Integrand size = 20, antiderivative size = 281 \[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a d e-\left (6 c d^2-5 a e^2\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]
-1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(12*c*d^ 2+5*a*e^2-18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(3/4)/(-e*a^(1/2)+d*c^(1/2))^( 3/2)+1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(12*c *d^2+5*a*e^2+18*d*e*a^(1/2)*c^(1/2))/a^(5/2)/c^(3/4)/(e*a^(1/2)+d*c^(1/2)) ^(3/2)+1/4*x*(e*x+d)^(1/2)/a/(-c*x^2+a)^2-1/16*(a*d*e-(-5*a*e^2+6*c*d^2)*x )*(e*x+d)^(1/2)/a^2/(-a*e^2+c*d^2)/(-c*x^2+a)
Time = 1.67 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (6 c^2 d^2 x^3+a^2 e (d+9 e x)-a c x \left (10 d^2+d e x+5 e^2 x^2\right )\right )}{\left (-c d^2+a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (12 c d^2+18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{c \left (\sqrt {c} d+\sqrt {a} e\right )^2}-\frac {\left (12 c d^2-18 \sqrt {a} \sqrt {c} d e+5 a e^2\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(6*c^2*d^2*x^3 + a^2*e*(d + 9*e*x) - a*c*x*(10*d ^2 + d*e*x + 5*e^2*x^2)))/((-(c*d^2) + a*e^2)*(a - c*x^2)^2) - (Sqrt[-(c*d ) - Sqrt[a]*Sqrt[c]*e]*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c]*d*e + 5*a*e^2)*ArcTa n[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e) ])/(c*(Sqrt[c]*d + Sqrt[a]*e)^2) - ((12*c*d^2 - 18*Sqrt[a]*Sqrt[c]*d*e + 5 *a*e^2)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt[c]*d - Sqrt[a]*e)])/(Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[-(c*d) + Sqrt[a]*Sqr t[c]*e]))/(32*a^(5/2))
Time = 0.50 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {494, 27, 686, 27, 654, 25, 27, 1480, 221}
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.
| \(\displaystyle \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 494 |
| \(\displaystyle \frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}-\frac {\int -\frac {6 d+5 e x}{2 \sqrt {d+e x} \left (a-c x^2\right )^2}dx}{4 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 d+5 e x}{\sqrt {d+e x} \left (a-c x^2\right )^2}dx}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (d \left (12 c d^2-13 a e^2\right )+e \left (6 c d^2-5 a e^2\right ) x\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {d \left (12 c d^2-13 a e^2\right )+e \left (6 c d^2-5 a e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int -\frac {e \left (2 d \left (3 c d^2-4 a e^2\right )+\left (6 c d^2-5 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {e \left (2 d \left (3 c d^2-4 a e^2\right )+\left (6 c d^2-5 a e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {2 d \left (3 c d^2-4 a e^2\right )+\left (6 c d^2-5 a e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {e \left (\frac {1}{2} \left (-\frac {12 c^{3/2} d^3}{\sqrt {a} e}+13 \sqrt {a} \sqrt {c} d e-5 a e^2+6 c d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}+\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {e \left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (18 \sqrt {a} \sqrt {c} d e+5 a e^2+12 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\left (-\frac {12 c^{3/2} d^3}{\sqrt {a} e}+13 \sqrt {a} \sqrt {c} d e-5 a e^2+6 c d^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 c^{3/4} \sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a d e-x \left (6 c d^2-5 a e^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a}+\frac {x \sqrt {d+e x}}{4 a \left (a-c x^2\right )^2}\) |
(x*Sqrt[d + e*x])/(4*a*(a - c*x^2)^2) + (-1/2*(Sqrt[d + e*x]*(a*d*e - (6*c *d^2 - 5*a*e^2)*x))/(a*(c*d^2 - a*e^2)*(a - c*x^2)) - (e*(-1/2*((6*c*d^2 - (12*c^(3/2)*d^3)/(Sqrt[a]*e) + 13*Sqrt[a]*Sqrt[c]*d*e - 5*a*e^2)*ArcTanh[ (c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(c^(3/4)*Sqrt[Sqrt[c ]*d - Sqrt[a]*e]) - ((Sqrt[c]*d - Sqrt[a]*e)*(12*c*d^2 + 18*Sqrt[a]*Sqrt[c ]*d*e + 5*a*e^2)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]* e]])/(2*Sqrt[a]*c^(3/4)*e*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(2*a*(c*d^2 - a*e ^2)))/(8*a)
3.7.41.3.1 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/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (-x)*(c + d*x)^n*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^(n - 1)*(a + b*x^2)^(p + 1)*(c*(2*p + 3) + d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d}, x] && LtQ[p, -1] && GtQ[n, 0] && (Lt Q[n, 1] || (ILtQ[n + 2*p + 3, 0] && NeQ[n, 2])) && IntQuadraticQ[a, 0, b, c , d, n, p, x]
Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Simp[2 Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d* x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ [p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Time = 2.72 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.31
| method | result | size |
| pseudoelliptic | \(\frac {13 c^{2} \left (\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\frac {\left (-5 e^{2} a +6 c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{13}+c d \left (e^{2} a -\frac {12 c \,d^{2}}{13}\right )\right ) e \left (-c \,x^{2}+a \right )^{2} \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (\left (\frac {\left (5 e^{2} a -6 c \,d^{2}\right ) \sqrt {a c \,e^{2}}}{13}+c d \left (e^{2} a -\frac {12 c \,d^{2}}{13}\right )\right ) e \left (-c \,x^{2}+a \right )^{2} \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\frac {2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, \left (\left (-5 a c \,x^{3}+9 a^{2} x \right ) e^{2}+a d \left (-c \,x^{2}+a \right ) e -10 x c \left (-\frac {3 c \,x^{2}}{5}+a \right ) d^{2}\right ) \sqrt {e x +d}}{13}\right )\right ) e^{4}}{32 \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a^{2} \left (c e x -\sqrt {a c \,e^{2}}\right )^{2} \left (c e x +\sqrt {a c \,e^{2}}\right )^{2} \left (e^{2} a -c \,d^{2}\right )}\) | \(369\) |
| derivativedivides | \(-2 e^{5} c^{3} \left (\frac {\frac {\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -5 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} \left (c d -\sqrt {a c \,e^{2}}\right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -7 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3}}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {\left (-5 e^{2} a -12 c \,d^{2}+18 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c^{2} a^{2} e^{4} \sqrt {a c \,e^{2}}}-\frac {\frac {-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +5 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} \left (c d +\sqrt {a c \,e^{2}}\right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +7 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3}}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {\left (5 e^{2} a +12 c \,d^{2}+18 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c^{2} a^{2} e^{4} \sqrt {a c \,e^{2}}}\right )\) | \(430\) |
| default | \(-2 e^{5} c^{3} \left (\frac {\frac {\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -5 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} \left (c d -\sqrt {a c \,e^{2}}\right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d -7 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3}}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {\left (-5 e^{2} a -12 c \,d^{2}+18 \sqrt {a c \,e^{2}}\, d \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c^{2} a^{2} e^{4} \sqrt {a c \,e^{2}}}-\frac {\frac {-\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +5 \sqrt {a c \,e^{2}}\right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} \left (c d +\sqrt {a c \,e^{2}}\right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 c d +7 \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{4 c^{3}}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {\left (5 e^{2} a +12 c \,d^{2}+18 \sqrt {a c \,e^{2}}\, d \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 c^{2} a^{2} e^{4} \sqrt {a c \,e^{2}}}\right )\) | \(430\) |
13/32/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*c^2/((c*d+(a*c*e^2)^(1/2))*c)^(1/2) *(((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(1/13*(-5*a*e^2+6*c*d^2)*(a*c*e^2)^(1/2) +c*d*(e^2*a-12/13*c*d^2))*e*(-c*x^2+a)^2*arctan(c*(e*x+d)^(1/2)/((-c*d+(a* c*e^2)^(1/2))*c)^(1/2))+((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*((1/13*(5*a*e^2-6 *c*d^2)*(a*c*e^2)^(1/2)+c*d*(e^2*a-12/13*c*d^2))*e*(-c*x^2+a)^2*arctanh(c* (e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+2/13*((c*d+(a*c*e^2)^(1/2)) *c)^(1/2)*(a*c*e^2)^(1/2)*((-5*a*c*x^3+9*a^2*x)*e^2+a*d*(-c*x^2+a)*e-10*x* c*(-3/5*c*x^2+a)*d^2)*(e*x+d)^(1/2)))/(a*c*e^2)^(1/2)*e^4/a^2/(c*e*x-(a*c* e^2)^(1/2))^2/(c*e*x+(a*c*e^2)^(1/2))^2/(a*e^2-c*d^2)
Leaf count of result is larger than twice the leaf count of optimal. 3787 vs. \(2 (223) = 446\).
Time = 1.24 (sec) , antiderivative size = 3787, normalized size of antiderivative = 13.48 \[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/64*((a^4*c*d^2 - a^5*e^2 + (a^2*c^3*d^2 - a^3*c^2*e^2)*x^4 - 2*(a^3*c^2 *d^2 - a^4*c*e^2)*x^2)*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d ^3*e^4 - 105*a^3*d*e^6 + (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2* e^4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14 )/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6 *e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^ 4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))*log(-(3024*c^3 *d^6*e^5 - 7884*a*c^2*d^4*e^7 + 5625*a^2*c*d^2*e^9 - 625*a^3*e^11)*sqrt(e* x + d) + (126*a^3*c^3*d^5*e^6 - 318*a^4*c^2*d^3*e^8 + 200*a^5*c*d*e^10 + ( 12*a^5*c^7*d^10 - 55*a^6*c^6*d^8*e^2 + 98*a^7*c^5*d^6*e^4 - 84*a^8*c^4*d^4 *e^6 + 34*a^9*c^3*d^2*e^8 - 5*a^10*c^2*e^10)*sqrt((441*c^2*d^4*e^10 - 1050 *a*c*d^2*e^12 + 625*a^2*e^14)/(a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7* c^7*d^8*e^4 - 20*a^8*c^6*d^6*e^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^1 0 + a^11*c^3*e^12)))*sqrt((144*c^3*d^7 - 420*a*c^2*d^5*e^2 + 385*a^2*c*d^3 *e^4 - 105*a^3*d*e^6 + (a^5*c^4*d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^ 4 - a^8*c*e^6)*sqrt((441*c^2*d^4*e^10 - 1050*a*c*d^2*e^12 + 625*a^2*e^14)/ (a^5*c^9*d^12 - 6*a^6*c^8*d^10*e^2 + 15*a^7*c^7*d^8*e^4 - 20*a^8*c^6*d^6*e ^6 + 15*a^9*c^5*d^4*e^8 - 6*a^10*c^4*d^2*e^10 + a^11*c^3*e^12)))/(a^5*c^4* d^6 - 3*a^6*c^3*d^4*e^2 + 3*a^7*c^2*d^2*e^4 - a^8*c*e^6))) - (a^4*c*d^2 - a^5*e^2 + (a^2*c^3*d^2 - a^3*c^2*e^2)*x^4 - 2*(a^3*c^2*d^2 - a^4*c*e^2)...
Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\int { -\frac {\sqrt {e x + d}}{{\left (c x^{2} - a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1064 vs. \(2 (223) = 446\).
Time = 0.47 (sec) , antiderivative size = 1064, normalized size of antiderivative = 3.79 \[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=-\frac {{\left ({\left (a^{2} c d^{2} e - a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e - 5 \, a e^{3}\right )} {\left | c \right |} + 2 \, {\left (3 \, \sqrt {a c} a c^{2} d^{5} e - 7 \, \sqrt {a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {a c} a^{3} d e^{5}\right )} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} - {\left (12 \, a^{3} c^{4} d^{8} e - 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} - 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} - a^{3} c d e^{2} + \sqrt {{\left (a^{2} c^{2} d^{3} - a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} - 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} - a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e - 2 \, a^{5} c^{2} d^{2} e^{3} + a^{6} c e^{5} - \sqrt {a c} a^{3} c^{3} d^{5} + 2 \, \sqrt {a c} a^{4} c^{2} d^{3} e^{2} - \sqrt {a c} a^{5} c d e^{4}\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {{\left ({\left (a^{2} c d^{2} e - a^{3} e^{3}\right )}^{2} {\left (6 \, c d^{2} e - 5 \, a e^{3}\right )} {\left | c \right |} - 2 \, {\left (3 \, \sqrt {a c} a c^{2} d^{5} e - 7 \, \sqrt {a c} a^{2} c d^{3} e^{3} + 4 \, \sqrt {a c} a^{3} d e^{5}\right )} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |} {\left | c \right |} - {\left (12 \, a^{3} c^{4} d^{8} e - 37 \, a^{4} c^{3} d^{6} e^{3} + 38 \, a^{5} c^{2} d^{4} e^{5} - 13 \, a^{6} c d^{2} e^{7}\right )} {\left | c \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {a^{2} c^{2} d^{3} - a^{3} c d e^{2} - \sqrt {{\left (a^{2} c^{2} d^{3} - a^{3} c d e^{2}\right )}^{2} - {\left (a^{2} c^{2} d^{4} - 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4}\right )} {\left (a^{2} c^{2} d^{2} - a^{3} c e^{2}\right )}}}{a^{2} c^{2} d^{2} - a^{3} c e^{2}}}}\right )}{32 \, {\left (a^{4} c^{3} d^{4} e - 2 \, a^{5} c^{2} d^{2} e^{3} + a^{6} c e^{5} + \sqrt {a c} a^{3} c^{3} d^{5} - 2 \, \sqrt {a c} a^{4} c^{2} d^{3} e^{2} + \sqrt {a c} a^{5} c d e^{4}\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | a^{2} c d^{2} e - a^{3} e^{3} \right |}} - \frac {6 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{2} d^{2} e - 18 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{2} d^{3} e + 18 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{2} d^{4} e - 6 \, \sqrt {e x + d} c^{2} d^{5} e - 5 \, {\left (e x + d\right )}^{\frac {7}{2}} a c e^{3} + 14 \, {\left (e x + d\right )}^{\frac {5}{2}} a c d e^{3} - 23 \, {\left (e x + d\right )}^{\frac {3}{2}} a c d^{2} e^{3} + 14 \, \sqrt {e x + d} a c d^{3} e^{3} + 9 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} e^{5} - 8 \, \sqrt {e x + d} a^{2} d e^{5}}{16 \, {\left (a^{2} c d^{2} - a^{3} e^{2}\right )} {\left ({\left (e x + d\right )}^{2} c - 2 \, {\left (e x + d\right )} c d + c d^{2} - a e^{2}\right )}^{2}} \]
-1/32*((a^2*c*d^2*e - a^3*e^3)^2*(6*c*d^2*e - 5*a*e^3)*abs(c) + 2*(3*sqrt( a*c)*a*c^2*d^5*e - 7*sqrt(a*c)*a^2*c*d^3*e^3 + 4*sqrt(a*c)*a^3*d*e^5)*abs( a^2*c*d^2*e - a^3*e^3)*abs(c) - (12*a^3*c^4*d^8*e - 37*a^4*c^3*d^6*e^3 + 3 8*a^5*c^2*d^4*e^5 - 13*a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-( a^2*c^2*d^3 - a^3*c*d*e^2 + sqrt((a^2*c^2*d^3 - a^3*c*d*e^2)^2 - (a^2*c^2* d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 - a^3*c*e^2)))/(a^2*c^2*d^2 - a^3*c*e^2)))/((a^4*c^3*d^4*e - 2*a^5*c^2*d^2*e^3 + a^6*c*e^5 - sqrt(a*c) *a^3*c^3*d^5 + 2*sqrt(a*c)*a^4*c^2*d^3*e^2 - sqrt(a*c)*a^5*c*d*e^4)*sqrt(- c^2*d - sqrt(a*c)*c*e)*abs(a^2*c*d^2*e - a^3*e^3)) - 1/32*((a^2*c*d^2*e - a^3*e^3)^2*(6*c*d^2*e - 5*a*e^3)*abs(c) - 2*(3*sqrt(a*c)*a*c^2*d^5*e - 7*s qrt(a*c)*a^2*c*d^3*e^3 + 4*sqrt(a*c)*a^3*d*e^5)*abs(a^2*c*d^2*e - a^3*e^3) *abs(c) - (12*a^3*c^4*d^8*e - 37*a^4*c^3*d^6*e^3 + 38*a^5*c^2*d^4*e^5 - 13 *a^6*c*d^2*e^7)*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^2*d^3 - a^3*c*d* e^2 - sqrt((a^2*c^2*d^3 - a^3*c*d*e^2)^2 - (a^2*c^2*d^4 - 2*a^3*c*d^2*e^2 + a^4*e^4)*(a^2*c^2*d^2 - a^3*c*e^2)))/(a^2*c^2*d^2 - a^3*c*e^2)))/((a^4*c ^3*d^4*e - 2*a^5*c^2*d^2*e^3 + a^6*c*e^5 + sqrt(a*c)*a^3*c^3*d^5 - 2*sqrt( a*c)*a^4*c^2*d^3*e^2 + sqrt(a*c)*a^5*c*d*e^4)*sqrt(-c^2*d + sqrt(a*c)*c*e) *abs(a^2*c*d^2*e - a^3*e^3)) - 1/16*(6*(e*x + d)^(7/2)*c^2*d^2*e - 18*(e*x + d)^(5/2)*c^2*d^3*e + 18*(e*x + d)^(3/2)*c^2*d^4*e - 6*sqrt(e*x + d)*c^2 *d^5*e - 5*(e*x + d)^(7/2)*a*c*e^3 + 14*(e*x + d)^(5/2)*a*c*d*e^3 - 23*...
Time = 12.19 (sec) , antiderivative size = 6163, normalized size of antiderivative = 21.93 \[ \int \frac {\sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
- atan(((((32768*a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3 *e^5)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/2)* (4096*a^7*c^4*d*e^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a ^5*c^5*d^7 - 25*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d ^5*e^2 + 385*a^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10* c^6*d^6 - a^13*c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)) /(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 25*a* e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7*c ^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13*c^3 *e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2) + ((d + e*x)^(1/2) *(25*a^3*c^2*e^8 + 144*c^5*d^6*e^2 - 276*a*c^4*d^4*e^4 + 109*a^2*c^3*d^2*e ^6))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*a^5*c^5*d^7 - 2 5*a*e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a ^7*c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13 *c^3*e^6 - 3*a^11*c^5*d^4*e^2 + 3*a^12*c^4*d^2*e^4)))^(1/2)*1i - (((32768* a^7*c^3*d*e^7 + 24576*a^5*c^5*d^5*e^3 - 57344*a^6*c^4*d^3*e^5)/(4096*(a^8* e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) + ((d + e*x)^(1/2)*(4096*a^7*c^4*d*e ^6 + 4096*a^5*c^6*d^5*e^2 - 8192*a^6*c^5*d^3*e^4)*((144*a^5*c^5*d^7 - 25*a *e^7*(a^15*c^3)^(1/2) - 105*a^8*c^2*d*e^6 - 420*a^6*c^4*d^5*e^2 + 385*a^7* c^3*d^3*e^4 + 21*c*d^2*e^5*(a^15*c^3)^(1/2))/(4096*(a^10*c^6*d^6 - a^13...