Integrand size = 25, antiderivative size = 417 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\frac {\sqrt {d+e x} (a (B d-A e)+(A c d-a B e) x)}{4 a \left (c d^2-a e^2\right ) \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e \left (A c d^2+6 a B d e-7 a A e^2\right )-\left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+5 a e^2\right )\right ) x\right )}{16 a^2 \left (c d^2-a e^2\right )^2 \left (a-c x^2\right )}-\frac {\left (a B e \left (2 \sqrt {c} d-5 \sqrt {a} e\right )+3 A \left (4 c^{3/2} d^2-10 \sqrt {a} c d e+7 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{5/2}}+\frac {\left (a B e \left (2 \sqrt {c} d+5 \sqrt {a} e\right )+3 A \left (4 c^{3/2} d^2+10 \sqrt {a} c d e+7 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{3/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{5/2}} \]
-1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(a*B*e*( -5*e*a^(1/2)+2*d*c^(1/2))+3*A*(4*c^(3/2)*d^2-10*c*d*e*a^(1/2)+7*a*e^2*c^(1 /2)))/a^(5/2)/c^(3/4)/(-e*a^(1/2)+d*c^(1/2))^(5/2)+1/32*arctanh(c^(1/4)*(e *x+d)^(1/2)/(e*a^(1/2)+d*c^(1/2))^(1/2))*(a*B*e*(5*e*a^(1/2)+2*d*c^(1/2))+ 3*A*(4*c^(3/2)*d^2+10*c*d*e*a^(1/2)+7*a*e^2*c^(1/2)))/a^(5/2)/c^(3/4)/(e*a ^(1/2)+d*c^(1/2))^(5/2)+1/4*(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)^2-1/16*(a*e*(-7*A*a*e^2+A*c*d^2+6*B*a*d*e)-(6*A *c*d*(-2*a*e^2+c*d^2)+a*B*e*(5*a*e^2+c*d^2))*x)*(e*x+d)^(1/2)/a^2/(-a*e^2+ c*d^2)^2/(-c*x^2+a)
Time = 1.89 (sec) , antiderivative size = 485, normalized size of antiderivative = 1.16 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\frac {-\frac {2 \sqrt {a} \sqrt {d+e x} \left (6 A c^3 d^3 x^3+a^3 e^2 (10 B d-11 A e-9 B e x)-a c^2 d x \left (-B d e x^2+A \left (10 d^2+d e x+12 e^2 x^2\right )\right )+a^2 c \left (A e \left (5 d^2+16 d e x+7 e^2 x^2\right )+B \left (-4 d^3+3 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )\right )\right )}{\left (c d^2-a e^2\right )^2 \left (a-c x^2\right )^2}-\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (a B e \left (2 \sqrt {c} d+5 \sqrt {a} e\right )+3 A \left (4 c^{3/2} d^2+10 \sqrt {a} c d e+7 a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{c \left (\sqrt {c} d+\sqrt {a} e\right )^3}-\frac {\left (a B e \left (2 \sqrt {c} d-5 \sqrt {a} e\right )+3 A \left (4 c^{3/2} d^2-10 \sqrt {a} c d e+7 a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )^2 \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2}} \]
((-2*Sqrt[a]*Sqrt[d + e*x]*(6*A*c^3*d^3*x^3 + a^3*e^2*(10*B*d - 11*A*e - 9 *B*e*x) - a*c^2*d*x*(-(B*d*e*x^2) + A*(10*d^2 + d*e*x + 12*e^2*x^2)) + a^2 *c*(A*e*(5*d^2 + 16*d*e*x + 7*e^2*x^2) + B*(-4*d^3 + 3*d^2*e*x - 6*d*e^2*x ^2 + 5*e^3*x^3))))/((c*d^2 - a*e^2)^2*(a - c*x^2)^2) - (Sqrt[-(c*d) - Sqrt [a]*Sqrt[c]*e]*(a*B*e*(2*Sqrt[c]*d + 5*Sqrt[a]*e) + 3*A*(4*c^(3/2)*d^2 + 1 0*Sqrt[a]*c*d*e + 7*a*Sqrt[c]*e^2))*ArcTan[(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)^3) - ((a*B*e*(2*Sqrt[c]*d - 5*Sqrt[a]*e) + 3*A*(4*c^(3/2)*d^2 - 10*Sqrt[a]*c* d*e + 7*a*Sqrt[c]*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)^2*Sqrt[-( c*d) + Sqrt[a]*Sqrt[c]*e]))/(32*a^(5/2))
Time = 0.89 (sec) , antiderivative size = 503, normalized size of antiderivative = 1.21, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {686, 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.
| \(\displaystyle \int \frac {A+B x}{\left (a-c x^2\right )^3 \sqrt {d+e x}} \, dx\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}-\frac {\int -\frac {c \left (6 A c d^2+a B e d-7 a A e^2+5 e (A c d-a B e) x\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )^2}dx}{4 a c \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 A c d^2+a B e d-7 a A e^2+5 e (A c d-a B e) x}{\sqrt {d+e x} \left (a-c x^2\right )^2}dx}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (2 a B d e \left (c d^2-4 a e^2\right )+3 A \left (4 c^2 d^4-9 a c e^2 d^2+7 a^2 e^4\right )+e \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+5 a e^2\right )\right ) x\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {2 a B d e \left (c d^2-4 a e^2\right )+3 A \left (4 c^2 d^4-9 a c e^2 d^2+7 a^2 e^4\right )+e \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+5 a e^2\right )\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int -\frac {e \left (a B d e \left (c d^2-13 a e^2\right )+3 A \left (2 c^2 d^4-5 a c e^2 d^2+7 a^2 e^4\right )+\left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+5 a e^2\right )\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {e \left (a B d e \left (c d^2-13 a e^2\right )+3 A \left (2 c^2 d^4-5 a c e^2 d^2+7 a^2 e^4\right )+\left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+5 a e^2\right )\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {a B d e \left (c d^2-13 a e^2\right )+3 A \left (2 c^2 d^4-5 a c e^2 d^2+7 a^2 e^4\right )+\left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (c d^2+5 a e^2\right )\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (3 A \left (10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (5 \sqrt {a} e+2 \sqrt {c} d\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (3 A \left (-10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (2 \sqrt {c} d-5 \sqrt {a} e\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\left (\sqrt {a} e+\sqrt {c} d\right )^2 \left (3 A \left (-10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (2 \sqrt {c} d-5 \sqrt {a} e\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {c} d-\sqrt {a} e}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2 \left (3 A \left (10 \sqrt {a} c d e+7 a \sqrt {c} e^2+4 c^{3/2} d^2\right )+a B e \left (5 \sqrt {a} e+2 \sqrt {c} d\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} c^{3/4} e \sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e \left (-7 a A e^2+6 a B d e+A c d^2\right )-x \left (6 A c d \left (c d^2-2 a e^2\right )+a B e \left (5 a e^2+c d^2\right )\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a \left (c d^2-a e^2\right )}+\frac {\sqrt {d+e x} (x (A c d-a B e)+a (B d-A e))}{4 a \left (a-c x^2\right )^2 \left (c d^2-a e^2\right )}\) |
(Sqrt[d + e*x]*(a*(B*d - A*e) + (A*c*d - a*B*e)*x))/(4*a*(c*d^2 - a*e^2)*( a - c*x^2)^2) + (-1/2*(Sqrt[d + e*x]*(a*e*(A*c*d^2 + 6*a*B*d*e - 7*a*A*e^2 ) - (6*A*c*d*(c*d^2 - 2*a*e^2) + a*B*e*(c*d^2 + 5*a*e^2))*x))/(a*(c*d^2 - a*e^2)*(a - c*x^2)) - (e*(((Sqrt[c]*d + Sqrt[a]*e)^2*(a*B*e*(2*Sqrt[c]*d - 5*Sqrt[a]*e) + 3*A*(4*c^(3/2)*d^2 - 10*Sqrt[a]*c*d*e + 7*a*Sqrt[c]*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]) - ((Sqrt[c]*d - Sqrt[a]*e)^2*(a*B*e* (2*Sqrt[c]*d + 5*Sqrt[a]*e) + 3*A*(4*c^(3/2)*d^2 + 10*Sqrt[a]*c*d*e + 7*a* Sqrt[c]*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*(c*d^2 - a*e^2))
3.15.62.3.1 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 = 2.28 (sec) , antiderivative size = 663, normalized size of antiderivative = 1.59
| method | result | size |
| derivativedivides | \(-2 e^{4} c^{3} \left (\frac {\frac {\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +5 B a \,e^{2}-9 A \sqrt {a c \,e^{2}}\, e -2 B \sqrt {a c \,e^{2}}\, d \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} e \left (e^{2} a +c \,d^{2}-2 d \sqrt {a c \,e^{2}}\right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +7 B a \,e^{2}-11 A \sqrt {a c \,e^{2}}\, e -2 B \sqrt {a c \,e^{2}}\, d \right ) \sqrt {e x +d}}{4 c^{2} e \left (c d -\sqrt {a c \,e^{2}}\right )}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {\left (-21 A a c \,e^{2}-12 A \,c^{2} d^{2}-2 B a c d e +30 A \sqrt {a c \,e^{2}}\, c d +5 B \sqrt {a c \,e^{2}}\, a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (-e^{2} a -c \,d^{2}+2 d \sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 \sqrt {a c \,e^{2}}\, a^{2} e^{3} c^{2}}-\frac {\frac {-\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +5 B a \,e^{2}+9 A \sqrt {a c \,e^{2}}\, e +2 B \sqrt {a c \,e^{2}}\, d \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} e \left (e^{2} a +c \,d^{2}+2 d \sqrt {a c \,e^{2}}\right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +7 B a \,e^{2}+11 A \sqrt {a c \,e^{2}}\, e +2 B \sqrt {a c \,e^{2}}\, d \right ) \sqrt {e x +d}}{4 c^{2} e \left (c d +\sqrt {a c \,e^{2}}\right )}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {\left (21 A a c \,e^{2}+12 A \,c^{2} d^{2}+2 B a c d e +30 A \sqrt {a c \,e^{2}}\, c d +5 B \sqrt {a c \,e^{2}}\, a e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (e^{2} a +c \,d^{2}+2 d \sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 \sqrt {a c \,e^{2}}\, a^{2} e^{3} c^{2}}\right )\) | \(663\) |
| default | \(2 e^{4} c^{3} \left (-\frac {\frac {\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +5 B a \,e^{2}-9 A \sqrt {a c \,e^{2}}\, e -2 B \sqrt {a c \,e^{2}}\, d \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} e \left (e^{2} a +c \,d^{2}-2 d \sqrt {a c \,e^{2}}\right )}-\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +7 B a \,e^{2}-11 A \sqrt {a c \,e^{2}}\, e -2 B \sqrt {a c \,e^{2}}\, d \right ) \sqrt {e x +d}}{4 c^{2} e \left (c d -\sqrt {a c \,e^{2}}\right )}}{{\left (-e x -\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}-\frac {\left (-21 A a c \,e^{2}-12 A \,c^{2} d^{2}-2 B a c d e +30 A \sqrt {a c \,e^{2}}\, c d +5 B \sqrt {a c \,e^{2}}\, a e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (-e^{2} a -c \,d^{2}+2 d \sqrt {a c \,e^{2}}\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 \sqrt {a c \,e^{2}}\, a^{2} e^{3} c^{2}}+\frac {\frac {-\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +5 B a \,e^{2}+9 A \sqrt {a c \,e^{2}}\, e +2 B \sqrt {a c \,e^{2}}\, d \right ) \left (e x +d \right )^{\frac {3}{2}}}{4 c^{2} e \left (e^{2} a +c \,d^{2}+2 d \sqrt {a c \,e^{2}}\right )}+\frac {\sqrt {a c \,e^{2}}\, \left (6 A c d e +7 B a \,e^{2}+11 A \sqrt {a c \,e^{2}}\, e +2 B \sqrt {a c \,e^{2}}\, d \right ) \sqrt {e x +d}}{4 c^{2} e \left (c d +\sqrt {a c \,e^{2}}\right )}}{{\left (-e x +\frac {\sqrt {a c \,e^{2}}}{c}\right )}^{2}}+\frac {\left (21 A a c \,e^{2}+12 A \,c^{2} d^{2}+2 B a c d e +30 A \sqrt {a c \,e^{2}}\, c d +5 B \sqrt {a c \,e^{2}}\, a e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{4 c \left (e^{2} a +c \,d^{2}+2 d \sqrt {a c \,e^{2}}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}}{16 \sqrt {a c \,e^{2}}\, a^{2} e^{3} c^{2}}\right )\) | \(663\) |
| pseudoelliptic | \(\text {Expression too large to display}\) | \(1259\) |
-2*e^4*c^3*(1/16/(a*c*e^2)^(1/2)/a^2/e^3/c^2*((1/4*(a*c*e^2)^(1/2)/c^2/e*( 6*A*c*d*e+5*B*a*e^2-9*A*(a*c*e^2)^(1/2)*e-2*B*(a*c*e^2)^(1/2)*d)/(e^2*a+c* d^2-2*d*(a*c*e^2)^(1/2))*(e*x+d)^(3/2)-1/4*(a*c*e^2)^(1/2)/c^2/e*(6*A*c*d* e+7*B*a*e^2-11*A*(a*c*e^2)^(1/2)*e-2*B*(a*c*e^2)^(1/2)*d)/(c*d-(a*c*e^2)^( 1/2))*(e*x+d)^(1/2))/(-e*x-(a*c*e^2)^(1/2)/c)^2-1/4*(-21*A*a*c*e^2-12*A*c^ 2*d^2-2*B*a*c*d*e+30*A*(a*c*e^2)^(1/2)*c*d+5*B*(a*c*e^2)^(1/2)*a*e)/c/(-e^ 2*a-c*d^2+2*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)))-1/16/(a*c*e^2)^(1/2)/a^2/e ^3/c^2*((-1/4*(a*c*e^2)^(1/2)/c^2/e*(6*A*c*d*e+5*B*a*e^2+9*A*(a*c*e^2)^(1/ 2)*e+2*B*(a*c*e^2)^(1/2)*d)/(e^2*a+c*d^2+2*d*(a*c*e^2)^(1/2))*(e*x+d)^(3/2 )+1/4*(a*c*e^2)^(1/2)/c^2/e*(6*A*c*d*e+7*B*a*e^2+11*A*(a*c*e^2)^(1/2)*e+2* B*(a*c*e^2)^(1/2)*d)/(c*d+(a*c*e^2)^(1/2))*(e*x+d)^(1/2))/(-e*x+(a*c*e^2)^ (1/2)/c)^2+1/4*(21*A*a*c*e^2+12*A*c^2*d^2+2*B*a*c*d*e+30*A*(a*c*e^2)^(1/2) *c*d+5*B*(a*c*e^2)^(1/2)*a*e)/c/(e^2*a+c*d^2+2*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))))
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\int { -\frac {B x + A}{{\left (c x^{2} - a\right )}^{3} \sqrt {e x + d}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 2380 vs. \(2 (347) = 694\).
Time = 0.61 (sec) , antiderivative size = 2380, normalized size of antiderivative = 5.71 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
-1/32*(6*(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(c^2*d^3*e - 2*a*c* d*e^3)*A*abs(c) + (a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)^2*(a*c*d^2*e ^2 + 5*a^2*e^4)*B*abs(c) + 3*(2*sqrt(a*c)*a*c^4*d^8*e - 9*sqrt(a*c)*a^2*c^ 3*d^6*e^3 + 19*sqrt(a*c)*a^3*c^2*d^4*e^5 - 19*sqrt(a*c)*a^4*c*d^2*e^7 + 7* sqrt(a*c)*a^5*e^9)*A*abs(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^4*e^5)*abs(c) + (sqrt(a*c)*a^2*c^3*d^7*e^2 - 15*sqrt(a*c)*a^3*c^2*d^5*e^4 + 27*sqrt(a*c )*a^4*c*d^3*e^6 - 13*sqrt(a*c)*a^5*d*e^8)*B*abs(a^2*c^2*d^4*e - 2*a^3*c*d^ 2*e^3 + a^4*e^5)*abs(c) - 3*(4*a^3*c^7*d^13*e - 25*a^4*c^6*d^11*e^3 + 67*a ^5*c^5*d^9*e^5 - 98*a^6*c^4*d^7*e^7 + 82*a^7*c^3*d^5*e^9 - 37*a^8*c^2*d^3* e^11 + 7*a^9*c*d*e^13)*A*abs(c) - 2*(a^4*c^6*d^12*e^2 - 8*a^5*c^5*d^10*e^4 + 22*a^6*c^4*d^8*e^6 - 28*a^7*c^3*d^6*e^8 + 17*a^8*c^2*d^4*e^10 - 4*a^9*c *d^2*e^12)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d^5 - 2*a^3*c^2*d ^3*e^2 + a^4*c*d*e^4 + sqrt((a^2*c^3*d^5 - 2*a^3*c^2*d^3*e^2 + a^4*c*d*e^4 )^2 - (a^2*c^3*d^6 - 3*a^3*c^2*d^4*e^2 + 3*a^4*c*d^2*e^4 - a^5*e^6)*(a^2*c ^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/(a^2*c^3*d^4 - 2*a^3*c^2*d^2*e^2 + a^4*c*e^4)))/((a^4*c^5*d^8*e - 4*a^5*c^4*d^6*e^3 + 6*a^6*c^3*d^4*e^5 - 4*a^7*c^2*d^2*e^7 + a^8*c*e^9 - sqrt(a*c)*a^3*c^5*d^9 + 4*sqrt(a*c)*a^4*c^ 4*d^7*e^2 - 6*sqrt(a*c)*a^5*c^3*d^5*e^4 + 4*sqrt(a*c)*a^6*c^2*d^3*e^6 - sq rt(a*c)*a^7*c*d*e^8)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(a^2*c^2*d^4*e - 2*a^ 3*c*d^2*e^3 + a^4*e^5)) - 1/32*(6*(a^2*c^2*d^4*e - 2*a^3*c*d^2*e^3 + a^...
Time = 18.09 (sec) , antiderivative size = 19125, normalized size of antiderivative = 45.86 \[ \int \frac {A+B x}{\sqrt {d+e x} \left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
- atan(((((86016*A*a^9*c^3*e^11 - 53248*B*a^9*c^3*d*e^10 + 24576*A*a^5*c^7 *d^8*e^3 - 110592*A*a^6*c^6*d^6*e^5 + 233472*A*a^7*c^5*d^4*e^7 - 233472*A* a^8*c^4*d^2*e^9 + 4096*B*a^6*c^6*d^7*e^4 - 61440*B*a^7*c^5*d^5*e^6 + 11059 2*B*a^8*c^4*d^3*e^8)/(4096*(a^10*e^8 + a^6*c^4*d^8 - 4*a^9*c*d^2*e^6 - 4*a ^7*c^3*d^6*e^2 + 6*a^8*c^2*d^4*e^4)) - ((d + e*x)^(1/2)*((144*A^2*a^5*c^7* d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 1701*A^2 *a^7*c^5*d^5*e^4 - 1890*A^2*a^8*c^4*d^3*e^6 + 4*B^2*a^7*c^5*d^7*e^2 - 35*B ^2*a^8*c^4*d^5*e^4 + 70*B^2*a^9*c^3*d^3*e^6 - 441*A^2*a^2*c*e^9*(a^15*c^3) ^(1/2) - 210*A*B*a^10*c^2*e^9 - 189*A^2*c^3*d^4*e^5*(a^15*c^3)^(1/2) + 945 *A^2*a^9*c^3*d*e^8 + 105*B^2*a^10*c^2*d*e^8 + 210*A*B*c^3*d^5*e^4*(a^15*c^ 3)^(1/2) + 48*A*B*a^6*c^6*d^8*e + 486*A^2*a*c^2*d^2*e^7*(a^15*c^3)^(1/2) - 336*A*B*a^7*c^5*d^6*e^3 + 630*A*B*a^8*c^4*d^4*e^5 - 420*A*B*a^9*c^3*d^2*e ^7 + 35*B^2*a*c^2*d^4*e^5*(a^15*c^3)^(1/2) - 154*B^2*a^2*c*d^2*e^7*(a^15*c ^3)^(1/2) + 666*A*B*a^2*c*d*e^8*(a^15*c^3)^(1/2) - 588*A*B*a*c^2*d^3*e^6*( a^15*c^3)^(1/2))/(4096*(a^10*c^8*d^10 - a^15*c^3*e^10 - 5*a^11*c^7*d^8*e^2 + 10*a^12*c^6*d^6*e^4 - 10*a^13*c^5*d^4*e^6 + 5*a^14*c^4*d^2*e^8)))^(1/2) *(4096*a^9*c^4*d*e^10 + 4096*a^5*c^8*d^9*e^2 - 16384*a^6*c^7*d^7*e^4 + 245 76*a^7*c^6*d^5*e^6 - 16384*a^8*c^5*d^3*e^8))/(64*(a^8*e^8 + a^4*c^4*d^8 - 4*a^7*c*d^2*e^6 - 4*a^5*c^3*d^6*e^2 + 6*a^6*c^2*d^4*e^4)))*((144*A^2*a^5*c ^7*d^9 - 25*B^2*a^3*e^9*(a^15*c^3)^(1/2) - 756*A^2*a^6*c^6*d^7*e^2 + 17...