Integrand size = 15, antiderivative size = 177 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=\frac {5 \sqrt {x}}{2 b^2}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}+\frac {5 \sqrt [4]{a} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} b^{9/4}}-\frac {5 \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{4 \sqrt {2} b^{9/4}} \] Output:
5/2*x^(1/2)/b^2-1/2*x^(5/2)/b/(b*x^2+a)+5/8*a^(1/4)*arctan(1-2^(1/2)*b^(1/ 4)*x^(1/2)/a^(1/4))*2^(1/2)/b^(9/4)-5/8*a^(1/4)*arctan(1+2^(1/2)*b^(1/4)*x ^(1/2)/a^(1/4))*2^(1/2)/b^(9/4)-5/8*a^(1/4)*arctanh(2^(1/2)*a^(1/4)*b^(1/4 )*x^(1/2)/(a^(1/2)+b^(1/2)*x))*2^(1/2)/b^(9/4)
Time = 0.26 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.78 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{b} \sqrt {x} \left (5 a+4 b x^2\right )}{a+b x^2}+5 \sqrt {2} \sqrt [4]{a} \arctan \left (\frac {\sqrt {a}-\sqrt {b} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}\right )-5 \sqrt {2} \sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}}{\sqrt {a}+\sqrt {b} x}\right )}{8 b^{9/4}} \] Input:
Integrate[x^(7/2)/(a + b*x^2)^2,x]
Output:
((4*b^(1/4)*Sqrt[x]*(5*a + 4*b*x^2))/(a + b*x^2) + 5*Sqrt[2]*a^(1/4)*ArcTa n[(Sqrt[a] - Sqrt[b]*x)/(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])] - 5*Sqrt[2]*a^( 1/4)*ArcTanh[(Sqrt[2]*a^(1/4)*b^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[b]*x)])/(8* b^(9/4))
Time = 0.41 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.46, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {252, 262, 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.
| \(\displaystyle \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 252 |
| \(\displaystyle \frac {5 \int \frac {x^{3/2}}{b x^2+a}dx}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {a \int \frac {1}{\sqrt {x} \left (b x^2+a\right )}dx}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \int \frac {1}{b x^2+a}d\sqrt {x}}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x+\sqrt {a}}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x}{b x^2+a}d\sqrt {x}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{b} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {2 \sqrt {x}}{b}-\frac {2 a \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} \sqrt {x}+\sqrt {a}+\sqrt {b} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{b}\right )}{4 b}-\frac {x^{5/2}}{2 b \left (a+b x^2\right )}\) |
Input:
Int[x^(7/2)/(a + b*x^2)^2,x]
Output:
-1/2*x^(5/2)/(b*(a + b*x^2)) + (5*((2*Sqrt[x])/b - (2*a*((-(ArcTan[1 - (Sq rt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (S qrt[2]*b^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[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])))/b))/(4*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*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && 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 = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{4 \left (b \,x^{2}+a \right )}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{2}}\) | \(136\) |
| default | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {2 a \left (-\frac {\sqrt {x}}{4 \left (b \,x^{2}+a \right )}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}\right )}{b^{2}}\) | \(136\) |
| risch | \(\frac {2 \sqrt {x}}{b^{2}}-\frac {a \left (-\frac {\sqrt {x}}{2 \left (b \,x^{2}+a \right )}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{16 a}\right )}{b^{2}}\) | \(136\) |
Input:
int(x^(7/2)/(b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
2*x^(1/2)/b^2-2*a/b^2*(-1/4*x^(1/2)/(b*x^2+a)+5/32*(a/b)^(1/4)/a*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 = 209, normalized size of antiderivative = 1.18 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=-\frac {5 \, {\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} \log \left (5 \, b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {x}\right ) + 5 \, {\left (i \, b^{3} x^{2} + i \, a b^{2}\right )} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} \log \left (5 i \, b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {x}\right ) + 5 \, {\left (-i \, b^{3} x^{2} - i \, a b^{2}\right )} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} \log \left (-5 i \, b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {x}\right ) - 5 \, {\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} \log \left (-5 \, b^{2} \left (-\frac {a}{b^{9}}\right )^{\frac {1}{4}} + 5 \, \sqrt {x}\right ) - 4 \, {\left (4 \, b x^{2} + 5 \, a\right )} \sqrt {x}}{8 \, {\left (b^{3} x^{2} + a b^{2}\right )}} \] Input:
integrate(x^(7/2)/(b*x^2+a)^2,x, algorithm="fricas")
Output:
-1/8*(5*(b^3*x^2 + a*b^2)*(-a/b^9)^(1/4)*log(5*b^2*(-a/b^9)^(1/4) + 5*sqrt (x)) + 5*(I*b^3*x^2 + I*a*b^2)*(-a/b^9)^(1/4)*log(5*I*b^2*(-a/b^9)^(1/4) + 5*sqrt(x)) + 5*(-I*b^3*x^2 - I*a*b^2)*(-a/b^9)^(1/4)*log(-5*I*b^2*(-a/b^9 )^(1/4) + 5*sqrt(x)) - 5*(b^3*x^2 + a*b^2)*(-a/b^9)^(1/4)*log(-5*b^2*(-a/b ^9)^(1/4) + 5*sqrt(x)) - 4*(4*b*x^2 + 5*a)*sqrt(x))/(b^3*x^2 + a*b^2)
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (163) = 326\).
Time = 81.88 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.91 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 x^{\frac {9}{2}}}{9 a^{2}} & \text {for}\: b = 0 \\\frac {2 \sqrt {x}}{b^{2}} & \text {for}\: a = 0 \\\frac {20 a \sqrt {x}}{8 a b^{2} + 8 b^{3} x^{2}} + \frac {5 a \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {5 a \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {10 a \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} + \frac {16 b x^{\frac {5}{2}}}{8 a b^{2} + 8 b^{3} x^{2}} + \frac {5 b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {5 b x^{2} \sqrt [4]{- \frac {a}{b}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{b}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} - \frac {10 b x^{2} \sqrt [4]{- \frac {a}{b}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{b}}} \right )}}{8 a b^{2} + 8 b^{3} x^{2}} & \text {otherwise} \end {cases} \] Input:
integrate(x**(7/2)/(b*x**2+a)**2,x)
Output:
Piecewise((zoo*sqrt(x), Eq(a, 0) & Eq(b, 0)), (2*x**(9/2)/(9*a**2), Eq(b, 0)), (2*sqrt(x)/b**2, Eq(a, 0)), (20*a*sqrt(x)/(8*a*b**2 + 8*b**3*x**2) + 5*a*(-a/b)**(1/4)*log(sqrt(x) - (-a/b)**(1/4))/(8*a*b**2 + 8*b**3*x**2) - 5*a*(-a/b)**(1/4)*log(sqrt(x) + (-a/b)**(1/4))/(8*a*b**2 + 8*b**3*x**2) - 10*a*(-a/b)**(1/4)*atan(sqrt(x)/(-a/b)**(1/4))/(8*a*b**2 + 8*b**3*x**2) + 16*b*x**(5/2)/(8*a*b**2 + 8*b**3*x**2) + 5*b*x**2*(-a/b)**(1/4)*log(sqrt(x ) - (-a/b)**(1/4))/(8*a*b**2 + 8*b**3*x**2) - 5*b*x**2*(-a/b)**(1/4)*log(s qrt(x) + (-a/b)**(1/4))/(8*a*b**2 + 8*b**3*x**2) - 10*b*x**2*(-a/b)**(1/4) *atan(sqrt(x)/(-a/b)**(1/4))/(8*a*b**2 + 8*b**3*x**2), True))
Time = 0.11 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.16 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=\frac {a \sqrt {x}}{2 \, {\left (b^{3} x^{2} + a b^{2}\right )}} - \frac {5 \, {\left (\frac {2 \, \sqrt {2} \sqrt {a} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} + 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {a} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} - 2 \, \sqrt {b} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}} - \frac {\sqrt {2} a^{\frac {1}{4}} \log \left (-\sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} \sqrt {x} + \sqrt {b} x + \sqrt {a}\right )}{b^{\frac {1}{4}}}\right )}}{16 \, b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \] Input:
integrate(x^(7/2)/(b*x^2+a)^2,x, algorithm="maxima")
Output:
1/2*a*sqrt(x)/(b^3*x^2 + a*b^2) - 5/16*(2*sqrt(2)*sqrt(a)*arctan(1/2*sqrt( 2)*(sqrt(2)*a^(1/4)*b^(1/4) + 2*sqrt(b)*sqrt(x))/sqrt(sqrt(a)*sqrt(b)))/sq rt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*sqrt(a)*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(2)*a^(1/4)*log(sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqrt(b)*x + sqr t(a))/b^(1/4) - sqrt(2)*a^(1/4)*log(-sqrt(2)*a^(1/4)*b^(1/4)*sqrt(x) + sqr t(b)*x + sqrt(a))/b^(1/4))/b^2 + 2*sqrt(x)/b^2
Time = 0.13 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.11 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=-\frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{3}} - \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \, b^{3}} - \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{3}} + \frac {5 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{b}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{b}}\right )}{16 \, b^{3}} + \frac {a \sqrt {x}}{2 \, {\left (b x^{2} + a\right )} b^{2}} + \frac {2 \, \sqrt {x}}{b^{2}} \] Input:
integrate(x^(7/2)/(b*x^2+a)^2,x, algorithm="giac")
Output:
-5/8*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))/b^3 - 5/8*sqrt(2)*(a*b^3)^(1/4)*arctan(-1/2*sqrt(2)*(sq rt(2)*(a/b)^(1/4) - 2*sqrt(x))/(a/b)^(1/4))/b^3 - 5/16*sqrt(2)*(a*b^3)^(1/ 4)*log(sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 + 5/16*sqrt(2)*(a* b^3)^(1/4)*log(-sqrt(2)*sqrt(x)*(a/b)^(1/4) + x + sqrt(a/b))/b^3 + 1/2*a*s qrt(x)/((b*x^2 + a)*b^2) + 2*sqrt(x)/b^2
Time = 0.12 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.45 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=\frac {2\,\sqrt {x}}{b^2}-\frac {5\,{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}}{{\left (-a\right )}^{1/4}}\right )}{4\,b^{9/4}}+\frac {a\,\sqrt {x}}{2\,\left (b^3\,x^2+a\,b^2\right )}+\frac {{\left (-a\right )}^{1/4}\,\mathrm {atan}\left (\frac {b^{1/4}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/4}}\right )\,5{}\mathrm {i}}{4\,b^{9/4}} \] Input:
int(x^(7/2)/(a + b*x^2)^2,x)
Output:
(2*x^(1/2))/b^2 - (5*(-a)^(1/4)*atan((b^(1/4)*x^(1/2))/(-a)^(1/4)))/(4*b^( 9/4)) + ((-a)^(1/4)*atan((b^(1/4)*x^(1/2)*1i)/(-a)^(1/4))*5i)/(4*b^(9/4)) + (a*x^(1/2))/(2*(a*b^2 + b^3*x^2))
Time = 0.25 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.77 \[ \int \frac {x^{7/2}}{\left (a+b x^2\right )^2} \, dx=\frac {10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+10 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}-10 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-10 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {x}\, \sqrt {b}}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) x^{2}+5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )+5 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}-5 b^{\frac {3}{4}} a^{\frac {5}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right )-5 b^{\frac {7}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (\sqrt {x}\, b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+\sqrt {a}+\sqrt {b}\, x \right ) x^{2}+40 \sqrt {x}\, a b +32 \sqrt {x}\, b^{2} x^{2}}{16 b^{3} \left (b \,x^{2}+a \right )} \] Input:
int(x^(7/2)/(b*x^2+a)^2,x)
Output:
(10*b**(3/4)*a**(1/4)*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*b**(3/4)*a**(1/4)*sqrt(2)*ata n((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*b**(3/4)*a**(1/4)*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*b**(3/4)*a**(1/4 )*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*b**(3/4)*a**(1/4)*sqrt(2)*log( - sqrt(x)*b**( 1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*a + 5*b**(3/4)*a**(1/4)*sqrt( 2)*log( - sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 - 5*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**(1/4)*sqrt(2) + sqrt (a) + sqrt(b)*x)*a - 5*b**(3/4)*a**(1/4)*sqrt(2)*log(sqrt(x)*b**(1/4)*a**( 1/4)*sqrt(2) + sqrt(a) + sqrt(b)*x)*b*x**2 + 40*sqrt(x)*a*b + 32*sqrt(x)*b **2*x**2)/(16*b**3*(a + b*x**2))