Integrand size = 26, antiderivative size = 305 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\frac {(6 B c d-7 b B e+8 A c e) (d+e x)^2 \sqrt {b x+c x^2}}{24 c^2}+\frac {B (d+e x)^3 \sqrt {b x+c x^2}}{4 c}+\frac {\left (8 A c e \left (64 c^2 d^2-54 b c d e+15 b^2 e^2\right )+B \left (96 c^3 d^3-376 b c^2 d^2 e+360 b^2 c d e^2-105 b^3 e^3\right )+2 c e \left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c d e+35 b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{192 c^4}+\frac {\left (128 A c^4 d^3+35 b^4 B e^3+144 b^2 c^2 d e (B d+A e)-40 b^3 c e^2 (3 B d+A e)-64 b c^3 d^2 (B d+3 A e)\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{9/2}} \]
1/64*(128*A*c^4*d^3+35*b^4*B*e^3+144*b^2*c^2*d*e*(A*e+B*d)-40*b^3*c*e^2*(A *e+3*B*d)-64*b*c^3*d^2*(3*A*e+B*d))*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c ^(9/2)+1/24*(8*A*c*e-7*B*b*e+6*B*c*d)*(e*x+d)^2*(c*x^2+b*x)^(1/2)/c^2+1/4* B*(e*x+d)^3*(c*x^2+b*x)^(1/2)/c+1/192*(8*A*c*e*(15*b^2*e^2-54*b*c*d*e+64*c ^2*d^2)+B*(-105*b^3*e^3+360*b^2*c*d*e^2-376*b*c^2*d^2*e+96*c^3*d^3)+2*c*e* (40*A*c*e*(-b*e+2*c*d)+B*(35*b^2*e^2-64*b*c*d*e+24*c^2*d^2))*x)*(c*x^2+b*x )^(1/2)/c^4
Time = 0.50 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.94 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\frac {\sqrt {x} \left (\sqrt {c} \sqrt {x} (b+c x) \left (8 A c e \left (15 b^2 e^2-2 b c e (27 d+5 e x)+4 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+B \left (-105 b^3 e^3+10 b^2 c e^2 (36 d+7 e x)-8 b c^2 e \left (54 d^2+30 d e x+7 e^2 x^2\right )+48 c^3 \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )\right )+3 \left (-128 A c^4 d^3-35 b^4 B e^3-144 b^2 c^2 d e (B d+A e)+40 b^3 c e^2 (3 B d+A e)+64 b c^3 d^2 (B d+3 A e)\right ) \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{192 c^{9/2} \sqrt {x (b+c x)}} \]
(Sqrt[x]*(Sqrt[c]*Sqrt[x]*(b + c*x)*(8*A*c*e*(15*b^2*e^2 - 2*b*c*e*(27*d + 5*e*x) + 4*c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + B*(-105*b^3*e^3 + 10*b^2 *c*e^2*(36*d + 7*e*x) - 8*b*c^2*e*(54*d^2 + 30*d*e*x + 7*e^2*x^2) + 48*c^3 *(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))) + 3*(-128*A*c^4*d^3 - 35*b^ 4*B*e^3 - 144*b^2*c^2*d*e*(B*d + A*e) + 40*b^3*c*e^2*(3*B*d + A*e) + 64*b* c^3*d^2*(B*d + 3*A*e))*Sqrt[b + c*x]*Log[-(Sqrt[c]*Sqrt[x]) + Sqrt[b + c*x ]]))/(192*c^(9/2)*Sqrt[x*(b + c*x)])
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {\int -\frac {(d+e x)^2 ((b B-8 A c) d-(6 B c d-7 b B e+8 A c e) x)}{2 \sqrt {c x^2+b x}}dx}{4 c}+\frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\int \frac {(d+e x)^2 ((b B-8 A c) d-(6 B c d-7 b B e+8 A c e) x)}{\sqrt {c x^2+b x}}dx}{8 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int \frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{2 \sqrt {c x^2+b x}}dx}{3 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {-\frac {\int -\frac {(d+e x) \left (d \left (-7 B e b^2+12 B c d b+8 A c e b-48 A c^2 d\right )-\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {B \sqrt {b x+c x^2} (d+e x)^3}{4 c}-\frac {\frac {\int -\frac {(d+e x) \left (d \left (7 B e b^2-4 c (3 B d+2 A e) b+48 A c^2 d\right )+\left (40 A c e (2 c d-b e)+B \left (24 c^2 d^2-64 b c e d+35 b^2 e^2\right )\right ) x\right )}{\sqrt {c x^2+b x}}dx}{6 c}-\frac {\sqrt {b x+c x^2} (d+e x)^2 (8 A c e-7 b B e+6 B c d)}{3 c}}{8 c}\) |
3.12.91.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 0.53 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.74
| method | result | size |
| pseudoelliptic | \(\frac {\frac {5 \left (\left (-A \,b^{3} c +\frac {7}{8} b^{4} B \right ) e^{3}+\frac {18 c \left (A c -\frac {5 B b}{6}\right ) d \,b^{2} e^{2}}{5}-\frac {24 c^{2} d^{2} \left (A c -\frac {3 B b}{4}\right ) b e}{5}+\frac {16 c^{3} d^{3} \left (A c -\frac {B b}{2}\right )}{5}\right ) \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{8}+\frac {5 \sqrt {x \left (c x +b \right )}\, \left (\frac {4 \left (\frac {2 \left (\frac {3 B x}{4}+A \right ) x^{2} e^{3}}{3}+3 x d \left (\frac {2 B x}{3}+A \right ) e^{2}+6 \left (\frac {B x}{2}+A \right ) d^{2} e +2 B \,d^{3}\right ) c^{\frac {7}{2}}}{5}+\left (2 \left (-\frac {x \left (\frac {7 B x}{10}+A \right ) e^{2}}{3}-\frac {9 d \left (\frac {5 B x}{9}+A \right ) e}{5}-\frac {9 B \,d^{2}}{5}\right ) c^{\frac {5}{2}}+\left (\left (\left (\frac {7 B x}{12}+A \right ) e +3 B d \right ) c^{\frac {3}{2}}-\frac {7 B b e \sqrt {c}}{8}\right ) e b \right ) e b \right )}{8}}{c^{\frac {9}{2}}}\) | \(225\) |
| risch | \(\frac {\left (48 B \,c^{3} e^{3} x^{3}+64 A \,c^{3} e^{3} x^{2}-56 B b \,c^{2} e^{3} x^{2}+192 B \,c^{3} d \,e^{2} x^{2}-80 A b \,c^{2} e^{3} x +288 A \,c^{3} d \,e^{2} x +70 B \,b^{2} c \,e^{3} x -240 B b \,c^{2} d \,e^{2} x +288 B \,c^{3} d^{2} e x +120 A \,b^{2} c \,e^{3}-432 A b \,c^{2} d \,e^{2}+576 A \,c^{3} d^{2} e -105 B \,b^{3} e^{3}+360 B \,b^{2} c d \,e^{2}-432 B b \,c^{2} d^{2} e +192 B \,c^{3} d^{3}\right ) x \left (c x +b \right )}{192 c^{4} \sqrt {x \left (c x +b \right )}}-\frac {\left (40 A \,b^{3} c \,e^{3}-144 A \,b^{2} c^{2} d \,e^{2}+192 A b \,c^{3} d^{2} e -128 A \,c^{4} d^{3}-35 b^{4} B \,e^{3}+120 b^{3} B c d \,e^{2}-144 b^{2} B \,c^{2} d^{2} e +64 B b \,c^{3} d^{3}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {9}{2}}}\) | \(316\) |
| default | \(\frac {A \,d^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{\sqrt {c}}+B \,e^{3} \left (\frac {x^{3} \sqrt {c \,x^{2}+b x}}{4 c}-\frac {7 b \left (\frac {x^{2} \sqrt {c \,x^{2}+b x}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{6 c}\right )}{8 c}\right )+\left (A \,e^{3}+3 B d \,e^{2}\right ) \left (\frac {x^{2} \sqrt {c \,x^{2}+b x}}{3 c}-\frac {5 b \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )}{6 c}\right )+\left (3 A \,d^{2} e +B \,d^{3}\right ) \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )+\left (3 A d \,e^{2}+3 B \,d^{2} e \right ) \left (\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\right )\) | \(417\) |
5/8*(((-A*b^3*c+7/8*b^4*B)*e^3+18/5*c*(A*c-5/6*B*b)*d*b^2*e^2-24/5*c^2*d^2 *(A*c-3/4*B*b)*b*e+16/5*c^3*d^3*(A*c-1/2*B*b))*arctanh((x*(c*x+b))^(1/2)/x /c^(1/2))+(x*(c*x+b))^(1/2)*(4/5*(2/3*(3/4*B*x+A)*x^2*e^3+3*x*d*(2/3*B*x+A )*e^2+6*(1/2*B*x+A)*d^2*e+2*B*d^3)*c^(7/2)+(2*(-1/3*x*(7/10*B*x+A)*e^2-9/5 *d*(5/9*B*x+A)*e-9/5*B*d^2)*c^(5/2)+(((7/12*B*x+A)*e+3*B*d)*c^(3/2)-7/8*B* b*e*c^(1/2))*e*b)*e*b))/c^(9/2)
Time = 0.31 (sec) , antiderivative size = 621, normalized size of antiderivative = 2.04 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\left [\frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 72 \, {\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + 5 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{5}}, \frac {3 \, {\left (64 \, {\left (B b c^{3} - 2 \, A c^{4}\right )} d^{3} - 48 \, {\left (3 \, B b^{2} c^{2} - 4 \, A b c^{3}\right )} d^{2} e + 24 \, {\left (5 \, B b^{3} c - 6 \, A b^{2} c^{2}\right )} d e^{2} - 5 \, {\left (7 \, B b^{4} - 8 \, A b^{3} c\right )} e^{3}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (48 \, B c^{4} e^{3} x^{3} + 192 \, B c^{4} d^{3} - 144 \, {\left (3 \, B b c^{3} - 4 \, A c^{4}\right )} d^{2} e + 72 \, {\left (5 \, B b^{2} c^{2} - 6 \, A b c^{3}\right )} d e^{2} - 15 \, {\left (7 \, B b^{3} c - 8 \, A b^{2} c^{2}\right )} e^{3} + 8 \, {\left (24 \, B c^{4} d e^{2} - {\left (7 \, B b c^{3} - 8 \, A c^{4}\right )} e^{3}\right )} x^{2} + 2 \, {\left (144 \, B c^{4} d^{2} e - 24 \, {\left (5 \, B b c^{3} - 6 \, A c^{4}\right )} d e^{2} + 5 \, {\left (7 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} e^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{5}}\right ] \]
[1/384*(3*(64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*A*b*c^3)*d^2*e + 24*(5*B*b^3*c - 6*A*b^2*c^2)*d*e^2 - 5*(7*B*b^4 - 8*A*b^3*c)*e^3)*sqrt( c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) + 2*(48*B*c^4*e^3*x^3 + 19 2*B*c^4*d^3 - 144*(3*B*b*c^3 - 4*A*c^4)*d^2*e + 72*(5*B*b^2*c^2 - 6*A*b*c^ 3)*d*e^2 - 15*(7*B*b^3*c - 8*A*b^2*c^2)*e^3 + 8*(24*B*c^4*d*e^2 - (7*B*b*c ^3 - 8*A*c^4)*e^3)*x^2 + 2*(144*B*c^4*d^2*e - 24*(5*B*b*c^3 - 6*A*c^4)*d*e ^2 + 5*(7*B*b^2*c^2 - 8*A*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x))/c^5, 1/192*(3* (64*(B*b*c^3 - 2*A*c^4)*d^3 - 48*(3*B*b^2*c^2 - 4*A*b*c^3)*d^2*e + 24*(5*B *b^3*c - 6*A*b^2*c^2)*d*e^2 - 5*(7*B*b^4 - 8*A*b^3*c)*e^3)*sqrt(-c)*arctan (sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) + (48*B*c^4*e^3*x^3 + 192*B*c^4*d^3 - 1 44*(3*B*b*c^3 - 4*A*c^4)*d^2*e + 72*(5*B*b^2*c^2 - 6*A*b*c^3)*d*e^2 - 15*( 7*B*b^3*c - 8*A*b^2*c^2)*e^3 + 8*(24*B*c^4*d*e^2 - (7*B*b*c^3 - 8*A*c^4)*e ^3)*x^2 + 2*(144*B*c^4*d^2*e - 24*(5*B*b*c^3 - 6*A*c^4)*d*e^2 + 5*(7*B*b^2 *c^2 - 8*A*b*c^3)*e^3)*x)*sqrt(c*x^2 + b*x))/c^5]
Time = 0.83 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.70 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\begin {cases} \left (A d^{3} - \frac {b \left (3 A d^{2} e + B d^{3} - \frac {3 b \left (3 A d e^{2} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{6 c}\right )}{4 c}\right )}{2 c}\right ) \left (\begin {cases} \frac {\log {\left (b + 2 \sqrt {c} \sqrt {b x + c x^{2}} + 2 c x \right )}}{\sqrt {c}} & \text {for}\: \frac {b^{2}}{c} \neq 0 \\\frac {\left (\frac {b}{2 c} + x\right ) \log {\left (\frac {b}{2 c} + x \right )}}{\sqrt {c \left (\frac {b}{2 c} + x\right )^{2}}} & \text {otherwise} \end {cases}\right ) + \sqrt {b x + c x^{2}} \left (\frac {B e^{3} x^{3}}{4 c} + \frac {x^{2} \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{3 c} + \frac {x \left (3 A d e^{2} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{6 c}\right )}{2 c} + \frac {3 A d^{2} e + B d^{3} - \frac {3 b \left (3 A d e^{2} + 3 B d^{2} e - \frac {5 b \left (A e^{3} - \frac {7 B b e^{3}}{8 c} + 3 B d e^{2}\right )}{6 c}\right )}{4 c}}{c}\right ) & \text {for}\: c \neq 0 \\\frac {2 \left (A d^{3} \sqrt {b x} + \frac {B e^{3} \left (b x\right )^{\frac {9}{2}}}{9 b^{4}} + \frac {\left (b x\right )^{\frac {3}{2}} \cdot \left (3 A d^{2} e + B d^{3}\right )}{3 b} + \frac {\left (b x\right )^{\frac {5}{2}} \cdot \left (3 A d e^{2} + 3 B d^{2} e\right )}{5 b^{2}} + \frac {\left (b x\right )^{\frac {7}{2}} \left (A e^{3} + 3 B d e^{2}\right )}{7 b^{3}}\right )}{b} & \text {for}\: b \neq 0 \\\tilde {\infty } \left (A d^{3} x + \frac {B e^{3} x^{5}}{5} + \frac {x^{4} \left (A e^{3} + 3 B d e^{2}\right )}{4} + \frac {x^{3} \cdot \left (3 A d e^{2} + 3 B d^{2} e\right )}{3} + \frac {x^{2} \cdot \left (3 A d^{2} e + B d^{3}\right )}{2}\right ) & \text {otherwise} \end {cases} \]
Piecewise(((A*d**3 - b*(3*A*d**2*e + B*d**3 - 3*b*(3*A*d*e**2 + 3*B*d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e**2)/(6*c))/(4*c))/(2*c))*Piece wise((log(b + 2*sqrt(c)*sqrt(b*x + c*x**2) + 2*c*x)/sqrt(c), Ne(b**2/c, 0) ), ((b/(2*c) + x)*log(b/(2*c) + x)/sqrt(c*(b/(2*c) + x)**2), True)) + sqrt (b*x + c*x**2)*(B*e**3*x**3/(4*c) + x**2*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B* d*e**2)/(3*c) + x*(3*A*d*e**2 + 3*B*d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8*c ) + 3*B*d*e**2)/(6*c))/(2*c) + (3*A*d**2*e + B*d**3 - 3*b*(3*A*d*e**2 + 3* B*d**2*e - 5*b*(A*e**3 - 7*B*b*e**3/(8*c) + 3*B*d*e**2)/(6*c))/(4*c))/c), Ne(c, 0)), (2*(A*d**3*sqrt(b*x) + B*e**3*(b*x)**(9/2)/(9*b**4) + (b*x)**(3 /2)*(3*A*d**2*e + B*d**3)/(3*b) + (b*x)**(5/2)*(3*A*d*e**2 + 3*B*d**2*e)/( 5*b**2) + (b*x)**(7/2)*(A*e**3 + 3*B*d*e**2)/(7*b**3))/b, Ne(b, 0)), (zoo* (A*d**3*x + B*e**3*x**5/5 + x**4*(A*e**3 + 3*B*d*e**2)/4 + x**3*(3*A*d*e** 2 + 3*B*d**2*e)/3 + x**2*(3*A*d**2*e + B*d**3)/2), True))
Time = 0.20 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.55 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\frac {\sqrt {c x^{2} + b x} B e^{3} x^{3}}{4 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b e^{3} x^{2}}{24 \, c^{2}} + \frac {35 \, \sqrt {c x^{2} + b x} B b^{2} e^{3} x}{96 \, c^{3}} + \frac {A d^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{\sqrt {c}} + \frac {35 \, B b^{4} e^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {9}{2}}} - \frac {35 \, \sqrt {c x^{2} + b x} B b^{3} e^{3}}{64 \, c^{4}} + \frac {{\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} x^{2}}{3 \, c} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b x}{12 \, c^{2}} + \frac {3 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} x}{2 \, c} - \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} b \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{2 \, c^{\frac {3}{2}}} + \frac {5 \, {\left (3 \, B d e^{2} + A e^{3}\right )} \sqrt {c x^{2} + b x} b^{2}}{8 \, c^{3}} - \frac {9 \, {\left (B d^{2} e + A d e^{2}\right )} \sqrt {c x^{2} + b x} b}{4 \, c^{2}} + \frac {{\left (B d^{3} + 3 \, A d^{2} e\right )} \sqrt {c x^{2} + b x}}{c} \]
1/4*sqrt(c*x^2 + b*x)*B*e^3*x^3/c - 7/24*sqrt(c*x^2 + b*x)*B*b*e^3*x^2/c^2 + 35/96*sqrt(c*x^2 + b*x)*B*b^2*e^3*x/c^3 + A*d^3*log(2*c*x + b + 2*sqrt( c*x^2 + b*x)*sqrt(c))/sqrt(c) + 35/128*B*b^4*e^3*log(2*c*x + b + 2*sqrt(c* x^2 + b*x)*sqrt(c))/c^(9/2) - 35/64*sqrt(c*x^2 + b*x)*B*b^3*e^3/c^4 + 1/3* (3*B*d*e^2 + A*e^3)*sqrt(c*x^2 + b*x)*x^2/c - 5/12*(3*B*d*e^2 + A*e^3)*sqr t(c*x^2 + b*x)*b*x/c^2 + 3/2*(B*d^2*e + A*d*e^2)*sqrt(c*x^2 + b*x)*x/c - 5 /16*(3*B*d*e^2 + A*e^3)*b^3*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c ^(7/2) + 9/8*(B*d^2*e + A*d*e^2)*b^2*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*s qrt(c))/c^(5/2) - 1/2*(B*d^3 + 3*A*d^2*e)*b*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqrt(c))/c^(3/2) + 5/8*(3*B*d*e^2 + A*e^3)*sqrt(c*x^2 + b*x)*b^2/c^3 - 9/4*(B*d^2*e + A*d*e^2)*sqrt(c*x^2 + b*x)*b/c^2 + (B*d^3 + 3*A*d^2*e)*s qrt(c*x^2 + b*x)/c
Time = 0.32 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (\frac {6 \, B e^{3} x}{c} + \frac {24 \, B c^{3} d e^{2} - 7 \, B b c^{2} e^{3} + 8 \, A c^{3} e^{3}}{c^{4}}\right )} x + \frac {144 \, B c^{3} d^{2} e - 120 \, B b c^{2} d e^{2} + 144 \, A c^{3} d e^{2} + 35 \, B b^{2} c e^{3} - 40 \, A b c^{2} e^{3}}{c^{4}}\right )} x + \frac {3 \, {\left (64 \, B c^{3} d^{3} - 144 \, B b c^{2} d^{2} e + 192 \, A c^{3} d^{2} e + 120 \, B b^{2} c d e^{2} - 144 \, A b c^{2} d e^{2} - 35 \, B b^{3} e^{3} + 40 \, A b^{2} c e^{3}\right )}}{c^{4}}\right )} + \frac {{\left (64 \, B b c^{3} d^{3} - 128 \, A c^{4} d^{3} - 144 \, B b^{2} c^{2} d^{2} e + 192 \, A b c^{3} d^{2} e + 120 \, B b^{3} c d e^{2} - 144 \, A b^{2} c^{2} d e^{2} - 35 \, B b^{4} e^{3} + 40 \, A b^{3} c e^{3}\right )} \log \left ({\left | 2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} + b \right |}\right )}{128 \, c^{\frac {9}{2}}} \]
1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*e^3*x/c + (24*B*c^3*d*e^2 - 7*B*b*c^2*e ^3 + 8*A*c^3*e^3)/c^4)*x + (144*B*c^3*d^2*e - 120*B*b*c^2*d*e^2 + 144*A*c^ 3*d*e^2 + 35*B*b^2*c*e^3 - 40*A*b*c^2*e^3)/c^4)*x + 3*(64*B*c^3*d^3 - 144* B*b*c^2*d^2*e + 192*A*c^3*d^2*e + 120*B*b^2*c*d*e^2 - 144*A*b*c^2*d*e^2 - 35*B*b^3*e^3 + 40*A*b^2*c*e^3)/c^4) + 1/128*(64*B*b*c^3*d^3 - 128*A*c^4*d^ 3 - 144*B*b^2*c^2*d^2*e + 192*A*b*c^3*d^2*e + 120*B*b^3*c*d*e^2 - 144*A*b^ 2*c^2*d*e^2 - 35*B*b^4*e^3 + 40*A*b^3*c*e^3)*log(abs(2*(sqrt(c)*x - sqrt(c *x^2 + b*x))*sqrt(c) + b))/c^(9/2)
Timed out. \[ \int \frac {(A+B x) (d+e x)^3}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^3}{\sqrt {c\,x^2+b\,x}} \,d x \]