Integrand size = 20, antiderivative size = 304 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {5 B \sqrt {x}}{2 c^2}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}+\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{4 \sqrt {2} \sqrt [4]{a} c^{9/4}}+\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{a} c^{9/4}}-\frac {\left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {c} x\right )}{8 \sqrt {2} \sqrt [4]{a} c^{9/4}} \]
-1/2*x^(3/2)*(B*x+A)/c/(c*x^2+a)+1/8*arctan(1-c^(1/4)*2^(1/2)*x^(1/2)/a^(1 /4))*(5*B*a^(1/2)-3*A*c^(1/2))/a^(1/4)/c^(9/4)*2^(1/2)-1/8*arctan(1+c^(1/4 )*2^(1/2)*x^(1/2)/a^(1/4))*(5*B*a^(1/2)-3*A*c^(1/2))/a^(1/4)/c^(9/4)*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))*(5*B*a^(1/2) +3*A*c^(1/2))/a^(1/4)/c^(9/4)*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))*(5*B*a^(1/2)+3*A*c^(1/2))/a^(1/4)/c^(9/4)*2^(1/2)+5 /2*B*x^(1/2)/c^2
Time = 0.54 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.58 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {4 \sqrt [4]{c} \sqrt {x} (5 a B+c x (-A+4 B x))}{a+c x^2}+\frac {\sqrt {2} \left (5 \sqrt {a} B-3 A \sqrt {c}\right ) \arctan \left (\frac {\sqrt {a}-\sqrt {c} x}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}\right )}{\sqrt [4]{a}}-\frac {\sqrt {2} \left (5 \sqrt {a} B+3 A \sqrt {c}\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}}{\sqrt {a}+\sqrt {c} x}\right )}{\sqrt [4]{a}}}{8 c^{9/4}} \]
((4*c^(1/4)*Sqrt[x]*(5*a*B + c*x*(-A + 4*B*x)))/(a + c*x^2) + (Sqrt[2]*(5* Sqrt[a]*B - 3*A*Sqrt[c])*ArcTan[(Sqrt[a] - Sqrt[c]*x)/(Sqrt[2]*a^(1/4)*c^( 1/4)*Sqrt[x])])/a^(1/4) - (Sqrt[2]*(5*Sqrt[a]*B + 3*A*Sqrt[c])*ArcTanh[(Sq rt[2]*a^(1/4)*c^(1/4)*Sqrt[x])/(Sqrt[a] + Sqrt[c]*x)])/a^(1/4))/(8*c^(9/4) )
Time = 0.52 (sec) , antiderivative size = 297, normalized size of antiderivative = 0.98, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {549, 27, 552, 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.
| \(\displaystyle \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 549 |
| \(\displaystyle \frac {\int \frac {\sqrt {x} (3 A+5 B x)}{2 \left (c x^2+a\right )}dx}{2 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {x} (3 A+5 B x)}{c x^2+a}dx}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 552 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \int \frac {5 a B-3 A c x}{2 \sqrt {x} \left (c x^2+a\right )}dx}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {\int \frac {5 a B-3 A c x}{\sqrt {x} \left (c x^2+a\right )}dx}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 554 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \int \frac {5 a B-3 A c x}{c x^2+a}d\sqrt {x}}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \int \frac {\sqrt {c} \left (\sqrt {a}-\sqrt {c} x\right )}{c x^2+a}d\sqrt {x}-\frac {1}{2} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} \left (\sqrt {c} x+\sqrt {a}\right )}{c x^2+a}d\sqrt {x}\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \int \frac {\sqrt {c} x+\sqrt {a}}{c x^2+a}d\sqrt {x}\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}+\frac {\int \frac {1}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\int \frac {1}{-x-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int \frac {1}{-x-1}d\left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \int \frac {\sqrt {a}-\sqrt {c} x}{c x^2+a}d\sqrt {x}-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \left (-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{c} \left (x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}\right )}{\sqrt [4]{c} \left (x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}\right )}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \left (\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{c} \sqrt {x}}{x-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt {2} \sqrt [4]{a} \sqrt {c}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}+\sqrt [4]{a}}{x+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{c}}+\frac {\sqrt {a}}{\sqrt {c}}}d\sqrt {x}}{2 \sqrt [4]{a} \sqrt {c}}\right )-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\frac {10 B \sqrt {x}}{c}-\frac {2 \left (\frac {1}{2} \sqrt {c} \left (\frac {5 \sqrt {a} B}{\sqrt {c}}+3 A\right ) \left (\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} \sqrt {x}+\sqrt {a}+\sqrt {c} x\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )-\frac {1}{2} \sqrt {c} \left (3 A-\frac {5 \sqrt {a} B}{\sqrt {c}}\right ) \left (\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} \sqrt {x}}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{c}}\right )\right )}{c}}{4 c}-\frac {x^{3/2} (A+B x)}{2 c \left (a+c x^2\right )}\) |
-1/2*(x^(3/2)*(A + B*x))/(c*(a + c*x^2)) + ((10*B*Sqrt[x])/c - (2*(-1/2*(( 3*A - (5*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))) + ArcTan[1 + (Sqrt[2]*c^(1/4)*Sqrt[ x])/a^(1/4)]/(Sqrt[2]*a^(1/4)*c^(1/4)))) + ((3*A + (5*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))/c)/(4*c)
3.5.20.3.1 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[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[d*(e*x)^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[e /(b*(m + 2*p + 2)) Int[(e*x)^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && GtQ[m, 0] && NeQ[ m + 2*p + 2, 0] && (IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.12 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.82
| method | result | size |
| derivativedivides | \(\frac {2 B \sqrt {x}}{c^{2}}+\frac {\frac {2 \left (-\frac {A c \,x^{\frac {3}{2}}}{4}+\frac {a B \sqrt {x}}{4}\right )}{c \,x^{2}+a}-\frac {5 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16}+\frac {3 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c^{2}}\) | \(248\) |
| default | \(\frac {2 B \sqrt {x}}{c^{2}}+\frac {\frac {2 \left (-\frac {A c \,x^{\frac {3}{2}}}{4}+\frac {a B \sqrt {x}}{4}\right )}{c \,x^{2}+a}-\frac {5 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16}+\frac {3 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c^{2}}\) | \(248\) |
| risch | \(\frac {2 B \sqrt {x}}{c^{2}}+\frac {\frac {-\frac {A c \,x^{\frac {3}{2}}}{2}+\frac {a B \sqrt {x}}{2}}{c \,x^{2}+a}-\frac {5 B \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16}+\frac {3 A \sqrt {2}\, \left (\ln \left (\frac {x -\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}{x +\left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {x}\, \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {x}}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{16 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c^{2}}\) | \(248\) |
2*B*x^(1/2)/c^2+2/c^2*((-1/4*A*c*x^(3/2)+1/4*a*B*x^(1/2))/(c*x^2+a)-5/32*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))+3/32*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*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. 884 vs. \(2 (208) = 416\).
Time = 0.33 (sec) , antiderivative size = 884, normalized size of antiderivative = 2.91 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {{\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 125 \, B^{3} a^{2} c^{2} - 45 \, A^{2} B a c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}}\right ) - {\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 125 \, B^{3} a^{2} c^{2} - 45 \, A^{2} B a c^{3}\right )} \sqrt {\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} + 30 \, A B}{c^{4}}}\right ) - {\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} + {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 125 \, B^{3} a^{2} c^{2} + 45 \, A^{2} B a c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}}\right ) + {\left (c^{3} x^{2} + a c^{2}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}} \log \left (-{\left (625 \, B^{4} a^{2} - 81 \, A^{4} c^{2}\right )} \sqrt {x} - {\left (3 \, A a c^{7} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 125 \, B^{3} a^{2} c^{2} + 45 \, A^{2} B a c^{3}\right )} \sqrt {-\frac {c^{4} \sqrt {-\frac {625 \, B^{4} a^{2} - 450 \, A^{2} B^{2} a c + 81 \, A^{4} c^{2}}{a c^{9}}} - 30 \, A B}{c^{4}}}\right ) + 4 \, {\left (4 \, B c x^{2} - A c x + 5 \, B a\right )} \sqrt {x}}{8 \, {\left (c^{3} x^{2} + a c^{2}\right )}} \]
1/8*((c^3*x^2 + a*c^2)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81 *A^4*c^2)/(a*c^9)) + 30*A*B)/c^4)*log(-(625*B^4*a^2 - 81*A^4*c^2)*sqrt(x) + (3*A*a*c^7*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 125*B^3*a^2*c^2 - 45*A^2*B*a*c^3)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^2* B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 30*A*B)/c^4)) - (c^3*x^2 + a*c^2)*sqrt((c ^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 30*A*B)/c ^4)*log(-(625*B^4*a^2 - 81*A^4*c^2)*sqrt(x) - (3*A*a*c^7*sqrt(-(625*B^4*a^ 2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 125*B^3*a^2*c^2 - 45*A^2*B*a* c^3)*sqrt((c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) + 30*A*B)/c^4)) - (c^3*x^2 + a*c^2)*sqrt(-(c^4*sqrt(-(625*B^4*a^2 - 450*A ^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 30*A*B)/c^4)*log(-(625*B^4*a^2 - 81*A^ 4*c^2)*sqrt(x) + (3*A*a*c^7*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4* c^2)/(a*c^9)) - 125*B^3*a^2*c^2 + 45*A^2*B*a*c^3)*sqrt(-(c^4*sqrt(-(625*B^ 4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 30*A*B)/c^4)) + (c^3*x^2 + a*c^2)*sqrt(-(c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a* c^9)) - 30*A*B)/c^4)*log(-(625*B^4*a^2 - 81*A^4*c^2)*sqrt(x) - (3*A*a*c^7* sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81*A^4*c^2)/(a*c^9)) - 125*B^3*a^2* c^2 + 45*A^2*B*a*c^3)*sqrt(-(c^4*sqrt(-(625*B^4*a^2 - 450*A^2*B^2*a*c + 81 *A^4*c^2)/(a*c^9)) - 30*A*B)/c^4)) + 4*(4*B*c*x^2 - A*c*x + 5*B*a)*sqrt(x) )/(c^3*x^2 + a*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (284) = 568\).
Time = 125.87 (sec) , antiderivative size = 916, normalized size of antiderivative = 3.01 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\begin {cases} \tilde {\infty } \left (- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}\right ) & \text {for}\: a = 0 \wedge c = 0 \\\frac {\frac {2 A x^{\frac {7}{2}}}{7} + \frac {2 B x^{\frac {9}{2}}}{9}}{a^{2}} & \text {for}\: c = 0 \\\frac {- \frac {2 A}{\sqrt {x}} + 2 B \sqrt {x}}{c^{2}} & \text {for}\: a = 0 \\\frac {3 A a \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {3 A a \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {6 A a \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {4 A c x^{\frac {3}{2}} \sqrt [4]{- \frac {a}{c}}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {3 A c x^{2} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {3 A c x^{2} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {6 A c x^{2} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {20 B a \sqrt {x} \sqrt [4]{- \frac {a}{c}}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {5 B a \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {5 B a \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {10 B a \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {16 B c x^{\frac {5}{2}} \sqrt [4]{- \frac {a}{c}}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} + \frac {5 B c x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} - \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {5 B c x^{2} \sqrt {- \frac {a}{c}} \log {\left (\sqrt {x} + \sqrt [4]{- \frac {a}{c}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} - \frac {10 B c x^{2} \sqrt {- \frac {a}{c}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [4]{- \frac {a}{c}}} \right )}}{8 a c^{2} \sqrt [4]{- \frac {a}{c}} + 8 c^{3} x^{2} \sqrt [4]{- \frac {a}{c}}} & \text {otherwise} \end {cases} \]
Piecewise((zoo*(-2*A/sqrt(x) + 2*B*sqrt(x)), Eq(a, 0) & Eq(c, 0)), ((2*A*x **(7/2)/7 + 2*B*x**(9/2)/9)/a**2, Eq(c, 0)), ((-2*A/sqrt(x) + 2*B*sqrt(x)) /c**2, Eq(a, 0)), (3*A*a*log(sqrt(x) - (-a/c)**(1/4))/(8*a*c**2*(-a/c)**(1 /4) + 8*c**3*x**2*(-a/c)**(1/4)) - 3*A*a*log(sqrt(x) + (-a/c)**(1/4))/(8*a *c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) + 6*A*a*atan(sqrt(x)/(-a/ c)**(1/4))/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) - 4*A*c*x* *(3/2)*(-a/c)**(1/4)/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) + 3*A*c*x**2*log(sqrt(x) - (-a/c)**(1/4))/(8*a*c**2*(-a/c)**(1/4) + 8*c**3 *x**2*(-a/c)**(1/4)) - 3*A*c*x**2*log(sqrt(x) + (-a/c)**(1/4))/(8*a*c**2*( -a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) + 6*A*c*x**2*atan(sqrt(x)/(-a/c) **(1/4))/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) + 20*B*a*sqr t(x)*(-a/c)**(1/4)/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) + 5*B*a*sqrt(-a/c)*log(sqrt(x) - (-a/c)**(1/4))/(8*a*c**2*(-a/c)**(1/4) + 8* c**3*x**2*(-a/c)**(1/4)) - 5*B*a*sqrt(-a/c)*log(sqrt(x) + (-a/c)**(1/4))/( 8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) - 10*B*a*sqrt(-a/c)*at an(sqrt(x)/(-a/c)**(1/4))/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1 /4)) + 16*B*c*x**(5/2)*(-a/c)**(1/4)/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2 *(-a/c)**(1/4)) + 5*B*c*x**2*sqrt(-a/c)*log(sqrt(x) - (-a/c)**(1/4))/(8*a* c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)**(1/4)) - 5*B*c*x**2*sqrt(-a/c)*lo g(sqrt(x) + (-a/c)**(1/4))/(8*a*c**2*(-a/c)**(1/4) + 8*c**3*x**2*(-a/c)...
Time = 0.41 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.93 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=-\frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c^{3} x^{2} + a c^{2}\right )}} + \frac {2 \, B \sqrt {x}}{c^{2}} - \frac {\frac {2 \, \sqrt {2} {\left (5 \, B a \sqrt {c} - 3 \, A \sqrt {a} c\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} + 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (5 \, B a \sqrt {c} - 3 \, A \sqrt {a} c\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} - 2 \, \sqrt {c} \sqrt {x}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (5 \, B a \sqrt {c} + 3 \, A \sqrt {a} c\right )} \log \left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (5 \, B a \sqrt {c} + 3 \, A \sqrt {a} c\right )} \log \left (-\sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} \sqrt {x} + \sqrt {c} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{16 \, c^{2}} \]
-1/2*(A*c*x^(3/2) - B*a*sqrt(x))/(c^3*x^2 + a*c^2) + 2*B*sqrt(x)/c^2 - 1/1 6*(2*sqrt(2)*(5*B*a*sqrt(c) - 3*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(s qrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(5*B*a*sqrt(c) - 3*A*sqrt(a)*c)*arcta n(-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)*(5*B*a*sqrt(c) + 3*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)) - sqrt(2)*(5*B*a*sqrt(c) + 3*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)))/c^2
Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 0.93 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {2 \, B \sqrt {x}}{c^{2}} - \frac {A c x^{\frac {3}{2}} - B a \sqrt {x}}{2 \, {\left (c x^{2} + a\right )} c^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} + 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c - 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}} - 2 \, \sqrt {x}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a c^{4}} + \frac {\sqrt {2} {\left (5 \, \left (a c^{3}\right )^{\frac {1}{4}} B a c + 3 \, \left (a c^{3}\right )^{\frac {3}{4}} A\right )} \log \left (-\sqrt {2} \sqrt {x} \left (\frac {a}{c}\right )^{\frac {1}{4}} + x + \sqrt {\frac {a}{c}}\right )}{16 \, a c^{4}} \]
2*B*sqrt(x)/c^2 - 1/2*(A*c*x^(3/2) - B*a*sqrt(x))/((c*x^2 + a)*c^2) - 1/8* sqrt(2)*(5*(a*c^3)^(1/4)*B*a*c - 3*(a*c^3)^(3/4)*A)*arctan(1/2*sqrt(2)*(sq rt(2)*(a/c)^(1/4) + 2*sqrt(x))/(a/c)^(1/4))/(a*c^4) - 1/8*sqrt(2)*(5*(a*c^ 3)^(1/4)*B*a*c - 3*(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*c^4) - 1/16*sqrt(2)*(5*(a*c^3)^(1/4)*B*a*c + 3*(a*c^3)^(3/4)*A)*log(sqrt(2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a* c^4) + 1/16*sqrt(2)*(5*(a*c^3)^(1/4)*B*a*c + 3*(a*c^3)^(3/4)*A)*log(-sqrt( 2)*sqrt(x)*(a/c)^(1/4) + x + sqrt(a/c))/(a*c^4)
Time = 10.17 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.03 \[ \int \frac {x^{5/2} (A+B x)}{\left (a+c x^2\right )^2} \, dx=\frac {\frac {B\,a\,\sqrt {x}}{2}-\frac {A\,c\,x^{3/2}}{2}}{c^3\,x^2+a\,c^2}+\frac {2\,B\,\sqrt {x}}{c^2}-\mathrm {atan}\left (\frac {B^2\,a^2\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}-\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}+\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,50{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c}+\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^7}-\frac {75\,A\,B^2\,a^2}{4\,c^2}-\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^6}}-\frac {A^2\,a\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}-\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}+\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,18{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c^2}+\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^8}-\frac {75\,A\,B^2\,a^2}{4\,c^3}-\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^7}}\right )\,\sqrt {\frac {9\,A^2\,c\,\sqrt {-a\,c^9}-25\,B^2\,a\,\sqrt {-a\,c^9}+30\,A\,B\,a\,c^5}{64\,a\,c^9}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {B^2\,a^2\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}+\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}-\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,50{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c}-\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^7}-\frac {75\,A\,B^2\,a^2}{4\,c^2}+\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^6}}-\frac {A^2\,a\,\sqrt {x}\,\sqrt {\frac {15\,A\,B}{32\,c^4}+\frac {25\,B^2\,\sqrt {-a\,c^9}}{64\,c^9}-\frac {9\,A^2\,\sqrt {-a\,c^9}}{64\,a\,c^8}}\,18{}\mathrm {i}}{\frac {27\,A^3\,a}{4\,c^2}-\frac {125\,B^3\,a^2\,\sqrt {-a\,c^9}}{4\,c^8}-\frac {75\,A\,B^2\,a^2}{4\,c^3}+\frac {45\,A^2\,B\,a\,\sqrt {-a\,c^9}}{4\,c^7}}\right )\,\sqrt {\frac {25\,B^2\,a\,\sqrt {-a\,c^9}-9\,A^2\,c\,\sqrt {-a\,c^9}+30\,A\,B\,a\,c^5}{64\,a\,c^9}}\,2{}\mathrm {i} \]
((B*a*x^(1/2))/2 - (A*c*x^(3/2))/2)/(a*c^2 + c^3*x^2) - atan((B^2*a^2*x^(1 /2)*((15*A*B)/(32*c^4) + (25*B^2*(-a*c^9)^(1/2))/(64*c^9) - (9*A^2*(-a*c^9 )^(1/2))/(64*a*c^8))^(1/2)*50i)/((27*A^3*a)/(4*c) - (125*B^3*a^2*(-a*c^9)^ (1/2))/(4*c^7) - (75*A*B^2*a^2)/(4*c^2) + (45*A^2*B*a*(-a*c^9)^(1/2))/(4*c ^6)) - (A^2*a*x^(1/2)*((15*A*B)/(32*c^4) + (25*B^2*(-a*c^9)^(1/2))/(64*c^9 ) - (9*A^2*(-a*c^9)^(1/2))/(64*a*c^8))^(1/2)*18i)/((27*A^3*a)/(4*c^2) - (1 25*B^3*a^2*(-a*c^9)^(1/2))/(4*c^8) - (75*A*B^2*a^2)/(4*c^3) + (45*A^2*B*a* (-a*c^9)^(1/2))/(4*c^7)))*((25*B^2*a*(-a*c^9)^(1/2) - 9*A^2*c*(-a*c^9)^(1/ 2) + 30*A*B*a*c^5)/(64*a*c^9))^(1/2)*2i - atan((B^2*a^2*x^(1/2)*((15*A*B)/ (32*c^4) - (25*B^2*(-a*c^9)^(1/2))/(64*c^9) + (9*A^2*(-a*c^9)^(1/2))/(64*a *c^8))^(1/2)*50i)/((27*A^3*a)/(4*c) + (125*B^3*a^2*(-a*c^9)^(1/2))/(4*c^7) - (75*A*B^2*a^2)/(4*c^2) - (45*A^2*B*a*(-a*c^9)^(1/2))/(4*c^6)) - (A^2*a* x^(1/2)*((15*A*B)/(32*c^4) - (25*B^2*(-a*c^9)^(1/2))/(64*c^9) + (9*A^2*(-a *c^9)^(1/2))/(64*a*c^8))^(1/2)*18i)/((27*A^3*a)/(4*c^2) + (125*B^3*a^2*(-a *c^9)^(1/2))/(4*c^8) - (75*A*B^2*a^2)/(4*c^3) - (45*A^2*B*a*(-a*c^9)^(1/2) )/(4*c^7)))*((9*A^2*c*(-a*c^9)^(1/2) - 25*B^2*a*(-a*c^9)^(1/2) + 30*A*B*a* c^5)/(64*a*c^9))^(1/2)*2i + (2*B*x^(1/2))/c^2