((b^(1/4)*(d*x)^(1/2)) /((-a)^(1/4)*d^(1/2))))/(128*(-a)^(1/4)*b^(15/4))
Time = 0.18 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.55 \[ \int \frac {(d x)^{13/2}}{\left (a^2+2 a b x^2+b^2 x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*x)^(13/2)/(b^2*x^4+2*a*b*x^2+a^2)^2,x)
Output:
(sqrt(d)*d**6*( - 462*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sq rt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3 - 1386*b**(1/ 4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/( b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b*x**2 - 1386*b**(1/4)*a**(3/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr t(2)))*a*b**2*x**4 - 462*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4) *sqrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**3*x**6 + 462 *b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqr t(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a**3 + 1386*b**(1/4)*a**(3/4)*sqrt(2)*a tan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqr t(2)))*a**2*b*x**2 + 1386*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4 )*sqrt(2) + 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b**2*x**4 + 462*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(x)* sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**3*x**6 + 231*b**(1/4)*a**(3/4)*sq rt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**3 + 693*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a**2*b*x**2 + 693*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a*b**2*x**4 + 231 *b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt (a) + sqrt(b)*x)*b**3*x**6 - 231*b**(1/4)*a**(3/4)*sqrt(2)*log(sqrt(x)*...