Integrand size = 20, antiderivative size = 265 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 A}{a \sqrt {x}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{c}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{c}}-\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{5/4} \sqrt [4]{c}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{2 \sqrt {2} a^{5/4} \sqrt [4]{c}} \]
-1/2*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2)-A*c^(1/2))/a^(5/ 4)/c^(1/4)*2^(1/2)+1/2*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(B*a^(1/2 )-A*c^(1/2))/a^(5/4)/c^(1/4)*2^(1/2)-1/4*ln(a^(1/2)+x*c^(1/2)-a^(1/4)*c^(1 /4)*2^(1/2)*x^(1/2))*(B*a^(1/2)+A*c^(1/2))/a^(5/4)/c^(1/4)*2^(1/2)+1/4*ln( a^(1/2)+x*c^(1/2)+a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(B*a^(1/2)+A*c^(1/2))/a ^(5/4)/c^(1/4)*2^(1/2)-2*A/a/x^(1/2)
Time = 0.25 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.56 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 A}{a \sqrt {x}}-\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{c}}+\frac {\left (\sqrt {a} B+A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt {2} a^{5/4} \sqrt [4]{c}} \]
(-2*A)/(a*Sqrt[x]) - ((Sqrt[a]*B - A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt[c]*x) /(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/(Sqrt[2]*a^(5/4)*c^(1/4)) + ((Sqrt[a] *B + A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[ c]*x)])/(Sqrt[2]*a^(5/4)*c^(1/4))
Time = 0.47 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.97, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {553, 27, 554, 1482, 27, 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.
| \(\displaystyle \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 553 |
| \(\displaystyle -\frac {2 \int -\frac {a B-A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a B-A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {2 \int \frac {a B-A c x}{c x^2+a}d\sqrt {x}}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {\sqrt {a} B}{\sqrt {c}}+A\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (A-\frac {\sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{a}-\frac {2 A}{a \sqrt {x}}\) |
(-2*A)/(a*Sqrt[x]) + (2*(-1/2*((A - (Sqrt[a]*B)/Sqrt[c])*Sqrt[c]*(-(ArcTan [1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTa n[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))) + ((A + (Sqrt[a]*B)/Sqrt[c])*Sqrt[c]*(-1/2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4 )*Sqrt[x] + Sqrt[c]*x]/(Sqrt[2]*a^(1/4)*c^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a ^(1/4)*c^(1/4)*Sqrt[x] + Sqrt[c]*x]/(2*Sqrt[2]*a^(1/4)*c^(1/4))))/2))/a
3.5.16.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp [1/(a*e*(m + 1)) Int[(e*x)^(m + 1)*(a + b*x^2)^p*(a*d*(m + 1) - b*c*(m + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && LtQ[m, -1]
Int[((c_) + (d_.)*(x_))/(Sqrt[(e_.)*(x_)]*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*c + d*x^2)/(a*e^2 + b*x^4), x], x, Sqrt[e*x]], x] /; FreeQ[{a, b, c, d, e}, x]
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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(d*q - a*e)/(2*a*c) Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ[{a , c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(- a)*c]
Time = 0.07 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4}-\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}-\frac {2 A}{a \sqrt {x}}\) | \(223\) |
| default | \(\frac {\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4}-\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}-\frac {2 A}{a \sqrt {x}}\) | \(223\) |
| risch | \(-\frac {2 A}{a \sqrt {x}}-\frac {-\frac {B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4}+\frac {A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}\) | \(223\) |
2/a*(1/8*B*(a/c)^(1/4)*2^(1/2)*(ln((x+(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1 /2))/(x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^( 1/4)*x^(1/2)+1)+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1))-1/8*A/(a/c)^(1/4) *2^(1/2)*(ln((x-(a/c)^(1/4)*x^(1/2)*2^(1/2)+(a/c)^(1/2))/(x+(a/c)^(1/4)*x^ (1/2)*2^(1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x^(1/2)+1)+2*arct an(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1)))-2*A/a/x^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 767 vs. \(2 (180) = 360\).
Time = 0.29 (sec) , antiderivative size = 767, normalized size of antiderivative = 2.89 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=-\frac {a x \sqrt {\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + B^{3} a^{3} - A^{2} B a^{2} c\right )} \sqrt {\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}}\right ) - a x \sqrt {\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + B^{3} a^{3} - A^{2} B a^{2} c\right )} \sqrt {\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} + 2 \, A B}{a^{2}}}\right ) - a x \sqrt {-\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (A a^{4} c \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - B^{3} a^{3} + A^{2} B a^{2} c\right )} \sqrt {-\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}}\right ) + a x \sqrt {-\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}} \log \left (-{\left (B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (A a^{4} c \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - B^{3} a^{3} + A^{2} B a^{2} c\right )} \sqrt {-\frac {a^{2} \sqrt {-\frac {B^{4} a^{2} - 2 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{5} c}} - 2 \, A B}{a^{2}}}\right ) + 4 \, A \sqrt {x}}{2 \, a x} \]
-1/2*(a*x*sqrt((a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) + 2 *A*B)/a^2)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) + (A*a^4*c*sqrt(-(B^4*a^2 - 2* A^2*B^2*a*c + A^4*c^2)/(a^5*c)) + B^3*a^3 - A^2*B*a^2*c)*sqrt((a^2*sqrt(-( B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) + 2*A*B)/a^2)) - a*x*sqrt((a^2 *sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) + 2*A*B)/a^2)*log(-(B^ 4*a^2 - A^4*c^2)*sqrt(x) - (A*a^4*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c ^2)/(a^5*c)) + B^3*a^3 - A^2*B*a^2*c)*sqrt((a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2 *a*c + A^4*c^2)/(a^5*c)) + 2*A*B)/a^2)) - a*x*sqrt(-(a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) - 2*A*B)/a^2)*log(-(B^4*a^2 - A^4*c^2)*s qrt(x) + (A*a^4*c*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) - B^3 *a^3 + A^2*B*a^2*c)*sqrt(-(a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/( a^5*c)) - 2*A*B)/a^2)) + a*x*sqrt(-(a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A ^4*c^2)/(a^5*c)) - 2*A*B)/a^2)*log(-(B^4*a^2 - A^4*c^2)*sqrt(x) - (A*a^4*c *sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) - B^3*a^3 + A^2*B*a^2* c)*sqrt(-(a^2*sqrt(-(B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^5*c)) - 2*A*B)/ a^2)) + 4*A*sqrt(x))/(a*x)
Time = 4.81 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.92 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {- \frac {2 A}{5 x^{\frac {5}{2}}} - \frac {2 B}{3 x^{\frac {3}{2}}}}{c} & \text {for}\: a = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{a} & \text {for}\: c = 0 \\- \frac {A \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 a \sqrt [4]{- \frac {a}{c}}} + \frac {A \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 a \sqrt [4]{- \frac {a}{c}}} - \frac {A \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{a \sqrt [4]{- \frac {a}{c}}} - \frac {2 A}{a \sqrt {x}} - \frac {B \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{2 a} + \frac {B \sqrt [4]{- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{2 a} + \frac {B \sqrt [4]{- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{a} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2))), Eq(a, 0) & Eq(c, 0) ), ((-2*A/(5*x**(5/2)) - 2*B/(3*x**(3/2)))/c, Eq(a, 0)), ((-2*A/sqrt(x) + 2*B*sqrt(x))/a, Eq(c, 0)), (-A*log(sqrt(x) - (-a/c)**(1/4))/(2*a*(-a/c)**( 1/4)) + A*log(sqrt(x) + (-a/c)**(1/4))/(2*a*(-a/c)**(1/4)) - A*atan(sqrt(x )/(-a/c)**(1/4))/(a*(-a/c)**(1/4)) - 2*A/(a*sqrt(x)) - B*(-a/c)**(1/4)*log (sqrt(x) - (-a/c)**(1/4))/(2*a) + B*(-a/c)**(1/4)*log(sqrt(x) + (-a/c)**(1 /4))/(2*a) + B*(-a/c)**(1/4)*atan(sqrt(x)/(-a/c)**(1/4))/a, True))
Time = 0.36 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.93 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=\frac {\frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (B a \sqrt {c} - A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (B a \sqrt {c} + A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{4 \, a} - \frac {2 \, A}{a \sqrt {x}} \]
1/4*(2*sqrt(2)*(B*a*sqrt(c) - A*sqrt(a)*c)*arctan(1/2*sqrt(2)*(sqrt(2)*a^( 1/4)*c^(1/4) + 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqr t(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(B*a*sqrt(c) - A*sqrt(a)*c)*arctan(-1/2 *sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) - 2*sqrt(c)*sqrt(x))/sqrt(sqrt(a)*sqrt(c )))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + sqrt(2)*(B*a*sqrt(c) + A*sqr t(a)*c)*log(sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4 )*c^(3/4)) - sqrt(2)*(B*a*sqrt(c) + A*sqrt(a)*c)*log(-sqrt(2)*a^(1/4)*c^(1 /4)*sqrt(x) + sqrt(c)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/a - 2*A/(a*sqrt(x))
Time = 0.29 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=-\frac {2 \, A}{a \sqrt {x}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} c^{2}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c - \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, a^{2} c^{2}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, a^{2} c^{2}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} B a c + \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{4 \, a^{2} c^{2}} \]
-2*A/(a*sqrt(x)) + 1/2*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*arc tan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^2) + 1/2*sqrt(2)*((a*c^3)^(1/4)*B*a*c - (a*c^3)^(3/4)*A)*arctan(-1/2*sqrt(2)*( sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4))/(a^2*c^2) + 1/4*sqrt(2)*((a* c^3)^(1/4)*B*a*c + (a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^2) - 1/4*sqrt(2)*((a*c^3)^(1/4)*B*a*c + (a*c^3)^(3/4)*A) *log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a^2*c^2)
Time = 0.24 (sec) , antiderivative size = 602, normalized size of antiderivative = 2.27 \[ \int \frac {A+B x}{x^{3/2} \left (a+c x^2\right )} \, dx=2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^4\,c^4\,\sqrt {x}\,\sqrt {\frac {A\,B}{2\,a^2}-\frac {A^2\,\sqrt {-a^5\,c}}{4\,a^5}+\frac {B^2\,\sqrt {-a^5\,c}}{4\,a^4\,c}}}{16\,A^3\,a^3\,c^4-16\,B^3\,a^2\,c^2\,\sqrt {-a^5\,c}-16\,A\,B^2\,a^4\,c^3+16\,A^2\,B\,a\,c^3\,\sqrt {-a^5\,c}}-\frac {32\,B^2\,a^5\,c^3\,\sqrt {x}\,\sqrt {\frac {A\,B}{2\,a^2}-\frac {A^2\,\sqrt {-a^5\,c}}{4\,a^5}+\frac {B^2\,\sqrt {-a^5\,c}}{4\,a^4\,c}}}{16\,A^3\,a^3\,c^4-16\,B^3\,a^2\,c^2\,\sqrt {-a^5\,c}-16\,A\,B^2\,a^4\,c^3+16\,A^2\,B\,a\,c^3\,\sqrt {-a^5\,c}}\right )\,\sqrt {\frac {B^2\,a\,\sqrt {-a^5\,c}-A^2\,c\,\sqrt {-a^5\,c}+2\,A\,B\,a^3\,c}{4\,a^5\,c}}+2\,\mathrm {atanh}\left (\frac {32\,A^2\,a^4\,c^4\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c}}{4\,a^5}+\frac {A\,B}{2\,a^2}-\frac {B^2\,\sqrt {-a^5\,c}}{4\,a^4\,c}}}{16\,A^3\,a^3\,c^4+16\,B^3\,a^2\,c^2\,\sqrt {-a^5\,c}-16\,A\,B^2\,a^4\,c^3-16\,A^2\,B\,a\,c^3\,\sqrt {-a^5\,c}}-\frac {32\,B^2\,a^5\,c^3\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^5\,c}}{4\,a^5}+\frac {A\,B}{2\,a^2}-\frac {B^2\,\sqrt {-a^5\,c}}{4\,a^4\,c}}}{16\,A^3\,a^3\,c^4+16\,B^3\,a^2\,c^2\,\sqrt {-a^5\,c}-16\,A\,B^2\,a^4\,c^3-16\,A^2\,B\,a\,c^3\,\sqrt {-a^5\,c}}\right )\,\sqrt {\frac {A^2\,c\,\sqrt {-a^5\,c}-B^2\,a\,\sqrt {-a^5\,c}+2\,A\,B\,a^3\,c}{4\,a^5\,c}}-\frac {2\,A}{a\,\sqrt {x}} \]
2*atanh((32*A^2*a^4*c^4*x^(1/2)*((A*B)/(2*a^2) - (A^2*(-a^5*c)^(1/2))/(4*a ^5) + (B^2*(-a^5*c)^(1/2))/(4*a^4*c))^(1/2))/(16*A^3*a^3*c^4 - 16*B^3*a^2* c^2*(-a^5*c)^(1/2) - 16*A*B^2*a^4*c^3 + 16*A^2*B*a*c^3*(-a^5*c)^(1/2)) - ( 32*B^2*a^5*c^3*x^(1/2)*((A*B)/(2*a^2) - (A^2*(-a^5*c)^(1/2))/(4*a^5) + (B^ 2*(-a^5*c)^(1/2))/(4*a^4*c))^(1/2))/(16*A^3*a^3*c^4 - 16*B^3*a^2*c^2*(-a^5 *c)^(1/2) - 16*A*B^2*a^4*c^3 + 16*A^2*B*a*c^3*(-a^5*c)^(1/2)))*((B^2*a*(-a ^5*c)^(1/2) - A^2*c*(-a^5*c)^(1/2) + 2*A*B*a^3*c)/(4*a^5*c))^(1/2) + 2*ata nh((32*A^2*a^4*c^4*x^(1/2)*((A^2*(-a^5*c)^(1/2))/(4*a^5) + (A*B)/(2*a^2) - (B^2*(-a^5*c)^(1/2))/(4*a^4*c))^(1/2))/(16*A^3*a^3*c^4 + 16*B^3*a^2*c^2*( -a^5*c)^(1/2) - 16*A*B^2*a^4*c^3 - 16*A^2*B*a*c^3*(-a^5*c)^(1/2)) - (32*B^ 2*a^5*c^3*x^(1/2)*((A^2*(-a^5*c)^(1/2))/(4*a^5) + (A*B)/(2*a^2) - (B^2*(-a ^5*c)^(1/2))/(4*a^4*c))^(1/2))/(16*A^3*a^3*c^4 + 16*B^3*a^2*c^2*(-a^5*c)^( 1/2) - 16*A*B^2*a^4*c^3 - 16*A^2*B*a*c^3*(-a^5*c)^(1/2)))*((A^2*c*(-a^5*c) ^(1/2) - B^2*a*(-a^5*c)^(1/2) + 2*A*B*a^3*c)/(4*a^5*c))^(1/2) - (2*A)/(a*x ^(1/2))