Integrand size = 27, antiderivative size = 545 \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x)^2 \sqrt {a+b x+c x^2}}+\frac {e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+b x+c x^2}}{2 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}-\frac {e \left (3 b^3 e^2 (B d-5 A e)-2 b^2 e \left (5 B c d^2-19 A c d e-6 a B e^2\right )-4 b c \left (2 B c d^3+6 A c d^2 e+9 a B d e^2-13 a A e^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )\right ) \sqrt {a+b x+c x^2}}{4 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {3 e \left (A e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right )-B \left (8 c^2 d^3-4 c d e (b d+3 a e)+b e^2 (b d+4 a e)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{8 \left (c d^2-b d e+a e^2\right )^{7/2}} \]
3/8*e*(A*e*(16*c^2*d^2+5*b^2*e^2-4*c*e*(a*e+4*b*d))-B*(8*c^2*d^3-4*c*d*e*( 3*a*e+b*d)+b*e^2*(4*a*e+b*d)))*arctanh(1/2*(b*d-2*a*e+(-b*e+2*c*d)*x)/(a*e ^2-b*d*e+c*d^2)^(1/2)/(c*x^2+b*x+a)^(1/2))/(a*e^2-b*d*e+c*d^2)^(7/2)+2*(a* B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d)+c*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*x)/ (-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^2/(c*x^2+b*x+a)^(1/2)+1/2*e*(b^2* e*(-5*A*e+B*d)-4*c*(-3*A*a*e^2+2*A*c*d^2+5*B*a*d*e)+4*b*(2*A*c*d*e+B*a*e^2 +B*c*d^2))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^ 2-1/4*e*(3*b^3*e^2*(-5*A*e+B*d)-2*b^2*e*(-19*A*c*d*e-6*B*a*e^2+5*B*c*d^2)- 4*b*c*(-13*A*a*e^3+6*A*c*d^2*e+9*B*a*d*e^2+2*B*c*d^3)+8*c*(A*c*d*(-13*a*e^ 2+2*c*d^2)+a*B*e*(-4*a*e^2+11*c*d^2)))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a *e^2-b*d*e+c*d^2)^3/(e*x+d)
Time = 12.13 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (\frac {e \left (b^2 e (B d-5 A e)-4 c \left (2 A c d^2+5 a B d e-3 a A e^2\right )+4 b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)^2}-\frac {e \left (3 b^3 e^2 (B d-5 A e)+2 b^2 e \left (-5 B c d^2+19 A c d e+6 a B e^2\right )-4 b c \left (2 B c d^3+6 A c d^2 e+9 a B d e^2-13 a A e^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )\right ) \sqrt {a+x (b+c x)}}{8 \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {A b^2 e+b B c d x-2 A c (a e+c d x)+A b c (-d+e x)+a B (-b e+2 c (d-e x))}{(d+e x)^2 \sqrt {a+x (b+c x)}}+\frac {3 \left (b^2-4 a c\right ) e \left (A e \left (-16 c^2 d^2-5 b^2 e^2+4 c e (4 b d+a e)\right )+B \left (8 c^2 d^3-4 c d e (b d+3 a e)+b e^2 (b d+4 a e)\right )\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{16 \left (c d^2+e (-b d+a e)\right )^{5/2}}\right )}{\left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]
(2*((e*(b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B*d*e - 3*a*A*e^2) + 4* b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(- (b*d) + a*e))*(d + e*x)^2) - (e*(3*b^3*e^2*(B*d - 5*A*e) + 2*b^2*e*(-5*B*c *d^2 + 19*A*c*d*e + 6*a*B*e^2) - 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d* e^2 - 13*a*A*e^3) + 8*c*(A*c*d*(2*c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4* a*e^2)))*Sqrt[a + x*(b + c*x)])/(8*(c*d^2 + e*(-(b*d) + a*e))^2*(d + e*x)) + (A*b^2*e + b*B*c*d*x - 2*A*c*(a*e + c*d*x) + A*b*c*(-d + e*x) + a*B*(-( b*e) + 2*c*(d - e*x)))/((d + e*x)^2*Sqrt[a + x*(b + c*x)]) + (3*(b^2 - 4*a *c)*e*(A*e*(-16*c^2*d^2 - 5*b^2*e^2 + 4*c*e*(4*b*d + a*e)) + B*(8*c^2*d^3 - 4*c*d*e*(b*d + 3*a*e) + b*e^2*(b*d + 4*a*e)))*ArcTanh[(-(b*d) + 2*a*e - 2*c*d*x + b*e*x)/(2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sqrt[a + x*(b + c*x)])] )/(16*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/((b^2 - 4*a*c)*(c*d^2 + e*(-(b*d ) + a*e)))
Time = 1.08 (sec) , antiderivative size = 583, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1235, 27, 1237, 27, 1228, 1154, 219}
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.
| \(\displaystyle \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {e \left ((B d-5 A e) b^2+4 (A c d+a B e) b-12 a c (B d-A e)-4 c (b B d-2 A c d+A b e-2 a B e) x\right )}{2 (d+e x)^3 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \int \frac {(B d-5 A e) b^2+4 (A c d+a B e) b-12 a c (B d-A e)-4 c (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^3 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {\int \frac {3 e (B d-5 A e) b^3-2 \left (4 B c d^2-14 A c e d-6 a B e^2\right ) b^2-4 c \left (2 A c d^2+7 a B e d-13 a A e^2\right ) b+16 a c \left (3 B c d^2-5 A c e d-2 a B e^2\right )+2 c \left (e (B d-5 A e) b^2+4 \left (B c d^2+2 A c e d+a B e^2\right ) b-4 c \left (2 A c d^2+5 a B e d-3 a A e^2\right )\right ) x}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {\int \frac {3 e (B d-5 A e) b^3-4 \left (2 B c d^2-7 A c e d-3 a B e^2\right ) b^2-4 c \left (2 A c d^2+7 a B e d-13 a A e^2\right ) b+16 a c \left (3 B c d^2-5 A c e d-2 a B e^2\right )+2 c \left (e (B d-5 A e) b^2+4 \left (B c d^2+2 A c e d+a B e^2\right ) b-4 c \left (2 A c d^2+5 a B e d-3 a A e^2\right )\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {\frac {3 \left (b^2-4 a c\right ) \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {-\frac {3 \left (b^2-4 a c\right ) \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right )}{a e^2-b d e+c d^2}-\frac {\sqrt {a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) (d+e x)^2 \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {\frac {3 \left (b^2-4 a c\right ) \left (A e \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )-B \left (-4 c d e (3 a e+b d)+b e^2 (4 a e+b d)+8 c^2 d^3\right )\right ) \text {arctanh}\left (\frac {-2 a e+x (2 c d-b e)+b d}{2 \sqrt {a+b x+c x^2} \sqrt {a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac {\sqrt {a+b x+c x^2} \left (-2 b^2 e \left (-6 a B e^2-19 A c d e+5 B c d^2\right )-4 b c \left (-13 a A e^3+9 a B d e^2+6 A c d^2 e+2 B c d^3\right )+8 c \left (A c d \left (2 c d^2-13 a e^2\right )+a B e \left (11 c d^2-4 a e^2\right )\right )+3 b^3 e^2 (B d-5 A e)\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}}{4 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (4 b \left (a B e^2+2 A c d e+B c d^2\right )-4 c \left (-3 a A e^2+5 a B d e+2 A c d^2\right )+b^2 e (B d-5 A e)\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2*S qrt[a + b*x + c*x^2]) - (e*(-1/2*((b^2*e*(B*d - 5*A*e) - 4*c*(2*A*c*d^2 + 5*a*B*d*e - 3*a*A*e^2) + 4*b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (-(((3*b^3*e^2*(B*d - 5 *A*e) - 2*b^2*e*(5*B*c*d^2 - 19*A*c*d*e - 6*a*B*e^2) - 4*b*c*(2*B*c*d^3 + 6*A*c*d^2*e + 9*a*B*d*e^2 - 13*a*A*e^3) + 8*c*(A*c*d*(2*c*d^2 - 13*a*e^2) + a*B*e*(11*c*d^2 - 4*a*e^2)))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a* e^2)*(d + e*x))) + (3*(b^2 - 4*a*c)*(A*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*( 4*b*d + a*e)) - B*(8*c^2*d^3 - 4*c*d*e*(b*d + 3*a*e) + b*e^2*(b*d + 4*a*e) ))*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e + a*e^2]* Sqrt[a + b*x + c*x^2])])/(2*(c*d^2 - b*d*e + a*e^2)^(3/2)))/(4*(c*d^2 - b* d*e + a*e^2))))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
3.25.79.3.1 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[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Leaf count of result is larger than twice the leaf count of optimal. \(1852\) vs. \(2(525)=1050\).
Time = 0.61 (sec) , antiderivative size = 1853, normalized size of antiderivative = 3.40
B/e^3*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/ e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1 /(a*e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c *d^2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d )/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c *d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a *e^2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*( x+d/e)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e )+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(x+d/e)))-4*c/(a*e^2-b*d*e+c*d^2)*e^2*(2 *c*(x+d/e)+(b*e-2*c*d)/e)/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/ ((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))+(A*e-B* d)/e^4*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2/((x+d/e)^2*c+(b*e-2*c*d)/e* (x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-5/4*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^ 2)*(-1/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+ (a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-3/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(1/(a *e^2-b*d*e+c*d^2)*e^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^ 2)/e^2)^(1/2)-(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2*c*(x+d/e)+(b*e-2*c*d)/e )/(4*c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)/((x+d/e)^2*c+(b*e-2*c*d) /e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-1/(a*e^2-b*d*e+c*d^2)*e^2/((a*e^ 2-b*d*e+c*d^2)/e^2)^(1/2)*ln((2*(a*e^2-b*d*e+c*d^2)/e^2+(b*e-2*c*d)/e*(...
Leaf count of result is larger than twice the leaf count of optimal. 4006 vs. \(2 (525) = 1050\).
Time = 60.99 (sec) , antiderivative size = 8054, normalized size of antiderivative = 14.78 \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a*e^2-b*d*e>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 4205 vs. \(2 (525) = 1050\).
Time = 0.39 (sec) , antiderivative size = 4205, normalized size of antiderivative = 7.72 \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
2*((B*b*c^6*d^9 - 2*A*c^7*d^9 - 3*B*b^2*c^5*d^8*e - 6*B*a*c^6*d^8*e + 9*A* b*c^6*d^8*e + 3*B*b^3*c^4*d^7*e^2 + 24*B*a*b*c^5*d^7*e^2 - 18*A*b^2*c^5*d^ 7*e^2 - B*b^4*c^3*d^6*e^3 - 34*B*a*b^2*c^4*d^6*e^3 + 21*A*b^3*c^4*d^6*e^3 - 16*B*a^2*c^5*d^6*e^3 + 21*B*a*b^3*c^3*d^5*e^4 - 15*A*b^4*c^3*d^5*e^4 + 4 2*B*a^2*b*c^4*d^5*e^4 - 6*A*a*b^2*c^4*d^5*e^4 + 12*A*a^2*c^5*d^5*e^4 - 6*B *a*b^4*c^2*d^4*e^5 + 6*A*b^5*c^2*d^4*e^5 - 36*B*a^2*b^2*c^3*d^4*e^5 + 15*A *a*b^3*c^3*d^4*e^5 - 12*B*a^3*c^4*d^4*e^5 - 30*A*a^2*b*c^4*d^4*e^5 + B*a*b ^5*c*d^3*e^6 - A*b^6*c*d^3*e^6 + 13*B*a^2*b^3*c^2*d^3*e^6 - 12*A*a*b^4*c^2 *d^3*e^6 + 16*B*a^3*b*c^3*d^3*e^6 + 18*A*a^2*b^2*c^3*d^3*e^6 + 16*A*a^3*c^ 4*d^3*e^6 - 3*B*a^2*b^4*c*d^2*e^7 + 3*A*a*b^5*c*d^2*e^7 - 6*B*a^3*b^2*c^2* d^2*e^7 + 3*A*a^2*b^3*c^2*d^2*e^7 - 24*A*a^3*b*c^3*d^2*e^7 + 3*B*a^3*b^3*c *d*e^8 - 3*A*a^2*b^4*c*d*e^8 - 3*B*a^4*b*c^2*d*e^8 + 6*A*a^3*b^2*c^2*d*e^8 + 6*A*a^4*c^3*d*e^8 - B*a^4*b^2*c*e^9 + A*a^3*b^3*c*e^9 + 2*B*a^5*c^2*e^9 - 3*A*a^4*b*c^2*e^9)*x/(b^2*c^6*d^12 - 4*a*c^7*d^12 - 6*b^3*c^5*d^11*e + 24*a*b*c^6*d^11*e + 15*b^4*c^4*d^10*e^2 - 54*a*b^2*c^5*d^10*e^2 - 24*a^2*c ^6*d^10*e^2 - 20*b^5*c^3*d^9*e^3 + 50*a*b^3*c^4*d^9*e^3 + 120*a^2*b*c^5*d^ 9*e^3 + 15*b^6*c^2*d^8*e^4 - 225*a^2*b^2*c^4*d^8*e^4 - 60*a^3*c^5*d^8*e^4 - 6*b^7*c*d^7*e^5 - 36*a*b^5*c^2*d^7*e^5 + 180*a^2*b^3*c^3*d^7*e^5 + 240*a ^3*b*c^4*d^7*e^5 + b^8*d^6*e^6 + 26*a*b^6*c*d^6*e^6 - 30*a^2*b^4*c^2*d^6*e ^6 - 340*a^3*b^2*c^3*d^6*e^6 - 80*a^4*c^4*d^6*e^6 - 6*a*b^7*d^5*e^7 - 3...
Timed out. \[ \int \frac {A+B x}{(d+e x)^3 \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^3\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]