Integrand size = 27, antiderivative size = 746 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (b c d-b^2 e+2 a c e\right ) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (b^2 e (3 B d-5 A e)+8 c \left (A c d^2-a B d e+2 a A e^2\right )-2 b \left (2 B c d^2+A c d e-a B e^2\right )\right )\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 (d+e x) \sqrt {a+b x+c x^2}}+\frac {e \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 e^2 \left (9 B c d^2-10 A c d e-3 a B e^2\right )+4 b^2 c e \left (10 B c d^3+3 A c d^2 e-14 a B d e^2+25 a A e^3\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )\right )-8 b c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c d^2 e^2+5 a^2 e^4\right )\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}+\frac {e^3 \left (5 A e (2 c d-b e)-B \left (8 c d^2-e (3 b d+2 a e)\right )\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{2 \left (c d^2-b d e+a e^2\right )^{7/2}} \]
2/3*(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)/(c*x^2+b*x+a)^(3/2)+1/2*e^ 3*(5*A*e*(-b*e+2*c*d)-B*(8*c*d^2-e*(2*a*e+3*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)+2/3*(6*a*c*e*(-b*e+2*c*d)*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)+(2* a*c*e-b^2*e+b*c*d)*(b^2*e*(-5*A*e+3*B*d)+8*c*(2*A*a*e^2+A*c*d^2-B*a*d*e)-2 *b*(A*c*d*e-B*a*e^2+2*B*c*d^2))+c*(6*c*e*(-2*a*e+b*d)*(A*b*e-2*A*c*d-2*B*a *e+B*b*d)+(-b*e+2*c*d)*(b^2*e*(-5*A*e+3*B*d)+8*c*(2*A*a*e^2+A*c*d^2-B*a*d* e)-2*b*(A*c*d*e-B*a*e^2+2*B*c*d^2)))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2) ^2/(e*x+d)/(c*x^2+b*x+a)^(1/2)+1/3*e*(3*b^4*e^3*(-5*A*e+3*B*d)-2*b^3*e^2*( -10*A*c*d*e-3*B*a*e^2+9*B*c*d^2)+4*b^2*c*e*(25*A*a*e^3+3*A*c*d^2*e-14*B*a* d*e^2+10*B*c*d^3)-16*c^2*(a*B*d*e*(-13*a*e^2+2*c*d^2)-A*(-8*a^2*e^4+9*a*c* d^2*e^2+2*c^2*d^4))-8*b*c*(2*A*c*d*e*(9*a*e^2+4*c*d^2)+B*(5*a^2*e^4+3*a*c* d^2*e^2+2*c^2*d^4)))*(c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2 )^3/(e*x+d)
Time = 13.01 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.01 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\frac {2 \left (\frac {e \left (3 b^4 e^3 (3 B d-5 A e)+2 b^3 e^2 \left (-9 B c d^2+10 A c d e+3 a B e^2\right )+4 b^2 c e \left (10 B c d^3+3 A c d^2 e-14 a B d e^2+25 a A e^3\right )+16 c^2 \left (a B d e \left (-2 c d^2+13 a e^2\right )+A \left (2 c^2 d^4+9 a c d^2 e^2-8 a^2 e^4\right )\right )-8 b c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c d^2 e^2+5 a^2 e^4\right )\right )\right ) \sqrt {a+x (b+c x)}}{2 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )^2 (d+e x)}+\frac {A b^2 e+b B c d x-2 A c (a e+c d x)+A b c (-d+e x)+a B (-b e+2 c (d-e x))}{(d+e x) (a+x (b+c x))^{3/2}}+\frac {b^4 e^2 (3 B d-5 A e)+2 b^2 c \left (16 a A e^3+2 B c d^2 (d-4 e x)+a B e^2 (-5 d+e x)+A c d e (5 d+e x)\right )+b^3 e \left (2 a B e^2+A c e (3 d-5 e x)+B c d (-7 d+3 e x)\right )-8 c^2 \left (2 A c^2 d^3 x-a c d e (A d+2 B d x-7 A e x)+a^2 e^2 (-5 B d+4 A e+3 B e x)\right )-4 b c \left (A c \left (a e^2 (9 d-7 e x)+2 c d^2 (d-3 e x)\right )-B \left (-4 a^2 e^3+2 c^2 d^3 x+a c d e (d+3 e x)\right )\right )}{\left (b^2-4 a c\right ) \left (-c d^2+e (b d-a e)\right ) (d+e x) \sqrt {a+x (b+c x)}}+\frac {3 \left (b^2-4 a c\right ) e^3 \left (8 B c d^2-B e (3 b d+2 a e)+5 A e (-2 c d+b e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{4 \left (c d^2+e (-b d+a e)\right )^{5/2}}\right )}{3 \left (b^2-4 a c\right ) \left (c d^2+e (-b d+a e)\right )} \]
(2*((e*(3*b^4*e^3*(3*B*d - 5*A*e) + 2*b^3*e^2*(-9*B*c*d^2 + 10*A*c*d*e + 3 *a*B*e^2) + 4*b^2*c*e*(10*B*c*d^3 + 3*A*c*d^2*e - 14*a*B*d*e^2 + 25*a*A*e^ 3) + 16*c^2*(a*B*d*e*(-2*c*d^2 + 13*a*e^2) + A*(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)) - 8*b*c*(2*A*c*d*e*(4*c*d^2 + 9*a*e^2) + B*(2*c^2*d^4 + 3*a* c*d^2*e^2 + 5*a^2*e^4)))*Sqrt[a + x*(b + c*x)])/(2*(b^2 - 4*a*c)*(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)*(a + x*(b + c*x))^(3/2)) + (b^4*e^2*(3*B*d - 5*A*e) + 2*b^2*c*(16*a*A*e^3 + 2*B*c*d^ 2*(d - 4*e*x) + a*B*e^2*(-5*d + e*x) + A*c*d*e*(5*d + e*x)) + b^3*e*(2*a*B *e^2 + A*c*e*(3*d - 5*e*x) + B*c*d*(-7*d + 3*e*x)) - 8*c^2*(2*A*c^2*d^3*x - a*c*d*e*(A*d + 2*B*d*x - 7*A*e*x) + a^2*e^2*(-5*B*d + 4*A*e + 3*B*e*x)) - 4*b*c*(A*c*(a*e^2*(9*d - 7*e*x) + 2*c*d^2*(d - 3*e*x)) - B*(-4*a^2*e^3 + 2*c^2*d^3*x + a*c*d*e*(d + 3*e*x))))/((b^2 - 4*a*c)*(-(c*d^2) + e*(b*d - a*e))*(d + e*x)*Sqrt[a + x*(b + c*x)]) + (3*(b^2 - 4*a*c)*e^3*(8*B*c*d^2 - B*e*(3*b*d + 2*a*e) + 5*A*e*(-2*c*d + b*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)])])/( 4*(c*d^2 + e*(-(b*d) + a*e))^(5/2))))/(3*(b^2 - 4*a*c)*(c*d^2 + e*(-(b*d) + a*e)))
Time = 1.62 (sec) , antiderivative size = 803, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1235, 27, 1235, 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.
| \(\displaystyle \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )-6 c e (b B d-2 A c d+A b e-2 a B e) x}{2 (d+e x)^2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )-6 c e (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^2 \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {2 \int -\frac {e \left (6 c e \left (d b^2+2 a e b-8 a c d\right ) (b B d-2 A c d+A b e-2 a B e)-\left (-3 e b^2+2 c d b+8 a c e\right ) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )-2 c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )\right ) x\right )}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {e \int \frac {6 c e \left (d b^2+2 a e b-8 a c d\right ) (b B d-2 A c d+A b e-2 a B e)-\left (-3 e b^2+2 c d b+8 a c e\right ) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )-2 c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{3 \left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\frac {e \left (\frac {3 e^2 \left (b^2-4 a c\right )^2 \left (-B e (2 a e+3 b d)-5 A e (2 c d-b e)+8 B c d^2\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {a+b x+c x^2} \left (-8 b c \left (B \left (5 a^2 e^4+3 a c d^2 e^2+2 c^2 d^4\right )+2 A c d e \left (9 a e^2+4 c d^2\right )\right )-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (-8 a^2 e^4+9 a c d^2 e^2+2 c^2 d^4\right )\right )-2 b^3 e^2 \left (-3 a B e^2-10 A c d e+9 B c d^2\right )+4 b^2 c e \left (25 a A e^3-14 a B d e^2+3 A c d^2 e+10 B c d^3\right )+3 b^4 e^3 (3 B d-5 A e)\right )}{(d+e x) \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (c x \left ((2 c d-b e) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 c e (b d-2 a e) (-2 a B e+A b e-2 A c d+b B d)\right )+\left (2 a c e+b^2 (-e)+b c d\right ) \left (-2 b \left (-a B e^2+A c d e+2 B c d^2\right )+8 c \left (2 a A e^2-a B d e+A c d^2\right )+b^2 e (3 B d-5 A e)\right )+6 a c e (2 c d-b e) (-2 a B e+A b e-2 A c d+b B d)\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {e \left (-\frac {3 \left (b^2-4 a c\right )^2 \left (8 B c d^2-B e (3 b d+2 a e)-5 A e (2 c d-b e)\right ) \int \frac {1}{4 \left (c d^2-b e d+a e^2\right )-\frac {(b d-2 a e+(2 c d-b e) x)^2}{c x^2+b x+a}}d\left (-\frac {b d-2 a e+(2 c d-b e) x}{\sqrt {c x^2+b x+a}}\right ) e^2}{c d^2-b e d+a e^2}-\frac {\left (3 e^3 (3 B d-5 A e) b^4-2 e^2 \left (9 B c d^2-10 A c e d-3 a B e^2\right ) b^3+4 c e \left (10 B c d^3+3 A c e d^2-14 a B e^2 d+25 a A e^3\right ) b^2-8 c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c e^2 d^2+5 a^2 e^4\right )\right ) b-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (d+e x)}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {e \left (\frac {3 \left (b^2-4 a c\right )^2 e^2 \left (8 B c d^2-B e (3 b d+2 a e)-5 A e (2 c d-b e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b e d+a e^2} \sqrt {c x^2+b x+a}}\right )}{2 \left (c d^2-b e d+a e^2\right )^{3/2}}-\frac {\left (3 e^3 (3 B d-5 A e) b^4-2 e^2 \left (9 B c d^2-10 A c e d-3 a B e^2\right ) b^3+4 c e \left (10 B c d^3+3 A c e d^2-14 a B e^2 d+25 a A e^3\right ) b^2-8 c \left (2 A c d e \left (4 c d^2+9 a e^2\right )+B \left (2 c^2 d^4+3 a c e^2 d^2+5 a^2 e^4\right )\right ) b-16 c^2 \left (a B d e \left (2 c d^2-13 a e^2\right )-A \left (2 c^2 d^4+9 a c e^2 d^2-8 a^2 e^4\right )\right )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) (d+e x)}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}-\frac {2 \left (6 a c e (2 c d-b e) (b B d-2 A c d+A b e-2 a B e)+\left (-e b^2+c d b+2 a c e\right ) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )+c \left (6 c e (b d-2 a e) (b B d-2 A c d+A b e-2 a B e)+(2 c d-b e) \left (e (3 B d-5 A e) b^2-2 \left (2 B c d^2+A c e d-a B e^2\right ) b+8 c \left (A c d^2-a B e d+2 a A e^2\right )\right )\right ) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x) \sqrt {c x^2+b x+a}}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
(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))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*( a + b*x + c*x^2)^(3/2)) - ((-2*(6*a*c*e*(2*c*d - b*e)*(b*B*d - 2*A*c*d + A *b*e - 2*a*B*e) + (b*c*d - b^2*e + 2*a*c*e)*(b^2*e*(3*B*d - 5*A*e) + 8*c*( A*c*d^2 - a*B*d*e + 2*a*A*e^2) - 2*b*(2*B*c*d^2 + A*c*d*e - a*B*e^2)) + c* (6*c*e*(b*d - 2*a*e)*(b*B*d - 2*A*c*d + A*b*e - 2*a*B*e) + (2*c*d - b*e)*( b^2*e*(3*B*d - 5*A*e) + 8*c*(A*c*d^2 - a*B*d*e + 2*a*A*e^2) - 2*b*(2*B*c*d ^2 + A*c*d*e - a*B*e^2)))*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(d + e*x)*Sqrt[a + b*x + c*x^2]) + (e*(-(((3*b^4*e^3*(3*B*d - 5*A*e) - 2*b^3*e^ 2*(9*B*c*d^2 - 10*A*c*d*e - 3*a*B*e^2) + 4*b^2*c*e*(10*B*c*d^3 + 3*A*c*d^2 *e - 14*a*B*d*e^2 + 25*a*A*e^3) - 16*c^2*(a*B*d*e*(2*c*d^2 - 13*a*e^2) - A *(2*c^2*d^4 + 9*a*c*d^2*e^2 - 8*a^2*e^4)) - 8*b*c*(2*A*c*d*e*(4*c*d^2 + 9* a*e^2) + B*(2*c^2*d^4 + 3*a*c*d^2*e^2 + 5*a^2*e^4)))*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)^2*e^2*(8*B*c*d^2 - B*e*(3*b*d + 2*a*e) - 5*A*e*(2*c*d - b*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))))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(3* (b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
3.25.86.3.1 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] )
Leaf count of result is larger than twice the leaf count of optimal. \(1917\) vs. \(2(724)=1448\).
Time = 0.54 (sec) , antiderivative size = 1918, normalized size of antiderivative = 2.57
B/e^2*(1/3/(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)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(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)^(3/2)+16/3*c/(4* c*(a*e^2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*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))+1/(a*e ^2-b*d*e+c*d^2)*e^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))))+(A*e-B* d)/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)^(3/2)-5/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*( 1/3/(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)^(3/2)-1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^2)*(2/3*(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)^(3/2)+16/3*c/(4*c*(a*e^ 2-b*d*e+c*d^2)/e^2-(b*e-2*c*d)^2/e^2)^2*(2*c*(x+d/e)+(b*e-2*c*d)/e)/((x...
Leaf count of result is larger than twice the leaf count of optimal. 6324 vs. \(2 (724) = 1448\).
Time = 69.61 (sec) , antiderivative size = 12690, normalized size of antiderivative = 17.01 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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. 9472 vs. \(2 (724) = 1448\).
Time = 1.13 (sec) , antiderivative size = 9472, normalized size of antiderivative = 12.70 \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \]
-1/6*((24*B*b^4*c^(3/2)*d^2*e^6*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b *d*e + a*e^2)*sqrt(c)*abs(e))) - 192*B*a*b^2*c^(5/2)*d^2*e^6*log(abs(2*c*d *e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 384*B*a^2*c^ (7/2)*d^2*e^6*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt (c)*abs(e))) - 9*B*b^5*sqrt(c)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^ 2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 72*B*a*b^3*c^(3/2)*d*e^7*log(abs(2*c *d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 30*A*b^4*c ^(3/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt( c)*abs(e))) - 144*B*a^2*b*c^(5/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c *d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 240*A*a*b^2*c^(5/2)*d*e^7*log(abs (2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 480*A* a^2*c^(7/2)*d*e^7*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)* sqrt(c)*abs(e))) - 6*B*a*b^4*sqrt(c)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt( c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 15*A*b^5*sqrt(c)*e^8*log(abs(2*c *d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 48*B*a^2*b ^2*c^(3/2)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqr t(c)*abs(e))) - 120*A*a*b^3*c^(3/2)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c *d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) - 96*B*a^3*c^(5/2)*e^8*log(abs(2*c* d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*abs(e))) + 240*A*a^2*b *c^(5/2)*e^8*log(abs(2*c*d*e - b*e^2 - 2*sqrt(c*d^2 - b*d*e + a*e^2)*sq...
Timed out. \[ \int \frac {A+B x}{(d+e x)^2 \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \]