Integrand size = 25, antiderivative size = 250 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{2 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )}-\frac {\left (2 A c d+a B e-3 \sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}+\frac {\left (2 A c d+a B e+3 \sqrt {a} A \sqrt {c} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{4 a^{3/2} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]
-1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d+ B*a*e-3*A*e*a^(1/2)*c^(1/2))/a^(3/2)/c^(3/4)/(-e*a^(1/2)+d*c^(1/2))^(3/2)+ 1/4*arctanh(c^(1/4)*(e*x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(2*A*c*d+B* a*e+3*A*e*a^(1/2)*c^(1/2))/a^(3/2)/c^(3/4)/(e*a^(1/2)+d*c^(1/2))^(3/2)+1/2 *(a*(-A*e+B*d)+(A*c*d-B*a*e)*x)*(e*x+d)^(1/2)/a/(-a*e^2+c*d^2)/(-c*x^2+a)
Time = 0.83 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.24 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} (-A c d x+a (-B d+A e+B e x))}{\left (-c d^2+a e^2\right ) \left (a-c x^2\right )}-\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (2 A c d+a B e+3 \sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{c \left (\sqrt {c} d+\sqrt {a} e\right )^2}-\frac {\left (2 A c d+a B e-3 \sqrt {a} A \sqrt {c} e\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{4 a^{3/2}} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(-(A*c*d*x) + a*(-(B*d) + A*e + B*e*x)))/((-(c*d ^2) + a*e^2)*(a - c*x^2)) - (Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*(2*A*c*d + a *B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqr t[d + e*x])/(Sqrt[c]*d + Sqrt[a]*e)])/(c*(Sqrt[c]*d + Sqrt[a]*e)^2) - ((2* A*c*d + a*B*e - 3*Sqrt[a]*A*Sqrt[c]*e)*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]*Sqrt[c]*e]))/(4*a^(3/2))
Time = 0.53 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.24, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {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.
| \(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right )^2 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}-\frac {\int -\frac {c \left (2 A c d^2+a B e d-3 a A e^2+e (A c d-a B e) x\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {2 A c d^2+a B e d-3 a A e^2+e (A c d-a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\int -\frac {e \left (A c d^2+2 a B e d-3 a A e^2+(A c d-a B e) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}-\frac {\int \frac {e \left (A c d^2+2 a B e d-3 a A e^2+(A c d-a B e) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}-\frac {e \int \frac {A c d^2+2 a B e d-3 a A e^2+(A c d-a B e) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}-\frac {e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \left (-3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}-\frac {e \left (\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \left (-3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (3 \sqrt {a} A \sqrt {c} e+a B e+2 A c d\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 a \left (c d^2-a e^2\right )}\) |
(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(2*a*(c*d^2 - a*e^2)*( a - c*x^2)) - (e*(((Sqrt[c]*d + Sqrt[a]*e)*(2*A*c*d + a*B*e - 3*Sqrt[a]*A* Sqrt[c]*e)*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]) - ((Sqrt[c]*d - Sqrt[a]*e )*(2*A*c*d + a*B*e + 3*Sqrt[a]*A*Sqrt[c]*e)*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))
3.15.56.3.1 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[((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 = 0.64 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.34
| method | result | size |
| derivativedivides | \(2 e^{2} c^{2} \left (\frac {-\frac {\left (B a e -A \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{2 c \left (c d -\sqrt {a c \,e^{2}}\right ) \left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (-2 A c d -B a e +3 A \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 e c a \sqrt {a c \,e^{2}}}+\frac {\frac {\left (B a e +A \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{2 c \left (c d +\sqrt {a c \,e^{2}}\right ) \left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (2 A c d +B a e +3 A \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 e c a \sqrt {a c \,e^{2}}}\right )\) | \(336\) |
| default | \(2 e^{2} c^{2} \left (\frac {-\frac {\left (B a e -A \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{2 c \left (c d -\sqrt {a c \,e^{2}}\right ) \left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (-2 A c d -B a e +3 A \sqrt {a c \,e^{2}}\right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (-c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 e c a \sqrt {a c \,e^{2}}}+\frac {\frac {\left (B a e +A \sqrt {a c \,e^{2}}\right ) \sqrt {e x +d}}{2 c \left (c d +\sqrt {a c \,e^{2}}\right ) \left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}+\frac {\left (2 A c d +B a e +3 A \sqrt {a c \,e^{2}}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \left (c d +\sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{4 e c a \sqrt {a c \,e^{2}}}\right )\) | \(336\) |
| pseudoelliptic | \(\frac {c e \left (\frac {\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c^{2} e^{2} x^{2} \left (A c d -B a e \right ) \sqrt {a c \,e^{2}}+2 A \,c^{4} d^{2} e^{2} x^{2}-2 \left (\frac {3}{2} A \,e^{2} x^{2}-\frac {1}{2} B d e \,x^{2}+A \,d^{2}\right ) e^{2} a \,c^{3}+a^{2} \left (3 A \,e^{4}-B d \,e^{3}\right ) c^{2}+\left (a c \,e^{2}\right )^{\frac {3}{2}} A c d -\left (a c \,e^{2}\right )^{\frac {3}{2}} B a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\left (\left (\frac {c^{2} e^{2} x^{2} \left (A c d -B a e \right ) \sqrt {a c \,e^{2}}}{2}+A \,c^{4} d^{2} e^{2} x^{2}-\left (\frac {3}{2} A \,e^{2} x^{2}-\frac {1}{2} B d e \,x^{2}+A \,d^{2}\right ) e^{2} a \,c^{3}+\frac {3 \left (A e -\frac {B d}{3}\right ) e^{3} a^{2} c^{2}}{2}-\frac {\left (a c \,e^{2}\right )^{\frac {3}{2}} A c d}{2}+\frac {\left (a c \,e^{2}\right )^{\frac {3}{2}} B a e}{2}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )+\sqrt {e x +d}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (-c e \left (A c d x +B a \left (-e x +d \right )\right ) \sqrt {a c \,e^{2}}+\left (a c \,e^{2}\right )^{\frac {3}{2}} A \right )\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\right )}{2 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a \left (c d +\sqrt {a c \,e^{2}}\right ) \left (x c e -\sqrt {a c \,e^{2}}\right ) \left (c d -\sqrt {a c \,e^{2}}\right ) \left (x c e +\sqrt {a c \,e^{2}}\right )}\) | \(509\) |
2*e^2*c^2*(1/4/e/c/a/(a*c*e^2)^(1/2)*(-1/2/c/(c*d-(a*c*e^2)^(1/2))*(B*a*e- A*(a*c*e^2)^(1/2))*(e*x+d)^(1/2)/(-e*x-(a*c*e^2)^(1/2)/c)+1/2*(-2*A*c*d-B* a*e+3*A*(a*c*e^2)^(1/2))/(-c*d+(a*c*e^2)^(1/2))/((-c*d+(a*c*e^2)^(1/2))*c) ^(1/2)*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)))+1/4/e/c/a /(a*c*e^2)^(1/2)*(1/2/c/(c*d+(a*c*e^2)^(1/2))*(B*a*e+A*(a*c*e^2)^(1/2))*(e *x+d)^(1/2)/(-e*x+(a*c*e^2)^(1/2)/c)+1/2*(2*A*c*d+B*a*e+3*A*(a*c*e^2)^(1/2 ))/(c*d+(a*c*e^2)^(1/2))/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d) ^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 7506 vs. \(2 (195) = 390\).
Time = 65.90 (sec) , antiderivative size = 7506, normalized size of antiderivative = 30.02 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\int { \frac {B x + A}{{\left (c x^{2} - a\right )}^{2} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1182 vs. \(2 (195) = 390\).
Time = 0.45 (sec) , antiderivative size = 1182, normalized size of antiderivative = 4.73 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
-1/4*((a*c*d^2*e - a^2*e^3)^2*A*c*d*e*abs(c) - (a*c*d^2*e - a^2*e^3)^2*B*a *e^2*abs(c) + (sqrt(a*c)*c^2*d^4*e - 4*sqrt(a*c)*a*c*d^2*e^3 + 3*sqrt(a*c) *a^2*e^5)*A*abs(a*c*d^2*e - a^2*e^3)*abs(c) + 2*(sqrt(a*c)*a*c*d^3*e^2 - s qrt(a*c)*a^2*d*e^4)*B*abs(a*c*d^2*e - a^2*e^3)*abs(c) - (2*a*c^4*d^7*e - 7 *a^2*c^3*d^5*e^3 + 8*a^3*c^2*d^3*e^5 - 3*a^4*c*d*e^7)*A*abs(c) - (a^2*c^3* d^6*e^2 - 2*a^3*c^2*d^4*e^4 + a^4*c*d^2*e^6)*B*abs(c))*arctan(sqrt(e*x + d )/sqrt(-(a*c^2*d^3 - a^2*c*d*e^2 + sqrt((a*c^2*d^3 - a^2*c*d*e^2)^2 - (a*c ^2*d^4 - 2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2 - a^2*c*e^2)))/(a*c^2*d^2 - a^2*c*e^2)))/((a^2*c^3*d^4*e - 2*a^3*c^2*d^2*e^3 + a^4*c*e^5 - sqrt(a*c)* a*c^3*d^5 + 2*sqrt(a*c)*a^2*c^2*d^3*e^2 - sqrt(a*c)*a^3*c*d*e^4)*sqrt(-c^2 *d - sqrt(a*c)*c*e)*abs(a*c*d^2*e - a^2*e^3)) - 1/4*((a*c*d^2*e - a^2*e^3) ^2*sqrt(a*c)*A*c*d*e*abs(c) - (a*c*d^2*e - a^2*e^3)^2*sqrt(a*c)*B*a*e^2*ab s(c) - (a*c^3*d^4*e - 4*a^2*c^2*d^2*e^3 + 3*a^3*c*e^5)*A*abs(a*c*d^2*e - a ^2*e^3)*abs(c) - 2*(a^2*c^2*d^3*e^2 - a^3*c*d*e^4)*B*abs(a*c*d^2*e - a^2*e ^3)*abs(c) - (2*sqrt(a*c)*a*c^4*d^7*e - 7*sqrt(a*c)*a^2*c^3*d^5*e^3 + 8*sq rt(a*c)*a^3*c^2*d^3*e^5 - 3*sqrt(a*c)*a^4*c*d*e^7)*A*abs(c) - (sqrt(a*c)*a ^2*c^3*d^6*e^2 - 2*sqrt(a*c)*a^3*c^2*d^4*e^4 + sqrt(a*c)*a^4*c*d^2*e^6)*B* abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a*c^2*d^3 - a^2*c*d*e^2 - sqrt((a*c^2* d^3 - a^2*c*d*e^2)^2 - (a*c^2*d^4 - 2*a^2*c*d^2*e^2 + a^3*e^4)*(a*c^2*d^2 - a^2*c*e^2)))/(a*c^2*d^2 - a^2*c*e^2)))/((a^2*c^4*d^5 - 2*a^3*c^3*d^3*...
Time = 14.94 (sec) , antiderivative size = 10862, normalized size of antiderivative = 43.45 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^2} \, dx=\text {Too large to display} \]
atan(((((192*A*a^5*c^3*e^7 - 128*B*a^5*c^3*d*e^6 + 64*A*a^3*c^5*d^4*e^3 - 256*A*a^4*c^4*d^2*e^5 + 128*B*a^4*c^4*d^3*e^4)/(8*(a^5*e^4 + a^3*c^2*d^4 - 2*a^4*c*d^2*e^2)) + ((d + e*x)^(1/2)*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a ^9*c^3)^(1/2) - 15*A^2*a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c ^2*e^5 - 5*A^2*c^2*d^2*e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2* a^6*c^2*d*e^4 + 9*A^2*a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3 )^(1/2) + 4*A*B*a^4*c^4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B* a^5*c^3*d^2*e^3 - 14*A*B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9 *c^3*e^6 - 3*a^7*c^5*d^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2)*(64*a^5*c^4*d*e^ 6 + 64*a^3*c^6*d^5*e^2 - 128*a^4*c^5*d^3*e^4))/(a^4*e^4 + a^2*c^2*d^4 - 2* a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/2) - 15*A^2 *a^4*c^4*d^3*e^2 + B^2*a^5*c^3*d^3*e^2 - 6*A*B*a^6*c^2*e^5 - 5*A^2*c^2*d^2 *e^3*(a^9*c^3)^(1/2) + 15*A^2*a^5*c^3*d*e^4 + 3*B^2*a^6*c^2*d*e^4 + 9*A^2* a*c*e^5*(a^9*c^3)^(1/2) + 6*A*B*c^2*d^3*e^2*(a^9*c^3)^(1/2) + 4*A*B*a^4*c^ 4*d^4*e + 3*B^2*a*c*d^2*e^3*(a^9*c^3)^(1/2) - 6*A*B*a^5*c^3*d^2*e^3 - 14*A *B*a*c*d*e^4*(a^9*c^3)^(1/2))/(64*(a^6*c^6*d^6 - a^9*c^3*e^6 - 3*a^7*c^5*d ^4*e^2 + 3*a^8*c^4*d^2*e^4)))^(1/2) - ((d + e*x)^(1/2)*(9*A^2*a^2*c^3*e^6 + B^2*a^3*c^2*e^6 + 4*A^2*c^5*d^4*e^2 + B^2*a^2*c^3*d^2*e^4 - 11*A^2*a*c^4 *d^2*e^4 + 4*A*B*a*c^4*d^3*e^3 - 8*A*B*a^2*c^3*d*e^5))/(a^4*e^4 + a^2*c^2* d^4 - 2*a^3*c*d^2*e^2))*((4*A^2*a^3*c^5*d^5 + B^2*a^2*e^5*(a^9*c^3)^(1/...