Integrand size = 20, antiderivative size = 289 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}-\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B+A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} a^{3/4} c^{7/4}}+\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{3/4} c^{7/4}}-\frac {\left (3 \sqrt {a} B-A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} a^{3/4} c^{7/4}} \]
1/16*ln(a^(1/2)+x*c^(1/2)-a^(1/4)*c^(1/4)*2^(1/2)*x^(1/2))*(3*B*a^(1/2)-A* c^(1/2))/a^(3/4)/c^(7/4)*2^(1/2)-1/16*ln(a^(1/2)+x*c^(1/2)+a^(1/4)*c^(1/4) *2^(1/2)*x^(1/2))*(3*B*a^(1/2)-A*c^(1/2))/a^(3/4)/c^(7/4)*2^(1/2)-1/8*arct an(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(3*B*a^(1/2)+A*c^(1/2))/a^(3/4)/c^(7 /4)*2^(1/2)+1/8*arctan(1+c^(1/4)*2^(1/2)*x^(1/2)/a^(1/4))*(3*B*a^(1/2)+A*c ^(1/2))/a^(3/4)/c^(7/4)*2^(1/2)-1/2*(B*x+A)*x^(1/2)/c/(c*x^2+a)
Time = 0.51 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.57 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {-\frac {4 c^{3/4} \sqrt {x} (A+B x)}{a+c x^2}-\frac {\sqrt {2} \left (3 \sqrt {a} B+A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{a^{3/4}}-\frac {\sqrt {2} \left (3 \sqrt {a} B-A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{a^{3/4}}}{8 c^{7/4}} \]
((-4*c^(3/4)*Sqrt[x]*(A + B*x))/(a + c*x^2) - (Sqrt[2]*(3*Sqrt[a]*B + A*Sq rt[c])*ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^(1/4)*Sqrt[x])])/a^ (3/4) - (Sqrt[2]*(3*Sqrt[a]*B - A*Sqrt[c])*ArcTanh[(Sqrt[2]*a^(1/4)*c^(1/4 )*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/a^(3/4))/(8*c^(7/4))
Time = 0.50 (sec) , antiderivative size = 279, 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 = {549, 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.
| \(\displaystyle \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {\int \frac {A+3 B x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {A+3 B x}{\sqrt {x} \left (c x^2+a\right )}dx}{4 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\int \frac {A+3 B x}{c x^2+a}d\sqrt {x}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}}{2 c}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}}{2 c}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {\left (\frac {A \sqrt {c}}{\sqrt {a}}+3 B\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}-\frac {\left (3 B-\frac {A \sqrt {c}}{\sqrt {a}}\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )}{2 \sqrt {c}}}{2 c}-\frac {\sqrt {x} (A+B x)}{2 c \left (a+c x^2\right )}\) |
-1/2*(Sqrt[x]*(A + B*x))/(c*(a + c*x^2)) + (((3*B + (A*Sqrt[c])/Sqrt[a])*( -(ArcTan[1 - (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)) ))/(2*Sqrt[c]) - ((3*B - (A*Sqrt[c])/Sqrt[a])*(-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*Sqrt[c]))/(2*c)
3.5.21.3.1 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[e*(e*x)^(m - 1)*(c + d*x)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[e^2/(2*b*(p + 1)) Int[(e*x)^(m - 2)*(c*(m - 1) + d*m*x)*(a + b *x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[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 = 250, normalized size of antiderivative = 0.87
| method | result | size |
| derivativedivides | \(\frac {-\frac {B \,x^{\frac {3}{2}}}{2 c}-\frac {A \sqrt {x}}{2 c}}{c \,x^{2}+a}+\frac {\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {3 B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{2 c}\) | \(250\) |
| default | \(\frac {-\frac {B \,x^{\frac {3}{2}}}{2 c}-\frac {A \sqrt {x}}{2 c}}{c \,x^{2}+a}+\frac {\frac {A \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {3 B \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{2 c}\) | \(250\) |
2*(-1/4*B*x^(3/2)/c-1/4*A*x^(1/2)/c)/(c*x^2+a)+1/2/c*(1/8*A*(a/c)^(1/4)/a* 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*arcta n(2^(1/2)/(a/c)^(1/4)*x^(1/2)-1))+3/8*B/c/(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)))
Leaf count of result is larger than twice the leaf count of optimal. 888 vs. \(2 (197) = 394\).
Time = 0.31 (sec) , antiderivative size = 888, normalized size of antiderivative = 3.07 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {{\left (c^{2} x^{2} + a c\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 9 \, A B^{2} a^{2} c^{2} + A^{3} a c^{3}\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}}\right ) - {\left (c^{2} x^{2} + a c\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 9 \, A B^{2} a^{2} c^{2} + A^{3} a c^{3}\right )} \sqrt {-\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 6 \, A B}{a c^{3}}}\right ) - {\left (c^{2} x^{2} + a c\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 9 \, A B^{2} a^{2} c^{2} - A^{3} a c^{3}\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}}\right ) + {\left (c^{2} x^{2} + a c\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}} \log \left (-{\left (81 \, B^{4} a^{2} - A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, B a^{3} c^{5} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} + 9 \, A B^{2} a^{2} c^{2} - A^{3} a c^{3}\right )} \sqrt {\frac {a c^{3} \sqrt {-\frac {81 \, B^{4} a^{2} - 18 \, A^{2} B^{2} a c + A^{4} c^{2}}{a^{3} c^{7}}} - 6 \, A B}{a c^{3}}}\right ) - 4 \, {\left (B x + A\right )} \sqrt {x}}{8 \, {\left (c^{2} x^{2} + a c\right )}} \]
1/8*((c^2*x^2 + a*c)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4 *c^2)/(a^3*c^7)) + 6*A*B)/(a*c^3))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) + ( 3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 9*A *B^2*a^2*c^2 + A^3*a*c^3)*sqrt(-(a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*c^3))) - (c^2*x^2 + a*c)*sqrt(-(a*c^3*sq rt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*c^3))*l og(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) - (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A ^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 9*A*B^2*a^2*c^2 + A^3*a*c^3)*sqrt(-(a*c ^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 6*A*B)/(a*c^ 3))) - (c^2*x^2 + a*c)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^ 4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))*log(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) + (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 9* A*B^2*a^2*c^2 - A^3*a*c^3)*sqrt((a*c^3*sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))) + (c^2*x^2 + a*c)*sqrt((a*c^3*sqr t(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3))*lo g(-(81*B^4*a^2 - A^4*c^2)*sqrt(x) - (3*B*a^3*c^5*sqrt(-(81*B^4*a^2 - 18*A^ 2*B^2*a*c + A^4*c^2)/(a^3*c^7)) + 9*A*B^2*a^2*c^2 - A^3*a*c^3)*sqrt((a*c^3 *sqrt(-(81*B^4*a^2 - 18*A^2*B^2*a*c + A^4*c^2)/(a^3*c^7)) - 6*A*B)/(a*c^3) )) - 4*(B*x + A)*sqrt(x))/(c^2*x^2 + a*c)
Leaf count of result is larger than twice the leaf count of optimal. 937 vs. \(2 (265) = 530\).
Time = 60.42 (sec) , antiderivative size = 937, normalized size of antiderivative = 3.24 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {5}{2}}}{5} + \frac {2 B x^{\frac {7}{2}}}{7}}{a^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{3 x^{\frac {3}{2}}} - \frac {2 B}{\sqrt {x}}}{c^{2}} & \text {for}\: a = 0 \\- \frac {4 A a c \sqrt {x} \sqrt [4]{- \frac {a}{c}}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A a c \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {A a c \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A a c \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {A c^{2} x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {A c^{2} x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {2 A c^{2} x^{2} \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {3 B a^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {3 B a^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {6 B a^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {4 B a c x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{c}}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {3 B a c x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {3 B a c x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {6 B a c x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a^{2} c^{2} \sqrt [4]{- \frac {a}{c}} + 8 a c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/(3*x**(3/2)) - 2*B/sqrt(x)), Eq(a, 0) & Eq(c, 0)), (( 2*A*x**(5/2)/5 + 2*B*x**(7/2)/7)/a**2, Eq(c, 0)), ((-2*A/(3*x**(3/2)) - 2* B/sqrt(x))/c**2, Eq(a, 0)), (-4*A*a*c*sqrt(x)*(-a/c)**(1/4)/(8*a**2*c**2*( -a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) - A*a*c*sqrt(-a/c)*log(sqrt(x) - (-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4) ) + A*a*c*sqrt(-a/c)*log(sqrt(x) + (-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/ 4) + 8*a*c**3*x**2*(-a/c)**(1/4)) + 2*A*a*c*sqrt(-a/c)*atan(sqrt(x)/(-a/c) **(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) - A*c** 2*x**2*sqrt(-a/c)*log(sqrt(x) - (-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) + A*c**2*x**2*sqrt(-a/c)*log(sqrt(x) + (-a/ c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) + 2*A *c**2*x**2*sqrt(-a/c)*atan(sqrt(x)/(-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/ 4) + 8*a*c**3*x**2*(-a/c)**(1/4)) + 3*B*a**2*log(sqrt(x) - (-a/c)**(1/4))/ (8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) - 3*B*a**2*log(s qrt(x) + (-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)* *(1/4)) + 6*B*a**2*atan(sqrt(x)/(-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) - 4*B*a*c*x**(3/2)*(-a/c)**(1/4)/(8*a**2*c* *2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4)) + 3*B*a*c*x**2*log(sqrt(x) - (-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4) + 8*a*c**3*x**2*(-a/c)**(1/4) ) - 3*B*a*c*x**2*log(sqrt(x) + (-a/c)**(1/4))/(8*a**2*c**2*(-a/c)**(1/4...
Time = 0.46 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.90 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=-\frac {B x^{\frac {3}{2}} + A \sqrt {x}}{2 \, {\left (c^{2} x^{2} + a c\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (3 \, B \sqrt {a} + A \sqrt {c}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} - \frac {\sqrt {2} {\left (3 \, B \sqrt {a} - A \sqrt {c}\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (3 \, B \sqrt {a} - A \sqrt {c}\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{16 \, c} \]
-1/2*(B*x^(3/2) + A*sqrt(x))/(c^2*x^2 + a*c) + 1/16*(2*sqrt(2)*(3*B*sqrt(a ) + A*sqrt(c))*arctan(1/2*sqrt(2)*(sqrt(2)*a^(1/4)*c^(1/4) + 2*sqrt(c)*sqr t(x))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*s qrt(2)*(3*B*sqrt(a) + A*sqrt(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)*(3*B*sqrt(a) - A*sqrt(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)*(3*B*sqrt(a ) - A*sqrt(c))*log(-sqrt(2)*a^(1/4)*c^(1/4)*sqrt(x) + sqrt(c)*x + sqrt(a)) /(a^(3/4)*c^(3/4)))/c
Time = 0.29 (sec) , antiderivative size = 271, normalized size of antiderivative = 0.94 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=-\frac {B x^{\frac {3}{2}} + A \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a c^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} A c^{2} - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} B\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a c^{4}} \]
-1/2*(B*x^(3/2) + A*sqrt(x))/((c*x^2 + a)*c) + 1/8*sqrt(2)*((a*c^3)^(1/4)* A*c^2 + 3*(a*c^3)^(3/4)*B)*arctan(1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) + 2*sqr t(x))/(a/c)^(1/4))/(a*c^4) + 1/8*sqrt(2)*((a*c^3)^(1/4)*A*c^2 + 3*(a*c^3)^ (3/4)*B)*arctan(-1/2*sqrt(2)*(sqrt(2)*(a/c)^(1/4) - 2*sqrt(x))/(a/c)^(1/4) )/(a*c^4) + 1/16*sqrt(2)*((a*c^3)^(1/4)*A*c^2 - 3*(a*c^3)^(3/4)*B)*log(sqr t(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^4) - 1/16*sqrt(2)*((a*c^3)^ (1/4)*A*c^2 - 3*(a*c^3)^(3/4)*B)*log(-sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sq rt(a/c))/(a*c^4)
Time = 10.20 (sec) , antiderivative size = 656, normalized size of antiderivative = 2.27 \[ \int \frac {x^{3/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=2\,\mathrm {atanh}\left (\frac {18\,B^2\,a\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}-\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}+\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}-\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}-\frac {2\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}-\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}+\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}-\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}\right )\,\sqrt {-\frac {A^2\,c\,\sqrt {-a^3\,c^7}-9\,B^2\,a\,\sqrt {-a^3\,c^7}+6\,A\,B\,a^2\,c^4}{64\,a^3\,c^7}}+2\,\mathrm {atanh}\left (\frac {18\,B^2\,a\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}-\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}-\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}+\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}-\frac {2\,A^2\,c\,\sqrt {x}\,\sqrt {\frac {A^2\,\sqrt {-a^3\,c^7}}{64\,a^3\,c^6}-\frac {3\,A\,B}{32\,a\,c^3}-\frac {9\,B^2\,\sqrt {-a^3\,c^7}}{64\,a^2\,c^7}}}{\frac {3\,A^2\,B}{4\,c}-\frac {27\,B^3\,a}{4\,c^2}-\frac {A^3\,\sqrt {-a^3\,c^7}}{4\,a^2\,c^4}+\frac {9\,A\,B^2\,\sqrt {-a^3\,c^7}}{4\,a\,c^5}}\right )\,\sqrt {-\frac {9\,B^2\,a\,\sqrt {-a^3\,c^7}-A^2\,c\,\sqrt {-a^3\,c^7}+6\,A\,B\,a^2\,c^4}{64\,a^3\,c^7}}-\frac {\frac {A\,\sqrt {x}}{2\,c}+\frac {B\,x^{3/2}}{2\,c}}{c\,x^2+a} \]
2*atanh((18*B^2*a*x^(1/2)*((9*B^2*(-a^3*c^7)^(1/2))/(64*a^2*c^7) - (A^2*(- a^3*c^7)^(1/2))/(64*a^3*c^6) - (3*A*B)/(32*a*c^3))^(1/2))/((3*A^2*B)/(4*c) - (27*B^3*a)/(4*c^2) + (A^3*(-a^3*c^7)^(1/2))/(4*a^2*c^4) - (9*A*B^2*(-a^ 3*c^7)^(1/2))/(4*a*c^5)) - (2*A^2*c*x^(1/2)*((9*B^2*(-a^3*c^7)^(1/2))/(64* a^2*c^7) - (A^2*(-a^3*c^7)^(1/2))/(64*a^3*c^6) - (3*A*B)/(32*a*c^3))^(1/2) )/((3*A^2*B)/(4*c) - (27*B^3*a)/(4*c^2) + (A^3*(-a^3*c^7)^(1/2))/(4*a^2*c^ 4) - (9*A*B^2*(-a^3*c^7)^(1/2))/(4*a*c^5)))*(-(A^2*c*(-a^3*c^7)^(1/2) - 9* B^2*a*(-a^3*c^7)^(1/2) + 6*A*B*a^2*c^4)/(64*a^3*c^7))^(1/2) + 2*atanh((18* B^2*a*x^(1/2)*((A^2*(-a^3*c^7)^(1/2))/(64*a^3*c^6) - (3*A*B)/(32*a*c^3) - (9*B^2*(-a^3*c^7)^(1/2))/(64*a^2*c^7))^(1/2))/((3*A^2*B)/(4*c) - (27*B^3*a )/(4*c^2) - (A^3*(-a^3*c^7)^(1/2))/(4*a^2*c^4) + (9*A*B^2*(-a^3*c^7)^(1/2) )/(4*a*c^5)) - (2*A^2*c*x^(1/2)*((A^2*(-a^3*c^7)^(1/2))/(64*a^3*c^6) - (3* A*B)/(32*a*c^3) - (9*B^2*(-a^3*c^7)^(1/2))/(64*a^2*c^7))^(1/2))/((3*A^2*B) /(4*c) - (27*B^3*a)/(4*c^2) - (A^3*(-a^3*c^7)^(1/2))/(4*a^2*c^4) + (9*A*B^ 2*(-a^3*c^7)^(1/2))/(4*a*c^5)))*(-(9*B^2*a*(-a^3*c^7)^(1/2) - A^2*c*(-a^3* c^7)^(1/2) + 6*A*B*a^2*c^4)/(64*a^3*c^7))^(1/2) - ((A*x^(1/2))/(2*c) + (B* x^(3/2))/(2*c))/(a + c*x^2)