Integrand size = 15, antiderivative size = 251 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {45}{16 a^3 \sqrt {x}}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}+\frac {9}{16 a^2 \sqrt {x} \left (a+b x^2\right )}+\frac {45 \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{13/4}}-\frac {45 \sqrt [4]{b} \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}}+\frac {45 \sqrt [4]{b} \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {b} x\right )}{64 \sqrt {2} a^{13/4}} \]
45/64*b^(1/4)*arctan(1-b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)*2^(1/2)-4 5/64*b^(1/4)*arctan(1+b^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))/a^(13/4)*2^(1/2)-45 /128*b^(1/4)*ln(a^(1/2)+x*b^(1/2)-a^(1/4)*b^(1/4)*2^(1/2)*x^(1/2))/a^(13/4 )*2^(1/2)+45/128*b^(1/4)*ln(a^(1/2)+x*b^(1/2)+a^(1/4)*b^(1/4)*2^(1/2)*x^(1 /2))/a^(13/4)*2^(1/2)-45/16/a^3/x^(1/2)+1/4/a/(b*x^2+a)^2/x^(1/2)+9/16/a^2 /(b*x^2+a)/x^(1/2)
Time = 0.38 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.59 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} \left (32 a^2+81 a b x^2+45 b^2 x^4\right )}{\sqrt {x} \left (a+b x^2\right )^2}+45 \sqrt {2} \sqrt [4]{b} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+45 \sqrt {2} \sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{13/4}} \]
((-4*a^(1/4)*(32*a^2 + 81*a*b*x^2 + 45*b^2*x^4))/(Sqrt[x]*(a + b*x^2)^2) + 45*Sqrt[2]*b^(1/4)*ArcTan[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)* Sqrt[x])] + 45*Sqrt[2]*b^(1/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/( Sqrt[a] + Sqrt[b]*x)])/(64*a^(13/4))
Time = 0.44 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {253, 253, 264, 266, 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.
| \(\displaystyle \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {9 \int \frac {1}{x^{3/2} \left (b x^2+a\right )^2}dx}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {9 \left (\frac {5 \int \frac {1}{x^{3/2} \left (b x^2+a\right )}dx}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {b \int \frac {\sqrt {x}}{b x^2+a}dx}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \int \frac {x}{b x^2+a}d\sqrt {x}}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {9 \left (\frac {5 \left (-\frac {2 b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}-\frac {2}{a \sqrt {x}}\right )}{4 a}+\frac {1}{2 a \sqrt {x} \left (a+b x^2\right )}\right )}{8 a}+\frac {1}{4 a \sqrt {x} \left (a+b x^2\right )^2}\) |
1/(4*a*Sqrt[x]*(a + b*x^2)^2) + (9*(1/(2*a*Sqrt[x]*(a + b*x^2)) + (5*(-2/( a*Sqrt[x]) - (2*b*((-(ArcTan[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[ 2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt [2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b]) - (-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^ (1/4)*Sqrt[x] + Sqrt[b]*x]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[ 2]*a^(1/4)*b^(1/4)*Sqrt[x] + Sqrt[b]*x]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sq rt[b])))/a))/(4*a)))/(8*a)
3.4.9.3.1 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] :> 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[((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 = 1.83 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.58
| method | result | size |
| derivativedivides | \(-\frac {2 b \left (\frac {\frac {13 b \,x^{\frac {7}{2}}}{32}+\frac {17 a \,x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}-\frac {2}{a^{3} \sqrt {x}}\) | \(145\) |
| default | \(-\frac {2 b \left (\frac {\frac {13 b \,x^{\frac {7}{2}}}{32}+\frac {17 a \,x^{\frac {3}{2}}}{32}}{\left (b \,x^{2}+a \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{256 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}-\frac {2}{a^{3} \sqrt {x}}\) | \(145\) |
| risch | \(-\frac {2}{a^{3} \sqrt {x}}-\frac {b \left (\frac {\frac {13 b \,x^{\frac {7}{2}}}{16}+\frac {17 a \,x^{\frac {3}{2}}}{16}}{\left (b \,x^{2}+a \right )^{2}}+\frac {45 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a^{3}}\) | \(146\) |
-2/a^3*b*((13/32*b*x^(7/2)+17/32*a*x^(3/2))/(b*x^2+a)^2+45/256/b/(a/b)^(1/ 4)*2^(1/2)*(ln((x-(a/b)^(1/4)*x^(1/2)*2^(1/2)+(a/b)^(1/2))/(x+(a/b)^(1/4)* x^(1/2)*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)+1)+2*ar ctan(2^(1/2)/(a/b)^(1/4)*x^(1/2)-1)))-2/a^3/x^(1/2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.13 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {45 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 \, a^{10} \left (-\frac {b}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, b \sqrt {x}\right ) + 45 \, {\left (-i \, a^{3} b^{2} x^{5} - 2 i \, a^{4} b x^{3} - i \, a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \log \left (91125 i \, a^{10} \left (-\frac {b}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, b \sqrt {x}\right ) + 45 \, {\left (i \, a^{3} b^{2} x^{5} + 2 i \, a^{4} b x^{3} + i \, a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 i \, a^{10} \left (-\frac {b}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, b \sqrt {x}\right ) - 45 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{4}} \log \left (-91125 \, a^{10} \left (-\frac {b}{a^{13}}\right )^{\frac {3}{4}} + 91125 \, b \sqrt {x}\right ) + 4 \, {\left (45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}\right )} \sqrt {x}}{64 \, {\left (a^{3} b^{2} x^{5} + 2 \, a^{4} b x^{3} + a^{5} x\right )}} \]
-1/64*(45*(a^3*b^2*x^5 + 2*a^4*b*x^3 + a^5*x)*(-b/a^13)^(1/4)*log(91125*a^ 10*(-b/a^13)^(3/4) + 91125*b*sqrt(x)) + 45*(-I*a^3*b^2*x^5 - 2*I*a^4*b*x^3 - I*a^5*x)*(-b/a^13)^(1/4)*log(91125*I*a^10*(-b/a^13)^(3/4) + 91125*b*sqr t(x)) + 45*(I*a^3*b^2*x^5 + 2*I*a^4*b*x^3 + I*a^5*x)*(-b/a^13)^(1/4)*log(- 91125*I*a^10*(-b/a^13)^(3/4) + 91125*b*sqrt(x)) - 45*(a^3*b^2*x^5 + 2*a^4* b*x^3 + a^5*x)*(-b/a^13)^(1/4)*log(-91125*a^10*(-b/a^13)^(3/4) + 91125*b*s qrt(x)) + 4*(45*b^2*x^4 + 81*a*b*x^2 + 32*a^2)*sqrt(x))/(a^3*b^2*x^5 + 2*a ^4*b*x^3 + a^5*x)
Timed out. \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {45 \, b^{2} x^{4} + 81 \, a b x^{2} + 32 \, a^{2}}{16 \, {\left (a^{3} b^{2} x^{\frac {9}{2}} + 2 \, a^{4} b x^{\frac {5}{2}} + a^{5} \sqrt {x}\right )}} - \frac {45 \, b {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a^{3}} \]
-1/16*(45*b^2*x^4 + 81*a*b*x^2 + 32*a^2)/(a^3*b^2*x^(9/2) + 2*a^4*b*x^(5/2 ) + a^5*sqrt(x)) - 45/128*b*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4) *b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b) )*sqrt(b)) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sq rt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sq rt(2)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)* b^(3/4)) + sqrt(2)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt (a))/(a^(1/4)*b^(3/4)))/a^3
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=-\frac {2}{a^{3} \sqrt {x}} - \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{2}} - \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{4} b^{2}} + \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b^{2}} - \frac {45 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{4} b^{2}} - \frac {13 \, b^{2} x^{\frac {7}{2}} + 17 \, a b x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a^{3}} \]
-2/(a^3*sqrt(x)) - 45/64*sqrt(2)*(a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2) *(a/b)^(1/4) + 2*sqrt(x))/(a/b)^(1/4))/(a^4*b^2) - 45/64*sqrt(2)*(a*b^3)^( 3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a ^4*b^2) + 45/128*sqrt(2)*(a*b^3)^(3/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 45/128*sqrt(2)*(a*b^3)^(3/4)*log(-sqrt(2)*sqrt(x )*(a/b)^(1/4) + x + sqrt(a/b))/(a^4*b^2) - 1/16*(13*b^2*x^(7/2) + 17*a*b*x ^(3/2))/((b*x^2 + a)^2*a^3)
Time = 0.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.39 \[ \int \frac {1}{x^{3/2} \left (a+b x^2\right )^3} \, dx=\frac {45\,{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{32\,a^{13/4}}-\frac {45\,{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/4}\,\sqrt {x}}{a^{1/4}}\right )}{32\,a^{13/4}}-\frac {\frac {2}{a}+\frac {81\,b\,x^2}{16\,a^2}+\frac {45\,b^2\,x^4}{16\,a^3}}{a^2\,\sqrt {x}+b^2\,x^{9/2}+2\,a\,b\,x^{5/2}} \]