Integrand size = 15, antiderivative size = 189 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}+\frac {3 x^{3/2}}{16 a b \left (a+b x^2\right )}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{7/4}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{32 \sqrt {2} a^{5/4} b^{7/4}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{32 \sqrt {2} a^{5/4} b^{7/4}} \] Output:
-1/4*x^(3/2)/b/(b*x^2+a)^2+3/16*x^(3/2)/a/b/(b*x^2+a)-3/64*arctan(1-2^(1/2 )*b^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(5/4)/b^(7/4)+3/64*arctan(1+2^(1/2)*b ^(1/4)*x^(1/2)/a^(1/4))*2^(1/2)/a^(5/4)/b^(7/4)-3/64*arctanh(2^(1/2)*a^(1/ 4)*b^(1/4)*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/a^(5/4)/b^(7/4)
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.72 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=\frac {-\frac {4 \sqrt [4]{a} b^{3/4} x^{3/2} \left (a-3 b x^2\right )}{\left (a+b x^2\right )^2}-3 \sqrt {2} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{64 a^{5/4} b^{7/4}} \] Input:
Integrate[x^(5/2)/(a + b*x^2)^3,x]
Output:
((-4*a^(1/4)*b^(3/4)*x^(3/2)*(a - 3*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^(5/4 )*b^(7/4))
Time = 0.42 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.43, 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, 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.
| \(\displaystyle \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {3 \int \frac {\sqrt {x}}{\left (b x^2+a\right )^2}dx}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {\sqrt {x}}{b x^2+a}dx}{4 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {3 \left (\frac {\int \frac {x}{b x^2+a}d\sqrt {x}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {3 \left (\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}}{2 a}+\frac {x^{3/2}}{2 a \left (a+b x^2\right )}\right )}{8 b}-\frac {x^{3/2}}{4 b \left (a+b x^2\right )^2}\) |
Input:
Int[x^(5/2)/(a + b*x^2)^3,x]
Output:
-1/4*x^(3/2)/(b*(a + b*x^2)^2) + (3*(x^(3/2)/(2*a*(a + b*x^2)) + ((-(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[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*Sqrt[b]))/(2*a)))/(8*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] :> 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[(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 = 0.29 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.73
| method | result | size |
| derivativedivides | \(\frac {\frac {3 x^{\frac {7}{2}}}{16 a}-\frac {x^{\frac {3}{2}}}{16 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a \,b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(138\) |
| default | \(\frac {\frac {3 x^{\frac {7}{2}}}{16 a}-\frac {x^{\frac {3}{2}}}{16 b}}{\left (b \,x^{2}+a \right )^{2}}+\frac {3 \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{128 a \,b^{2} \left (\frac {a}{b}\right )^{\frac {1}{4}}}\) | \(138\) |
Input:
int(x^(5/2)/(b*x^2+a)^3,x,method=_RETURNVERBOSE)
Output:
2*(3/32/a*x^(7/2)-1/32*x^(3/2)/b)/(b*x^2+a)^2+3/128/a/b^2/(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*arctan(2^ (1/2)/(a/b)^(1/4)*x^(1/2)-1))
Result contains complex when optimal does not.
Time = 0.08 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.53 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (a^{4} b^{5} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 3 \, {\left (i \, a b^{3} x^{4} + 2 i \, a^{2} b^{2} x^{2} + i \, a^{3} b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (i \, a^{4} b^{5} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 3 \, {\left (-i \, a b^{3} x^{4} - 2 i \, a^{2} b^{2} x^{2} - i \, a^{3} b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (-i \, a^{4} b^{5} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) - 3 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {1}{4}} \log \left (-a^{4} b^{5} \left (-\frac {1}{a^{5} b^{7}}\right )^{\frac {3}{4}} + \sqrt {x}\right ) + 4 \, {\left (3 \, b x^{3} - a x\right )} \sqrt {x}}{64 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} \] Input:
integrate(x^(5/2)/(b*x^2+a)^3,x, algorithm="fricas")
Output:
1/64*(3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-1/(a^5*b^7))^(1/4)*log(a^4*b ^5*(-1/(a^5*b^7))^(3/4) + sqrt(x)) - 3*(I*a*b^3*x^4 + 2*I*a^2*b^2*x^2 + I* a^3*b)*(-1/(a^5*b^7))^(1/4)*log(I*a^4*b^5*(-1/(a^5*b^7))^(3/4) + sqrt(x)) - 3*(-I*a*b^3*x^4 - 2*I*a^2*b^2*x^2 - I*a^3*b)*(-1/(a^5*b^7))^(1/4)*log(-I *a^4*b^5*(-1/(a^5*b^7))^(3/4) + sqrt(x)) - 3*(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)*(-1/(a^5*b^7))^(1/4)*log(-a^4*b^5*(-1/(a^5*b^7))^(3/4) + sqrt(x)) + 4*(3*b*x^3 - a*x)*sqrt(x))/(a*b^3*x^4 + 2*a^2*b^2*x^2 + a^3*b)
Timed out. \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**(5/2)/(b*x**2+a)**3,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.17 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, b x^{\frac {7}{2}} - a x^{\frac {3}{2}}}{16 \, {\left (a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{2} + a^{3} b\right )}} + \frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{128 \, a b} \] Input:
integrate(x^(5/2)/(b*x^2+a)^3,x, algorithm="maxima")
Output:
1/16*(3*b*x^(7/2) - a*x^(3/2))/(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(sqrt(a)*sqrt(b))*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(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(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(-sqr t(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqrt(a))/(a^(1/4)*b^(3/4)))/(a* b)
Time = 0.13 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.12 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=\frac {3 \, b x^{\frac {7}{2}} - a x^{\frac {3}{2}}}{16 \, {\left (b x^{2} + a\right )}^{2} a b} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{128 \, a^{2} b^{4}} \] Input:
integrate(x^(5/2)/(b*x^2+a)^3,x, algorithm="giac")
Output:
1/16*(3*b*x^(7/2) - a*x^(3/2))/((b*x^2 + a)^2*a*b) + 3/64*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 ^2*b^4) + 3/64*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^2*b^4) - 3/128*sqrt(2)*(a*b^3)^(3/4)*log( sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^4) + 3/128*sqrt(2)*(a* b^3)^(3/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/(a^2*b^4)
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.45 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx=\frac {\frac {3\,x^{7/2}}{16\,a}-\frac {x^{3/2}}{16\,b}}{a^2+2\,a\,b\,x^2+b^2\,x^4}-\frac {3\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{5/4}\,b^{7/4}}+\frac {3\,\mathrm {atanh}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{32\,{\left (-a\right )}^{5/4}\,b^{7/4}} \] Input:
int(x^(5/2)/(a + b*x^2)^3,x)
Output:
((3*x^(7/2))/(16*a) - x^(3/2)/(16*b))/(a^2 + b^2*x^4 + 2*a*b*x^2) - (3*ata n((b^(1/4)*x^(1/2))/(-a)^(1/4)))/(32*(-a)^(5/4)*b^(7/4)) + (3*atanh((b^(1/ 4)*x^(1/2))/(-a)^(1/4)))/(32*(-a)^(5/4)*b^(7/4))
Time = 0.20 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.53 \[ \int \frac {x^{5/2}}{\left (a+b x^2\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^(5/2)/(b*x^2+a)^3,x)
Output:
( - 6*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 - 12*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*x**2 - 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(x)*sqrt(b))/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*x**4 + 6*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 + 12*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*x**2 + 6*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**2*x**4 + 3*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 + 6*b**(1/4)*a**(3/4)*sqrt(2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sq rt(2) + sqrt(a) + sqrt(b)*x)*a*b*x**2 + 3*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**2*x**4 - 3*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 - 6*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*x**2 - 3*b**(1/4)*a**(3/4)*sqrt(2)*lo g(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b**2*x**4 - 8*s qrt(x)*a**2*b*x + 24*sqrt(x)*a*b**2*x**3)/(128*a**2*b**2*(a**2 + 2*a*b*x** 2 + b**2*x**4))