Integrand size = 28, antiderivative size = 223 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {5}{2 a^2 d \sqrt {d x}}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}+\frac {5 \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}-\frac {5 \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}}+\frac {5 \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d x}}{\sqrt {d} \left (\sqrt {a}+\sqrt {b} x\right )}\right )}{4 \sqrt {2} a^{9/4} d^{3/2}} \] Output:
-5/2/a^2/d/(d*x)^(1/2)+1/2/a/d/(d*x)^(1/2)/(b*x^2+a)+5/8*b^(1/4)*arctan(1- 2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(9/4)/d^(3/2)-5/8*b ^(1/4)*arctan(1+2^(1/2)*b^(1/4)*(d*x)^(1/2)/a^(1/4)/d^(1/2))*2^(1/2)/a^(9/ 4)/d^(3/2)+5/8*b^(1/4)*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^(9/4)/d^(3/2)
Time = 0.29 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {x \left (-4 \sqrt [4]{a} \left (4 a+5 b x^2\right )+5 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+5 \sqrt {2} \sqrt [4]{b} \sqrt {x} \left (a+b x^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )\right )}{8 a^{9/4} (d x)^{3/2} \left (a+b x^2\right )} \] Input:
Integrate[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
Output:
(x*(-4*a^(1/4)*(4*a + 5*b*x^2) + 5*Sqrt[2]*b^(1/4)*Sqrt[x]*(a + b*x^2)*Arc Tan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 5*Sqrt[2]*b ^(1/4)*Sqrt[x]*(a + b*x^2)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt [a] + Sqrt[b]*x)]))/(8*a^(9/4)*(d*x)^(3/2)*(a + b*x^2))
Time = 0.90 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.46, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1380, 27, 253, 264, 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.
| \(\displaystyle \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx\) |
| \(\Big \downarrow \) 1380 |
| \(\displaystyle b^2 \int \frac {1}{b^2 (d x)^{3/2} \left (b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {1}{(d x)^{3/2} \left (a+b x^2\right )^2}dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {5 \int \frac {1}{(d x)^{3/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {5 \left (-\frac {b \int \frac {\sqrt {d x}}{b x^2+a}dx}{a d^2}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \int \frac {d^3 x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d^3}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \int \frac {d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x d+\sqrt {a} d}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-d x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int \frac {1}{-d x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a} d-\sqrt {b} d x}{b x^2 d^2+a d^2}d\sqrt {d x}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}\right )}{\sqrt [4]{b} \left (x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}\right )}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a} \sqrt {d}-2 \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b} \sqrt {d}}+\frac {\int \frac {\sqrt [4]{a} \sqrt {d}+\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{x d+\frac {\sqrt {a} d}{\sqrt {b}}+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {d x} \sqrt {d}}{\sqrt [4]{b}}}d\sqrt {d x}}{2 \sqrt [4]{a} \sqrt {b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {d x}}{\sqrt [4]{a} \sqrt {d}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d} \sqrt {d x}+\sqrt {a} d+\sqrt {b} d x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {d}}}{2 \sqrt {b}}\right )}{a d}-\frac {2}{a d \sqrt {d x}}\right )}{4 a}+\frac {1}{2 a d \sqrt {d x} \left (a+b x^2\right )}\) |
Input:
Int[1/((d*x)^(3/2)*(a^2 + 2*a*b*x^2 + b^2*x^4)),x]
Output:
1/(2*a*d*Sqrt[d*x]*(a + b*x^2)) + (5*(-2/(a*d*Sqrt[d*x]) - (2*b*((-(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)*Sqrt[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])))/(a*d)))/( 4*a)
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] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -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[(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 = 0.43 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {2}{a^{2} d \sqrt {d x}}-\frac {b \left (\frac {\left (d x \right )^{\frac {3}{2}}}{2 b \,d^{2} x^{2}+2 a \,d^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} d}\) | \(186\) |
| derivativedivides | \(2 d^{3} \left (-\frac {1}{a^{2} d^{4} \sqrt {d x}}-\frac {b \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} d^{4}}\right )\) | \(191\) |
| default | \(2 d^{3} \left (-\frac {1}{a^{2} d^{4} \sqrt {d x}}-\frac {b \left (\frac {\left (d x \right )^{\frac {3}{2}}}{4 b \,d^{2} x^{2}+4 a \,d^{2}}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 b \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{2} d^{4}}\right )\) | \(191\) |
| pseudoelliptic | \(-\frac {5 \left (\frac {\sqrt {2}\, \left (b \,x^{2}+a \right ) \left (\ln \left (\frac {d x -\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}{d x +\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, \sqrt {2}+\sqrt {\frac {a \,d^{2}}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}-\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d x}+\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}{\left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}}}\right )\right ) \sqrt {d x}}{2}+\frac {16 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \left (\frac {5 b \,x^{2}}{4}+a \right )}{5}\right )}{8 \left (\frac {a \,d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x}\, d \,a^{2} \left (b \,x^{2}+a \right )}\) | \(213\) |
Input:
int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2),x,method=_RETURNVERBOSE)
Output:
-2/a^2/d/(d*x)^(1/2)-1/a^2*b*(1/2*(d*x)^(3/2)/(b*d^2*x^2+a*d^2)+5/16/b/(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*arct an(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)))/d
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {5 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \log \left (125 \, a^{7} d^{5} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} b\right ) + 5 \, {\left (-i \, a^{2} b d^{2} x^{3} - i \, a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \log \left (125 i \, a^{7} d^{5} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} b\right ) + 5 \, {\left (i \, a^{2} b d^{2} x^{3} + i \, a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \log \left (-125 i \, a^{7} d^{5} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} b\right ) - 5 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {1}{4}} \log \left (-125 \, a^{7} d^{5} \left (-\frac {b}{a^{9} d^{6}}\right )^{\frac {3}{4}} + 125 \, \sqrt {d x} b\right ) + 4 \, {\left (5 \, b x^{2} + 4 \, a\right )} \sqrt {d x}}{8 \, {\left (a^{2} b d^{2} x^{3} + a^{3} d^{2} x\right )}} \] Input:
integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="fricas")
Output:
-1/8*(5*(a^2*b*d^2*x^3 + a^3*d^2*x)*(-b/(a^9*d^6))^(1/4)*log(125*a^7*d^5*( -b/(a^9*d^6))^(3/4) + 125*sqrt(d*x)*b) + 5*(-I*a^2*b*d^2*x^3 - I*a^3*d^2*x )*(-b/(a^9*d^6))^(1/4)*log(125*I*a^7*d^5*(-b/(a^9*d^6))^(3/4) + 125*sqrt(d *x)*b) + 5*(I*a^2*b*d^2*x^3 + I*a^3*d^2*x)*(-b/(a^9*d^6))^(1/4)*log(-125*I *a^7*d^5*(-b/(a^9*d^6))^(3/4) + 125*sqrt(d*x)*b) - 5*(a^2*b*d^2*x^3 + a^3* d^2*x)*(-b/(a^9*d^6))^(1/4)*log(-125*a^7*d^5*(-b/(a^9*d^6))^(3/4) + 125*sq rt(d*x)*b) + 4*(5*b*x^2 + 4*a)*sqrt(d*x))/(a^2*b*d^2*x^3 + a^3*d^2*x)
\[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\int \frac {1}{\left (d x\right )^{\frac {3}{2}} \left (a + b x^{2}\right )^{2}}\, dx \] Input:
integrate(1/(d*x)**(3/2)/(b**2*x**4+2*a*b*x**2+a**2),x)
Output:
Integral(1/((d*x)**(3/2)*(a + b*x**2)**2), x)
Time = 0.11 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {\frac {8 \, {\left (5 \, b d^{2} x^{2} + 4 \, a d^{2}\right )}}{\left (d x\right )^{\frac {5}{2}} a^{2} b + \sqrt {d x} a^{3} d^{2}} + \frac {5 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {d x} \sqrt {b}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b} d}}\right )}{\sqrt {\sqrt {a} \sqrt {b} d} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} d x + \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} d x - \sqrt {2} \left (a d^{2}\right )^{\frac {1}{4}} \sqrt {d x} b^{\frac {1}{4}} + \sqrt {a} d\right )}{\left (a d^{2}\right )^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{a^{2}}}{16 \, d} \] Input:
integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="maxima")
Output:
-1/16*(8*(5*b*d^2*x^2 + 4*a*d^2)/((d*x)^(5/2)*a^2*b + sqrt(d*x)*a^3*d^2) + 5*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)*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)*sqrt(b)) - s qrt(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)))/a^2)/d
Time = 0.14 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=-\frac {\frac {8 \, {\left (5 \, b d^{2} x^{2} + 4 \, a d^{2}\right )}}{{\left (\sqrt {d x} b d^{2} x^{2} + \sqrt {d x} a d^{2}\right )} a^{2}} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{2} d^{2}} + \frac {10 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {d x}\right )}}{2 \, \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}}}\right )}{a^{3} b^{2} d^{2}} - \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x + \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{2} d^{2}} + \frac {5 \, \sqrt {2} \left (a b^{3} d^{2}\right )^{\frac {3}{4}} \log \left (d x - \sqrt {2} \left (\frac {a d^{2}}{b}\right )^{\frac {1}{4}} \sqrt {d x} + \sqrt {\frac {a d^{2}}{b}}\right )}{a^{3} b^{2} d^{2}}}{16 \, d} \] Input:
integrate(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2),x, algorithm="giac")
Output:
-1/16*(8*(5*b*d^2*x^2 + 4*a*d^2)/((sqrt(d*x)*b*d^2*x^2 + sqrt(d*x)*a*d^2)* a^2) + 10*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^3*b^2*d^2) + 10*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^3*b^2*d^2) - 5*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^3*b^2*d^2) + 5*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^3*b^2*d^2))/d
Time = 0.13 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {5\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{9/4}\,d^{3/2}}-\frac {5\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {d\,x}}{a^{1/4}\,\sqrt {d}}\right )}{4\,a^{9/4}\,d^{3/2}}-\frac {\frac {2\,d}{a}+\frac {5\,b\,d\,x^2}{2\,a^2}}{b\,{\left (d\,x\right )}^{5/2}+a\,d^2\,\sqrt {d\,x}} \] Input:
int(1/((d*x)^(3/2)*(a^2 + b^2*x^4 + 2*a*b*x^2)),x)
Output:
(5*(-b)^(1/4)*atanh(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2))))/(4*a^(9/4 )*d^(3/2)) - (5*(-b)^(1/4)*atan(((-b)^(1/4)*(d*x)^(1/2))/(a^(1/4)*d^(1/2)) ))/(4*a^(9/4)*d^(3/2)) - ((2*d)/a + (5*b*d*x^2)/(2*a^2))/(b*(d*x)^(5/2) + a*d^2*(d*x)^(1/2))
Time = 0.17 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(d x)^{3/2} \left (a^2+2 a b x^2+b^2 x^4\right )} \, dx=\frac {\sqrt {d}\, \left (10 \sqrt {x}\, b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+10 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-10 \sqrt {x}\, b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-10 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-5 \sqrt {x}\, b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-5 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+5 \sqrt {x}\, b^{\frac {1}{4}} a^{\frac {7}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+5 \sqrt {x}\, b^{\frac {5}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-32 a^{2}-40 a b \,x^{2}\right )}{16 \sqrt {x}\, a^{3} d^{2} \left (b \,x^{2}+a \right )} \] Input:
int(1/(d*x)^(3/2)/(b^2*x^4+2*a*b*x^2+a^2),x)
Output:
(sqrt(d)*(10*sqrt(x)*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr t(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*a + 10*sqrt(x)*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*x**2 - 10*sqrt(x)*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 - 10*sqrt(x)*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*x**2 - 5*sqrt (x)*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + s qrt(a) + sqrt(b)*x)*a - 5*sqrt(x)*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*x**2 + 5*sqrt(x)*b**(1 /4)*a**(3/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqr t(b)*x)*a + 5*sqrt(x)*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*x**2 - 32*a**2 - 40*a*b*x**2))/(16*sq rt(x)*a**3*d**2*(a + b*x**2))