Integrand size = 25, antiderivative size = 372 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\frac {(a B+A c x) \sqrt {d+e x}}{4 a c \left (a-c x^2\right )^2}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c \left (6 A c d^2-a B d e-5 a A e^2\right ) x\right )}{16 a^2 c \left (c d^2-a e^2\right ) \left (a-c x^2\right )}+\frac {\left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt {c} d-\sqrt {a} e\right )^{3/2}}-\frac {\left (a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )-A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{32 a^{5/2} c^{5/4} \left (\sqrt {c} d+\sqrt {a} e\right )^{3/2}} \]
1/32*arctanh(c^(1/4)*(e*x+d)^(1/2)/(-e*a^(1/2)+d*c^(1/2))^(1/2))*(a*B*e*(- 3*e*a^(1/2)+2*d*c^(1/2))-A*(12*c^(3/2)*d^2-18*c*d*e*a^(1/2)+5*a*e^2*c^(1/2 )))/a^(5/2)/c^(5/4)/(-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))*(a*B*e*(3*e*a^(1/2)+2*d*c^(1/2))-A* (12*c^(3/2)*d^2+18*c*d*e*a^(1/2)+5*a*e^2*c^(1/2)))/a^(5/2)/c^(5/4)/(e*a^(1 /2)+d*c^(1/2))^(3/2)+1/4*(A*c*x+B*a)*(e*x+d)^(1/2)/a/c/(-c*x^2+a)^2-1/16*( a*e*(A*c*d-B*a*e)-c*(-5*A*a*e^2+6*A*c*d^2-B*a*d*e)*x)*(e*x+d)^(1/2)/a^2/c/ (-a*e^2+c*d^2)/(-c*x^2+a)
Time = 1.94 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.18 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\frac {\frac {2 \sqrt {a} \sqrt {d+e x} \left (3 a^3 B e^2+6 A c^3 d^2 x^3+a^2 c \left (A e (d+9 e x)+B \left (-4 d^2+d e x+e^2 x^2\right )\right )-a c^2 x \left (B d e x^2+A \left (10 d^2+d e x+5 e^2 x^2\right )\right )\right )}{\left (-c d^2+a e^2\right ) \left (a-c x^2\right )^2}+\frac {\left (-a B e \left (2 \sqrt {c} d+3 \sqrt {a} e\right )+A \left (12 c^{3/2} d^2+18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \sqrt {-c d-\sqrt {a} \sqrt {c} e}}-\frac {\left (a B e \left (-2 \sqrt {c} d+3 \sqrt {a} e\right )+A \left (12 c^{3/2} d^2-18 \sqrt {a} c d e+5 a \sqrt {c} e^2\right )\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {-c d+\sqrt {a} \sqrt {c} e}}}{32 a^{5/2} c} \]
((2*Sqrt[a]*Sqrt[d + e*x]*(3*a^3*B*e^2 + 6*A*c^3*d^2*x^3 + a^2*c*(A*e*(d + 9*e*x) + B*(-4*d^2 + d*e*x + e^2*x^2)) - a*c^2*x*(B*d*e*x^2 + A*(10*d^2 + d*e*x + 5*e^2*x^2))))/((-(c*d^2) + a*e^2)*(a - c*x^2)^2) + ((-(a*B*e*(2*S qrt[c]*d + 3*Sqrt[a]*e)) + A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e + 5*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]*d + Sqrt[a]*e)*Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e ]) - ((a*B*e*(-2*Sqrt[c]*d + 3*Sqrt[a]*e) + A*(12*c^(3/2)*d^2 - 18*Sqrt[a] *c*d*e + 5*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]*d - Sqrt[a]*e)*Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]))/(32*a^(5/2)*c)
Time = 0.73 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {685, 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.
| \(\displaystyle \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 685 |
| \(\displaystyle \frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}-\frac {\int -\frac {6 A c d-a B e+5 A c e x}{2 \sqrt {d+e x} \left (a-c x^2\right )^2}dx}{4 a c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {6 A c d-a B e+5 A c e x}{\sqrt {d+e x} \left (a-c x^2\right )^2}dx}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 686 |
| \(\displaystyle \frac {-\frac {\int -\frac {c \left (A c d \left (12 c d^2-13 a e^2\right )-a B e \left (2 c d^2-3 a e^2\right )+c e \left (6 A c d^2-a B e d-5 a A e^2\right ) x\right )}{2 \sqrt {d+e x} \left (a-c x^2\right )}dx}{2 a c \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {A c d \left (12 c d^2-13 a e^2\right )-a B e \left (2 c d^2-3 a e^2\right )+c e \left (6 A c d^2-a B e d-5 a A e^2\right ) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{4 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 654 |
| \(\displaystyle \frac {\frac {\int -\frac {e \left (2 A c d \left (3 c d^2-4 a e^2\right )-a B e \left (c d^2-3 a e^2\right )+c \left (6 A c d^2-a B e d-5 a A e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {\int \frac {e \left (2 A c d \left (3 c d^2-4 a e^2\right )-a B e \left (c d^2-3 a e^2\right )+c \left (6 A c d^2-a B e d-5 a A e^2\right ) (d+e x)\right )}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {2 A c d \left (3 c d^2-4 a e^2\right )-a B e \left (c d^2-3 a e^2\right )+c \left (6 A c d^2-a B e d-5 a A e^2\right ) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}-\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (a B e \left (3 \sqrt {a} e+2 \sqrt {c} d\right )-A \left (18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a} e}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {-\frac {e \left (\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (a B e \left (3 \sqrt {a} e+2 \sqrt {c} d\right )-A \left (18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c} e \sqrt {\sqrt {a} e+\sqrt {c} d}}-\frac {\left (\sqrt {a} e+\sqrt {c} d\right ) \left (a B e \left (2 \sqrt {c} d-3 \sqrt {a} e\right )-A \left (-18 \sqrt {a} c d e+5 a \sqrt {c} e^2+12 c^{3/2} d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c} e \sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 a \left (c d^2-a e^2\right )}-\frac {\sqrt {d+e x} \left (a e (A c d-a B e)-c x \left (-5 a A e^2-a B d e+6 A c d^2\right )\right )}{2 a \left (a-c x^2\right ) \left (c d^2-a e^2\right )}}{8 a c}+\frac {\sqrt {d+e x} (a B+A c x)}{4 a c \left (a-c x^2\right )^2}\) |
((a*B + A*c*x)*Sqrt[d + e*x])/(4*a*c*(a - c*x^2)^2) + (-1/2*(Sqrt[d + e*x] *(a*e*(A*c*d - a*B*e) - c*(6*A*c*d^2 - a*B*d*e - 5*a*A*e^2)*x))/(a*(c*d^2 - a*e^2)*(a - c*x^2)) - (e*(-1/2*((Sqrt[c]*d + Sqrt[a]*e)*(a*B*e*(2*Sqrt[c ]*d - 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 - 18*Sqrt[a]*c*d*e + 5*a*Sqrt[c]*e^ 2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]])/(Sqrt[a] *c^(1/4)*e*Sqrt[Sqrt[c]*d - Sqrt[a]*e]) + ((Sqrt[c]*d - Sqrt[a]*e)*(a*B*e* (2*Sqrt[c]*d + 3*Sqrt[a]*e) - A*(12*c^(3/2)*d^2 + 18*Sqrt[a]*c*d*e + 5*a*S qrt[c]*e^2))*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]]) /(2*Sqrt[a]*c^(1/4)*e*Sqrt[Sqrt[c]*d + Sqrt[a]*e])))/(2*a*(c*d^2 - a*e^2)) )/(8*a*c)
3.15.61.3.1 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*(a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c *(p + 1))), x] - Simp[1/(2*a*c*(p + 1)) Int[(d + e*x)^(m - 1)*(a + c*x^2) ^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.81 (sec) , antiderivative size = 433, normalized size of antiderivative = 1.16
| method | result | size |
| pseudoelliptic | \(\frac {\frac {13 c \left (-c \,x^{2}+a \right )^{2} \left (\frac {\left (6 A c \,d^{2}-5 \left (A e +\frac {B d}{5}\right ) e a \right ) \sqrt {a c \,e^{2}}}{13}-\frac {12 A \,d^{3} c^{2}}{13}+a d e \left (A e +\frac {2 B d}{13}\right ) c -\frac {3 a^{2} B \,e^{3}}{13}\right ) \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, e \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\left (\frac {13 c \left (-c \,x^{2}+a \right )^{2} e \left (\frac {\left (-6 A c \,d^{2}+5 \left (A e +\frac {B d}{5}\right ) e a \right ) \sqrt {a c \,e^{2}}}{13}-\frac {12 A \,d^{3} c^{2}}{13}+a d e \left (A e +\frac {2 B d}{13}\right ) c -\frac {3 a^{2} B \,e^{3}}{13}\right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2}+\sqrt {e x +d}\, \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \left (6 A \,c^{3} d^{2} x^{3}-10 x \left (A \,d^{2}+\frac {e x \left (B x +A \right ) d}{10}+\frac {A \,e^{2} x^{2}}{2}\right ) a \,c^{2}+\left (-4 B \,d^{2}+e \left (B x +A \right ) d +9 x \,e^{2} \left (\frac {B x}{9}+A \right )\right ) a^{2} c +3 B \,e^{2} a^{3}\right )\right ) \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}{16 \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}\, \sqrt {a c \,e^{2}}\, a^{2} \left (e^{2} a -c \,d^{2}\right ) c \left (-c \,x^{2}+a \right )^{2}}\) | \(433\) |
| default | \(2 e^{4} \left (\frac {-\frac {c \left (5 A a \,e^{2}-6 A c \,d^{2}+B a d e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}+\frac {\left (14 A a c d \,e^{2}-18 A \,d^{3} c^{2}+a^{2} B \,e^{3}+3 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}+\frac {\left (9 A \,a^{2} e^{4}-23 A a c \,d^{2} e^{2}+18 d^{4} A \,c^{2}-B \,a^{2} d \,e^{3}-3 B a c \,d^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}-\frac {\left (8 A a c d \,e^{2}-6 A \,d^{3} c^{2}-3 a^{2} B \,e^{3}+B a c \,d^{2} e \right ) \sqrt {e x +d}}{32 c \,a^{2} e^{3}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}+\frac {-\frac {\left (-13 A a c d \,e^{2}+12 A \,d^{3} c^{2}+3 a^{2} B \,e^{3}-2 B a c \,d^{2} e -5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+6 A \sqrt {a c \,e^{2}}\, c \,d^{2}-B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (13 A a c d \,e^{2}-12 A \,d^{3} c^{2}-3 a^{2} B \,e^{3}+2 B a c \,d^{2} e -5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+6 A \sqrt {a c \,e^{2}}\, c \,d^{2}-B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}\right )\) | \(580\) |
| derivativedivides | \(-2 e^{4} \left (-\frac {-\frac {c \left (5 A a \,e^{2}-6 A c \,d^{2}+B a d e \right ) \left (e x +d \right )^{\frac {7}{2}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}+\frac {\left (14 A a c d \,e^{2}-18 A \,d^{3} c^{2}+a^{2} B \,e^{3}+3 B a c \,d^{2} e \right ) \left (e x +d \right )^{\frac {5}{2}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}+\frac {\left (9 A \,a^{2} e^{4}-23 A a c \,d^{2} e^{2}+18 d^{4} A \,c^{2}-B \,a^{2} d \,e^{3}-3 B a c \,d^{3} e \right ) \left (e x +d \right )^{\frac {3}{2}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}-\frac {\left (8 A a c d \,e^{2}-6 A \,d^{3} c^{2}-3 a^{2} B \,e^{3}+B a c \,d^{2} e \right ) \sqrt {e x +d}}{32 c \,a^{2} e^{3}}}{\left (-c \left (e x +d \right )^{2}+2 c d \left (e x +d \right )+e^{2} a -c \,d^{2}\right )^{2}}-\frac {-\frac {\left (-13 A a c d \,e^{2}+12 A \,d^{3} c^{2}+3 a^{2} B \,e^{3}-2 B a c \,d^{2} e -5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+6 A \sqrt {a c \,e^{2}}\, c \,d^{2}-B \sqrt {a c \,e^{2}}\, a d e \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (13 A a c d \,e^{2}-12 A \,d^{3} c^{2}-3 a^{2} B \,e^{3}+2 B a c \,d^{2} e -5 A \sqrt {a c \,e^{2}}\, a \,e^{2}+6 A \sqrt {a c \,e^{2}}\, c \,d^{2}-B \sqrt {a c \,e^{2}}\, a d e \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{2 \sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}}{32 a^{2} e^{3} \left (e^{2} a -c \,d^{2}\right )}\right )\) | \(581\) |
1/16/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)/(a*c *e^2)^(1/2)*(13/2*c*(-c*x^2+a)^2*(1/13*(6*A*c*d^2-5*(A*e+1/5*B*d)*e*a)*(a* c*e^2)^(1/2)-12/13*A*d^3*c^2+a*d*e*(A*e+2/13*B*d)*c-3/13*a^2*B*e^3)*((c*d+ (a*c*e^2)^(1/2))*c)^(1/2)*e*arctan(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2)) *c)^(1/2))+(13/2*c*(-c*x^2+a)^2*e*(1/13*(-6*A*c*d^2+5*(A*e+1/5*B*d)*e*a)*( a*c*e^2)^(1/2)-12/13*A*d^3*c^2+a*d*e*(A*e+2/13*B*d)*c-3/13*a^2*B*e^3)*arct anh(c*(e*x+d)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))+(e*x+d)^(1/2)*(a*c*e^ 2)^(1/2)*((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*(6*A*c^3*d^2*x^3-10*x*(A*d^2+1/10 *e*x*(B*x+A)*d+1/2*A*e^2*x^2)*a*c^2+(-4*B*d^2+e*(B*x+A)*d+9*x*e^2*(1/9*B*x +A))*a^2*c+3*B*e^2*a^3))*((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))/a^2/(a*e^2-c*d^ 2)/c/(-c*x^2+a)^2
Leaf count of result is larger than twice the leaf count of optimal. 8803 vs. \(2 (302) = 604\).
Time = 140.76 (sec) , antiderivative size = 8803, normalized size of antiderivative = 23.66 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\int { -\frac {{\left (B x + A\right )} \sqrt {e x + d}}{{\left (c x^{2} - a\right )}^{3}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1633 vs. \(2 (302) = 604\).
Time = 0.51 (sec) , antiderivative size = 1633, normalized size of antiderivative = 4.39 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
1/32*((a^2*c^2*d^2*e - a^3*c*e^3)^2*B*a*d*e^2*abs(c) - (a^2*c^2*d^2*e - a^ 3*c*e^3)^2*(6*c*d^2*e - 5*a*e^3)*A*abs(c) - 2*(3*sqrt(a*c)*a*c^3*d^5*e - 7 *sqrt(a*c)*a^2*c^2*d^3*e^3 + 4*sqrt(a*c)*a^3*c*d*e^5)*A*abs(a^2*c^2*d^2*e - a^3*c*e^3)*abs(c) + (sqrt(a*c)*a^2*c^2*d^4*e^2 - 4*sqrt(a*c)*a^3*c*d^2*e ^4 + 3*sqrt(a*c)*a^4*e^6)*B*abs(a^2*c^2*d^2*e - a^3*c*e^3)*abs(c) + (12*a^ 3*c^6*d^8*e - 37*a^4*c^5*d^6*e^3 + 38*a^5*c^4*d^4*e^5 - 13*a^6*c^3*d^2*e^7 )*A*abs(c) - (2*a^4*c^5*d^7*e^2 - 7*a^5*c^4*d^5*e^4 + 8*a^6*c^3*d^3*e^6 - 3*a^7*c^2*d*e^8)*B*abs(c))*arctan(sqrt(e*x + d)/sqrt(-(a^2*c^3*d^3 - a^3*c ^2*d*e^2 + sqrt((a^2*c^3*d^3 - a^3*c^2*d*e^2)^2 - (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^2 - a^3*c^2*e^2)))/(a^2*c^3*d^2 - a^3*c^2 *e^2)))/((a^4*c^4*d^4*e - 2*a^5*c^3*d^2*e^3 + a^6*c^2*e^5 - sqrt(a*c)*a^3* c^4*d^5 + 2*sqrt(a*c)*a^4*c^3*d^3*e^2 - sqrt(a*c)*a^5*c^2*d*e^4)*sqrt(-c^2 *d - sqrt(a*c)*c*e)*abs(a^2*c^2*d^2*e - a^3*c*e^3)) + 1/32*((a^2*c^2*d^2*e - a^3*c*e^3)^2*B*a*d*e^2*abs(c) - (a^2*c^2*d^2*e - a^3*c*e^3)^2*(6*c*d^2* e - 5*a*e^3)*A*abs(c) + 2*(3*sqrt(a*c)*a*c^3*d^5*e - 7*sqrt(a*c)*a^2*c^2*d ^3*e^3 + 4*sqrt(a*c)*a^3*c*d*e^5)*A*abs(a^2*c^2*d^2*e - a^3*c*e^3)*abs(c) - (sqrt(a*c)*a^2*c^2*d^4*e^2 - 4*sqrt(a*c)*a^3*c*d^2*e^4 + 3*sqrt(a*c)*a^4 *e^6)*B*abs(a^2*c^2*d^2*e - a^3*c*e^3)*abs(c) + (12*a^3*c^6*d^8*e - 37*a^4 *c^5*d^6*e^3 + 38*a^5*c^4*d^4*e^5 - 13*a^6*c^3*d^2*e^7)*A*abs(c) - (2*a^4* c^5*d^7*e^2 - 7*a^5*c^4*d^5*e^4 + 8*a^6*c^3*d^3*e^6 - 3*a^7*c^2*d*e^8)*...
Time = 16.81 (sec) , antiderivative size = 13200, normalized size of antiderivative = 35.48 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\left (a-c x^2\right )^3} \, dx=\text {Too large to display} \]
atan(((((12288*B*a^8*c^2*e^8 - 32768*A*a^7*c^3*d*e^7 - 24576*A*a^5*c^5*d^5 *e^3 + 57344*A*a^6*c^4*d^3*e^5 + 4096*B*a^6*c^4*d^4*e^4 - 16384*B*a^7*c^3* d^2*e^6)/(4096*(a^8*e^4 + a^6*c^2*d^4 - 2*a^7*c*d^2*e^2)) - ((d + e*x)^(1/ 2)*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/2) - 420*A^2*a^6*c^ 6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5*e^2 - 15*B^2*a^8*c ^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a^15*c^5)^(1/2) - 10 5*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c*e^7*(a^15*c^5)^(1/ 2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c^6*d^6*e + 5*B^2*a* c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2 *e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c^8*d^6 - a^13*c^5*e ^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(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))/(64*(a^6*e^4 + a^4*c^2*d^4 - 2*a^5*c*d^2*e^2)))*((144*A^2*a^5*c^7*d^7 - 9*B^2*a^2*e^7*(a^15*c^5)^(1/ 2) - 420*A^2*a^6*c^6*d^5*e^2 + 385*A^2*a^7*c^5*d^3*e^4 + 4*B^2*a^7*c^5*d^5 *e^2 - 15*B^2*a^8*c^4*d^3*e^4 + 30*A*B*a^9*c^3*e^7 + 21*A^2*c^2*d^2*e^5*(a ^15*c^5)^(1/2) - 105*A^2*a^8*c^4*d*e^6 + 15*B^2*a^9*c^3*d*e^6 - 25*A^2*a*c *e^7*(a^15*c^5)^(1/2) - 30*A*B*c^2*d^3*e^4*(a^15*c^5)^(1/2) - 48*A*B*a^6*c ^6*d^6*e + 5*B^2*a*c*d^2*e^5*(a^15*c^5)^(1/2) + 160*A*B*a^7*c^5*d^4*e^3 - 150*A*B*a^8*c^4*d^2*e^5 + 38*A*B*a*c*d*e^6*(a^15*c^5)^(1/2))/(4096*(a^10*c ^8*d^6 - a^13*c^5*e^6 - 3*a^11*c^7*d^4*e^2 + 3*a^12*c^6*d^2*e^4)))^(1/2...