Integrand size = 15, antiderivative size = 139 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\frac {\sqrt {x} \left (-3 a+b x^2\right )}{16 a b \left (a+b x^2\right )^2}+\frac {3 \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}}{\sqrt {\frac {a}{b}}-x}\right )+\log \left (\frac {\sqrt {\frac {a}{b}}+\sqrt {2} \sqrt [4]{\frac {a}{b}} \sqrt {x}+x}{\sqrt {a+b x^2}}\right )\right )}{32 \sqrt {2} a \left (\frac {a}{b}\right )^{3/4} b^2} \]
1/16*(b*x^2-3*a)*x^(1/2)/a/b/(b*x^2+a)^2+3/64/a/b^2/(a/b)^(3/4)*2^(1/2)*(l n((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(b*x^2+a)^(1/2))+arctan((a/b )^(1/4)*2^(1/2)*x^(1/2)/((a/b)^(1/2)-x)))
Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.99 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {4 a^{3/4} \sqrt [4]{b} \sqrt {x} \left (-3 a+b x^2\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{7/4} b^{5/4}} \]
((4*a^(3/4)*b^(1/4)*Sqrt[x]*(-3*a + b*x^2))/(a + b*x^2)^2 - 3*Sqrt[2]*ArcT an[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] + 3*Sqrt[2]*Ar cTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(64*a^(7/4 )*b^(5/4))
Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {252, 253, 266, 755, 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.
| \(\displaystyle \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt {x} \left (b x^2+a\right )^2}dx}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{4 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\frac {3 \int \frac {1}{b x^2+a}d\sqrt {x}}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{2 a}+\frac {\sqrt {x}}{2 a \left (a+b x^2\right )}}{8 b}-\frac {\sqrt {x}}{4 b \left (a+b x^2\right )^2}\) |
-1/4*Sqrt[x]/(b*(a + b*x^2)^2) + (Sqrt[x]/(2*a*(a + b*x^2)) + (3*((-(ArcTa n[1 - (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcT an[1 + (Sqrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sq rt[a]) + (-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*Sqrt[a])))/(2*a))/(8*b)
3.2.31.3.1 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*(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] :> 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[((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 = 0.08 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.99
| method | result | size |
| derivativedivides | \(\frac {\frac {x^{\frac {5}{2}}}{16 a}-\frac {3 \sqrt {x}}{16 b}}{\left (x^{2} b +a \right )^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \,a^{2}}\) | \(138\) |
| default | \(\frac {\frac {x^{\frac {5}{2}}}{16 a}-\frac {3 \sqrt {x}}{16 b}}{\left (x^{2} b +a \right )^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \sqrt {x}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 b \,a^{2}}\) | \(138\) |
2*(1/32/a*x^(5/2)-3/32*x^(1/2)/b)/(b*x^2+a)^2+3/128/b/a^2*(a/b)^(1/4)*2^(1 /2)*(ln((x+(a/b)^(1/4)*2^(1/2)*x^(1/2)+(a/b)^(1/2))/(x-(a/b)^(1/4)*2^(1/2) *x^(1/2)+(a/b)^(1/2)))+2*arctan(1/(a/b)^(1/4)*2^(1/2)*x^(1/2)+1)+2*arctan( 1/(a/b)^(1/4)*2^(1/2)*x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.01 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 3 \, {\left (-i \, a b^{3} x^{4} - 2 i \, a^{2} b^{2} x^{2} - i \, a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 3 \, {\left (i \, a b^{3} x^{4} + 2 i \, a^{2} b^{2} x^{2} + i \, a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) - 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} b \left (-\frac {1}{a^{7} b^{5}}\right )^{\frac {1}{4}} + \sqrt {x}\right ) + 4 \, {\left (b x^{2} - 3 \, a\right )} \sqrt {x}}{64 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \]
1/64*(3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-1/(a^7*b^5))^(1/4)*log(a^2*b *(-1/(a^7*b^5))^(1/4) + sqrt(x)) - 3*(-I*a*b^3*x^4 - 2*I*a^2*b^2*x^2 - I*a ^3*b)*(-1/(a^7*b^5))^(1/4)*log(I*a^2*b*(-1/(a^7*b^5))^(1/4) + sqrt(x)) - 3 *(I*a*b^3*x^4 + 2*I*a^2*b^2*x^2 + I*a^3*b)*(-1/(a^7*b^5))^(1/4)*log(-I*a^2 *b*(-1/(a^7*b^5))^(1/4) + sqrt(x)) - 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b) *(-1/(a^7*b^5))^(1/4)*log(-a^2*b*(-1/(a^7*b^5))^(1/4) + sqrt(x)) + 4*(b*x^ 2 - 3*a)*sqrt(x))/(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)
Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (112) = 224\).
Time = 158.65 (sec) , antiderivative size = 666, normalized size of antiderivative = 4.79 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {7}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {5}{2}}}{5 a^{3}} & \text {for}\: b = 0 \\- \frac {2}{7 b^{3} x^{\frac {7}{2}}} & \text {for}\: a = 0 \\- \frac {12 a^{2} \sqrt {x}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {3 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {3 a^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 a^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {4 a b x^{\frac {5}{2}}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {6 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 a b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {12 a b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} - \frac {3 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {3 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} + \frac {6 b^{2} x^{4} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{64 a^{4} b + 128 a^{3} b^{2} x^{2} + 64 a^{2} b^{3} x^{4}} & \text {otherwise} \end {cases} \]
Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (2*x**(5/2)/(5*a**3), Eq(b, 0)), (-2/(7*b**3*x**(7/2)), Eq(a, 0)), (-12*a**2*sqrt(x)/(64*a**4*b + 128 *a**3*b**2*x**2 + 64*a**2*b**3*x**4) - 3*a**2*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 3*a* *2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x **2 + 64*a**2*b**3*x**4) + 6*a**2*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4) )/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 4*a*b*x**(5/2)/(6 4*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) - 6*a*b*x**2*(-a/b)**(1 /4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2 *b**3*x**4) + 6*a*b*x**2*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(64*a* *4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 12*a*b*x**2*(-a/b)**(1/4) *atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b** 3*x**4) - 3*b**2*x**4*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(64*a**4* b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4) + 3*b**2*x**4*(-a/b)**(1/4)*lo g(sqrt(x) + (-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3* x**4) + 6*b**2*x**4*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(64*a**4*b + 128*a**3*b**2*x**2 + 64*a**2*b**3*x**4), True))
Time = 0.30 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.59 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\frac {b x^{\frac {5}{2}} - 3 \, a \sqrt {x}}{16 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}\right )}}{128 \, a b} \]
1/16*(b*x^(5/2) - 3*a*sqrt(x))/(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b) + 3/128 *(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(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*arct an(-1/2*sqrt(2)*(sqrt(2)*a^(1/4)*b^(1/4) - 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a) *sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*log(sqrt(2)*a^(1/4)*b ^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*log(-sqr t(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a* b)
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.52 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{2}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{2}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{2}} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{2}} + \frac {b x^{\frac {5}{2}} - 3 \, a \sqrt {x}}{16 \, {\left (b x^{2} + a\right )}^{2} a b} \]
3/64*sqrt(2)*(a*b^3)^(1/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/b)^(1/4) + 2*sqr t(x))/(a/b)^(1/4))/(a^2*b^2) + 3/64*sqrt(2)*(a*b^3)^(1/4)*arctan(-1/2*sqrt (2)*(sqrt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/(a^2*b^2) + 3/128*sqrt( 2)*(a*b^3)^(1/4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^2 ) - 3/128*sqrt(2)*(a*b^3)^(1/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqr t(a/b))/(a^2*b^2) + 1/16*(b*x^(5/2) - 3*a*sqrt(x))/((b*x^2 + a)^2*a*b)
Time = 0.13 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.61 \[ \int \frac {x^{3/2}}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {x^{5/2}}{16\,a}-\frac {3\,\sqrt {x}}{16\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}+\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{7/4}\,b^{5/4}} \]