Integrand size = 28, antiderivative size = 328 \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {(2 c d-b e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^3}+\frac {\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{12 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+b x+c x^2}}{24 \left (c d^2-b d e+a e^2\right )^3 (d+e x)}-\frac {\left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {b d-2 a e+(2 c d-b e) x}{2 \sqrt {c d^2-b d e+a e^2} \sqrt {a+b x+c x^2}}\right )}{16 \left (c d^2-b d e+a e^2\right )^{7/2}} \] Output:
1/3*(-b*e+2*c*d)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^3+1/12*(8 *c^2*d^2+5*b^2*e^2-4*c*e*(3*a*e+2*b*d))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c *d^2)^2/(e*x+d)^2+1/24*(-b*e+2*c*d)*(8*c^2*d^2+15*b^2*e^2-4*c*e*(13*a*e+2* b*d))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^3/(e*x+d)-1/16*(-4*a*c+b^2)* e*(16*c^2*d^2+5*b^2*e^2-4*c*e*(a*e+4*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)
Time = 10.59 (sec) , antiderivative size = 323, normalized size of antiderivative = 0.98 \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\frac {\frac {2 (2 c d-b e) \left (c d^2+e (-b d+a e)\right ) \sqrt {a+x (b+c x)}}{(d+e x)^3}+\frac {\left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) \sqrt {a+x (b+c x)}}{2 (d+e x)^2}+\frac {(2 c d-b e) \left (8 c^2 d^2+15 b^2 e^2-4 c e (2 b d+13 a e)\right ) \sqrt {a+x (b+c x)}}{4 \left (c d^2+e (-b d+a e)\right ) (d+e x)}+\frac {3 \left (b^2-4 a c\right ) e \left (16 c^2 d^2+5 b^2 e^2-4 c e (4 b d+a e)\right ) \text {arctanh}\left (\frac {-b d+2 a e-2 c d x+b e x}{2 \sqrt {c d^2+e (-b d+a e)} \sqrt {a+x (b+c x)}}\right )}{8 \left (c d^2+e (-b d+a e)\right )^{3/2}}}{6 \left (c d^2+e (-b d+a e)\right )^2} \] Input:
Integrate[(b + 2*c*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
Output:
((2*(2*c*d - b*e)*(c*d^2 + e*(-(b*d) + a*e))*Sqrt[a + x*(b + c*x)])/(d + e *x)^3 + ((8*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(2*b*d + 3*a*e))*Sqrt[a + x*(b + c *x)])/(2*(d + e*x)^2) + ((2*c*d - b*e)*(8*c^2*d^2 + 15*b^2*e^2 - 4*c*e*(2* b*d + 13*a*e))*Sqrt[a + x*(b + c*x)])/(4*(c*d^2 + e*(-(b*d) + a*e))*(d + e *x)) + (3*(b^2 - 4*a*c)*e*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*A rcTanh[(-(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)])])/(8*(c*d^2 + e*(-(b*d) + a*e))^(3/2)))/(6*(c*d^2 + e*(-(b*d) + a*e))^2)
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.
| \(\displaystyle \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {-5 e b^2+4 c d b+12 a c e+4 c (2 c d-b e) x}{2 (d+e x)^3 \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {-5 e b^2+4 c d b+12 a c e+4 c (2 c d-b e) x}{(d+e x)^3 \sqrt {c x^2+b x+a}}dx}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{2 (d+e x)^2 \sqrt {c x^2+b x+a}}dx}{2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {15 e^2 b^3-28 c d e b^2+8 c^2 d^2 b-52 a c e^2 b+80 a c^2 d e+2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int -\frac {-15 e^2 b^3+28 c d e b^2-4 c \left (2 c d^2-13 a e^2\right ) b-80 a c^2 d e-2 c \left (8 c^2 d^2+5 b^2 e^2-4 c e (2 b d+3 a e)\right ) x}{(d+e x)^2 \sqrt {c x^2+b x+a}}dx}{4 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} \left (-4 c e (3 a e+2 b d)+5 b^2 e^2+8 c^2 d^2\right )}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )}}{6 \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^3 \left (a e^2-b d e+c d^2\right )}\) |
Input:
Int[(b + 2*c*x)/((d + e*x)^4*Sqrt[a + b*x + c*x^2]),x]
Output:
$Aborted
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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)/((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. \(1536\) vs. \(2(306)=612\).
Time = 2.16 (sec) , antiderivative size = 1537, normalized size of antiderivative = 4.69
Input:
int((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
(b*e-2*c*d)/e^5*(-1/3/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^3*(c*(x+d/e)^2+(b*e- 2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-5/6*(b*e-2*c*d)*e/(a*e^2-b *d*e+c*d^2)*(-1/2/(a*e^2-b*d*e+c*d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+(b*e-2*c* d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)-3/4*(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)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+ d/e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)*(c*(x+d/e)^2+(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)))+1/2*c/(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)*(c*(x+d/e)^2+(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)))-2/3*c/(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)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/ e)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)+1/2*(b*e-2*c*d)*e/(a*e^2-b*d*e+c*d^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)*(c*(x+d/e)^2+(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))))+2*c/e^4*(-1/2/(a*e^2-b*d*e+c *d^2)*e^2/(x+d/e)^2*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2) /e^2)^(1/2)-3/4*(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)*(c*(x+d/e)^2+(b*e-2*c*d)/e*(x+d/e)+(a*e^2-b*d*e+c*d^2)/e^2)...
Leaf count of result is larger than twice the leaf count of optimal. 1478 vs. \(2 (306) = 612\).
Time = 3.19 (sec) , antiderivative size = 2998, normalized size of antiderivative = 9.14 \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
[1/96*(3*(16*(b^2*c^2 - 4*a*c^3)*d^5*e - 16*(b^3*c - 4*a*b*c^2)*d^4*e^2 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d^3*e^3 + (16*(b^2*c^2 - 4*a*c^3)*d^2*e^ 4 - 16*(b^3*c - 4*a*b*c^2)*d*e^5 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*e^6)* x^3 + 3*(16*(b^2*c^2 - 4*a*c^3)*d^3*e^3 - 16*(b^3*c - 4*a*b*c^2)*d^2*e^4 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*d*e^5)*x^2 + 3*(16*(b^2*c^2 - 4*a*c^3)* d^4*e^2 - 16*(b^3*c - 4*a*b*c^2)*d^3*e^3 + (5*b^4 - 24*a*b^2*c + 16*a^2*c^ 2)*d^2*e^4)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*log((8*a*b*d*e - 8*a^2*e^2 - (b ^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 + 4*sqrt (c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)*(b*d - 2*a*e + (2*c*d - b*e) *x) - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)/(e^2*x^2 + 2*d*e* x + d^2)) + 4*(48*c^4*d^7 - 144*b*c^3*d^6*e - 8*a^3*b*e^7 + 32*(6*b^2*c^2 - a*c^3)*d^5*e^2 - (129*b^3*c + 4*a*b*c^2)*d^4*e^3 + 11*(3*b^4 + 10*a*b^2* c - 8*a^2*c^2)*d^3*e^4 - (59*a*b^3 - 12*a^2*b*c)*d^2*e^5 + 2*(17*a^2*b^2 - 4*a^3*c)*d*e^6 + (16*c^4*d^5*e^2 - 40*b*c^3*d^4*e^3 + 2*(31*b^2*c^2 - 44* a*c^3)*d^3*e^4 - (53*b^3*c - 132*a*b*c^2)*d^2*e^5 + (15*b^4 - 14*a*b^2*c - 104*a^2*c^2)*d*e^6 - (15*a*b^3 - 52*a^2*b*c)*e^7)*x^2 + 2*(24*c^4*d^6*e - 64*b*c^3*d^5*e^2 + 7*(13*b^2*c^2 - 12*a*c^3)*d^4*e^3 - (71*b^3*c - 124*a* b*c^2)*d^3*e^4 + 20*(b^4 - 6*a^2*c^2)*d^2*e^5 - (25*a*b^3 - 68*a^2*b*c)*d* e^6 + (5*a^2*b^2 - 12*a^3*c)*e^7)*x)*sqrt(c*x^2 + b*x + a))/(c^4*d^11 - 4* b*c^3*d^10*e - 4*a^3*b*d^4*e^7 + a^4*d^3*e^8 + 2*(3*b^2*c^2 + 2*a*c^3)*...
\[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {b + 2 c x}{\left (d + e x\right )^{4} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate((2*c*x+b)/(e*x+d)**4/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral((b + 2*c*x)/((d + e*x)**4*sqrt(a + b*x + c*x**2)), x)
Exception generated. \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
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. 2742 vs. \(2 (306) = 612\).
Time = 0.30 (sec) , antiderivative size = 2742, normalized size of antiderivative = 8.36 \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:
integrate((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
-1/8*(16*b^2*c^2*d^2*e - 64*a*c^3*d^2*e - 16*b^3*c*d*e^2 + 64*a*b*c^2*d*e^ 2 + 5*b^4*e^3 - 24*a*b^2*c*e^3 + 16*a^2*c^2*e^3)*arctan(-((sqrt(c)*x - sqr t(c*x^2 + b*x + a))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e - a*e^2))/((c^3*d^6 - 3*b*c^2*d^5*e + 3*b^2*c*d^4*e^2 + 3*a*c^2*d^4*e^2 - b^3*d^3*e^3 - 6*a*b *c*d^3*e^3 + 3*a*b^2*d^2*e^4 + 3*a^2*c*d^2*e^4 - 3*a^2*b*d*e^5 + a^3*e^6)* sqrt(-c*d^2 + b*d*e - a*e^2)) + 1/24*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a ))^5*b^2*c^2*d^2*e^4 - 192*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*c^3*d^2 *e^4 - 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^3*c*d*e^5 + 192*(sqrt(c) *x - sqrt(c*x^2 + b*x + a))^5*a*b*c^2*d*e^5 + 15*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*b^4*e^6 - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a*b^2*c*e^ 6 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*a^2*c^2*e^6 + 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^2*c^(5/2)*d^3*e^3 - 960*(sqrt(c)*x - sqrt(c*x ^2 + b*x + a))^4*a*c^(7/2)*d^3*e^3 - 240*(sqrt(c)*x - sqrt(c*x^2 + b*x + a ))^4*b^3*c^(3/2)*d^2*e^4 + 960*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b*c ^(5/2)*d^2*e^4 + 75*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*b^4*sqrt(c)*d*e^ 5 - 360*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a*b^2*c^(3/2)*d*e^5 + 240*(s qrt(c)*x - sqrt(c*x^2 + b*x + a))^4*a^2*c^(5/2)*d*e^5 + 128*(sqrt(c)*x - s qrt(c*x^2 + b*x + a))^3*c^5*d^6 - 384*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^ 3*b*c^4*d^5*e + 736*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*b^2*c^3*d^4*e^2 - 1024*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*a*c^4*d^4*e^2 - 352*(sqrt(...
Timed out. \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx=\int \frac {b+2\,c\,x}{{\left (d+e\,x\right )}^4\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int((b + 2*c*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)),x)
Output:
int((b + 2*c*x)/((d + e*x)^4*(a + b*x + c*x^2)^(1/2)), x)
Time = 1.45 (sec) , antiderivative size = 4974, normalized size of antiderivative = 15.16 \[ \int \frac {b+2 c x}{(d+e x)^4 \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:
int((2*c*x+b)/(e*x+d)^4/(c*x^2+b*x+a)^(1/2),x)
Output:
(48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e **2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**3*e**3 + 144*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt( a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d**2*e **4*x + 144*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)* sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c**2*d *e**5*x**2 + 48*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x* *2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a**2*c* *2*e**6*x**3 - 72*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c* x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b** 2*c*d**3*e**3 - 216*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x)*a*b **2*c*d**2*e**4*x - 216*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b* x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d*x) *a*b**2*c*d*e**5*x**2 - 72*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c*d *x)*a*b**2*c*e**6*x**3 + 192*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt(a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x + 2*c *d*x)*a*b*c**2*d**4*e**2 + 576*sqrt(a*e**2 - b*d*e + c*d**2)*log( - 2*sqrt (a + b*x + c*x**2)*sqrt(a*e**2 - b*d*e + c*d**2) - 2*a*e + b*d - b*e*x ...