Integrand size = 26, antiderivative size = 703 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {\left (3 \left (4 A c e \left (128 c^4 d^4-288 b c^3 d^3 e+176 b^2 c^2 d^2 e^2-10 b^3 c d e^3-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 d^4 e+704 b^2 c^3 d^3 e^2-40 b^3 c^2 d^2 e^3-12 b^4 c d e^4-5 b^5 e^5\right )\right )-2 c e \left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+\left (16 c^2 d^2-8 b c d e-3 b^2 e^2\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {b x+c x^2}}{1536 c^3 e^6}+\frac {\left (4 A c e \left (16 c^2 d^2-22 b c d e+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 d^2 e+12 b^2 c d e^2+5 b^3 e^3\right )-2 c e \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c d e-5 b^2 e^2\right )\right ) x\right ) \left (b x+c x^2\right )^{3/2}}{192 c^2 e^4}-\frac {(12 B c d-5 b B e-12 A c e-10 B c e x) \left (b x+c x^2\right )^{5/2}}{60 c e^2}-\frac {\left (4 A c e \left (256 c^5 d^5-640 b c^4 d^4 e+480 b^2 c^3 d^3 e^2-80 b^3 c^2 d^2 e^3-10 b^4 c d e^4-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 d^5 e+1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{7/2} e^7}-\frac {d^{5/2} (B d-A e) (c d-b e)^{5/2} \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{e^7} \]
1/192*(4*A*c*e*(3*b^2*e^2-22*b*c*d*e+16*c^2*d^2)-B*(5*b^3*e^3+12*b^2*c*d*e ^2-88*b*c^2*d^2*e+64*c^3*d^3)-2*c*e*(12*A*c*e*(-b*e+2*c*d)-B*(-5*b^2*e^2-1 2*b*c*d*e+24*c^2*d^2))*x)*(c*x^2+b*x)^(3/2)/c^2/e^4-1/60*(-10*B*c*e*x-12*A *c*e-5*B*b*e+12*B*c*d)*(c*x^2+b*x)^(5/2)/c/e^2-1/512*(4*A*c*e*(-3*b^5*e^5- 10*b^4*c*d*e^4-80*b^3*c^2*d^2*e^3+480*b^2*c^3*d^3*e^2-640*b*c^4*d^4*e+256* c^5*d^5)-B*(-5*b^6*e^6-12*b^5*c*d*e^5-40*b^4*c^2*d^2*e^4-320*b^3*c^3*d^3*e ^3+1920*b^2*c^4*d^4*e^2-2560*b*c^5*d^5*e+1024*c^6*d^6))*arctanh(x*c^(1/2)/ (c*x^2+b*x)^(1/2))/c^(7/2)/e^7-d^(5/2)*(-A*e+B*d)*(-b*e+c*d)^(5/2)*arctanh (1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/e^7+ 1/1536*(12*A*c*e*(-3*b^4*e^4-10*b^3*c*d*e^3+176*b^2*c^2*d^2*e^2-288*b*c^3* d^3*e+128*c^4*d^4)-3*B*(-5*b^5*e^5-12*b^4*c*d*e^4-40*b^3*c^2*d^2*e^3+704*b ^2*c^3*d^3*e^2-1152*b*c^4*d^4*e+512*c^5*d^5)-2*c*e*(8*b*c*d*e*(-b*e+2*c*d) *(-12*A*c*e-5*B*b*e+12*B*c*d)+(-3*b^2*e^2-8*b*c*d*e+16*c^2*d^2)*(12*A*c*e* (-b*e+2*c*d)-B*(-5*b^2*e^2-12*b*c*d*e+24*c^2*d^2)))*x)*(c*x^2+b*x)^(1/2)/c ^3/e^6
Result contains complex when optimal does not.
Time = 9.93 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\frac {(x (b+c x))^{5/2} \left (\sqrt {c} e \sqrt {x} \sqrt {b+c x} \left (-4 A c e \left (45 b^4 e^4-30 b^3 c e^3 (-5 d+e x)-4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (270 d^3-130 d^2 e x+85 d e^2 x^2-63 e^3 x^3\right )-32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+B \left (75 b^5 e^5+10 b^4 c e^4 (18 d-5 e x)+40 b^3 c^2 e^3 \left (15 d^2-3 d e x+e^2 x^2\right )+16 b^2 c^3 e^2 \left (-660 d^3+295 d^2 e x-186 d e^2 x^2+135 e^3 x^3\right )+64 b c^4 e \left (270 d^4-130 d^3 e x+85 d^2 e^2 x^2-63 d e^3 x^3+50 e^4 x^4\right )-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )\right )+15360 c^{5/2} d^{3/2} (B d-A e) (c d-b e)^2 \left (c d-b e-i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e-2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+15360 c^{5/2} d^{3/2} (B d-A e) (c d-b e)^2 \left (c d-b e+i \sqrt {b} \sqrt {e} \sqrt {c d-b e}\right ) \sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \arctan \left (\frac {\sqrt {-c d+2 b e+2 i \sqrt {b} \sqrt {e} \sqrt {c d-b e}} \sqrt {x}}{\sqrt {d} \left (-\sqrt {b}+\sqrt {b+c x}\right )}\right )+30 \left (4 A c e \left (-256 c^5 d^5+640 b c^4 d^4 e-480 b^2 c^3 d^3 e^2+80 b^3 c^2 d^2 e^3+10 b^4 c d e^4+3 b^5 e^5\right )+B \left (1024 c^6 d^6-2560 b c^5 d^5 e+1920 b^2 c^4 d^4 e^2-320 b^3 c^3 d^3 e^3-40 b^4 c^2 d^2 e^4-12 b^5 c d e^5-5 b^6 e^6\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {x}}{-\sqrt {b}+\sqrt {b+c x}}\right )\right )}{7680 c^{7/2} e^7 x^{5/2} (b+c x)^{5/2}} \]
((x*(b + c*x))^(5/2)*(Sqrt[c]*e*Sqrt[x]*Sqrt[b + c*x]*(-4*A*c*e*(45*b^4*e^ 4 - 30*b^3*c*e^3*(-5*d + e*x) - 4*b^2*c^2*e^2*(660*d^2 - 295*d*e*x + 186*e ^2*x^2) + 16*b*c^3*e*(270*d^3 - 130*d^2*e*x + 85*d*e^2*x^2 - 63*e^3*x^3) - 32*c^4*(60*d^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4) ) + B*(75*b^5*e^5 + 10*b^4*c*e^4*(18*d - 5*e*x) + 40*b^3*c^2*e^3*(15*d^2 - 3*d*e*x + e^2*x^2) + 16*b^2*c^3*e^2*(-660*d^3 + 295*d^2*e*x - 186*d*e^2*x ^2 + 135*e^3*x^3) + 64*b*c^4*e*(270*d^4 - 130*d^3*e*x + 85*d^2*e^2*x^2 - 6 3*d*e^3*x^3 + 50*e^4*x^4) - 128*c^5*(60*d^5 - 30*d^4*e*x + 20*d^3*e^2*x^2 - 15*d^2*e^3*x^3 + 12*d*e^4*x^4 - 10*e^5*x^5))) + 15360*c^(5/2)*d^(3/2)*(B *d - A*e)*(c*d - b*e)^2*(c*d - b*e - I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sq rt[-(c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-( c*d) + 2*b*e - (2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(- Sqrt[b] + Sqrt[b + c*x]))] + 15360*c^(5/2)*d^(3/2)*(B*d - A*e)*(c*d - b*e) ^2*(c*d - b*e + I*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e])*Sqrt[-(c*d) + 2*b*e + ( 2*I)*Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*ArcTan[(Sqrt[-(c*d) + 2*b*e + (2*I)* Sqrt[b]*Sqrt[e]*Sqrt[c*d - b*e]]*Sqrt[x])/(Sqrt[d]*(-Sqrt[b] + Sqrt[b + c* x]))] + 30*(4*A*c*e*(-256*c^5*d^5 + 640*b*c^4*d^4*e - 480*b^2*c^3*d^3*e^2 + 80*b^3*c^2*d^2*e^3 + 10*b^4*c*d*e^4 + 3*b^5*e^5) + B*(1024*c^6*d^6 - 256 0*b*c^5*d^5*e + 1920*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 - 40*b^4*c^2*d^ 2*e^4 - 12*b^5*c*d*e^5 - 5*b^6*e^6))*ArcTanh[(Sqrt[c]*Sqrt[x])/(-Sqrt[b...
Time = 1.53 (sec) , antiderivative size = 740, normalized size of antiderivative = 1.05, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {1231, 27, 1231, 27, 1231, 27, 1269, 1091, 219, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle -\frac {\int -\frac {\left (b d (12 B c d-5 b B e-12 A c e)-\left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c e d-5 b^2 e^2\right )\right ) x\right ) \left (c x^2+b x\right )^{3/2}}{2 (d+e x)}dx}{12 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\left (b d (12 B c d-5 b B e-12 A c e)-\left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c e d-5 b^2 e^2\right )\right ) x\right ) \left (c x^2+b x\right )^{3/2}}{d+e x}dx}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\int \frac {\left (3 b d \left (4 A c e \left (16 c^2 d^2-22 b c e d+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 e d^2+12 b^2 c e^2 d+5 b^3 e^3\right )\right )+\left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c e d-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {c x^2+b x}}{2 (d+e x)}dx}{8 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\int \frac {\left (3 b d \left (4 A c e \left (16 c^2 d^2-22 b c e d+3 b^2 e^2\right )-B \left (64 c^3 d^3-88 b c^2 e d^2+12 b^2 c e^2 d+5 b^3 e^3\right )\right )+\left (8 b c d e (2 c d-b e) (12 B c d-5 b B e-12 A c e)+2 \left (8 c^2 d^2-4 b c e d-\frac {3 b^2 e^2}{2}\right ) \left (12 A c e (2 c d-b e)-B \left (24 c^2 d^2-12 b c e d-5 b^2 e^2\right )\right )\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {-\frac {\int -\frac {3 \left (b d \left (4 A c e \left (128 c^4 d^4-288 b c^3 e d^3+176 b^2 c^2 e^2 d^2-10 b^3 c e^3 d-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 e d^4+704 b^2 c^3 e^2 d^3-40 b^3 c^2 e^3 d^2-12 b^4 c e^4 d-5 b^5 e^5\right )\right )+\left (4 A c e \left (256 c^5 d^5-640 b c^4 e d^4+480 b^2 c^3 e^2 d^3-80 b^3 c^2 e^3 d^2-10 b^4 c e^4 d-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 e d^5+1920 b^2 c^4 e^2 d^4-320 b^3 c^3 e^3 d^3-40 b^4 c^2 e^4 d^2-12 b^5 c e^5 d-5 b^6 e^6\right )\right ) x\right )}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\frac {3 \int \frac {b d \left (4 A c e \left (128 c^4 d^4-288 b c^3 e d^3+176 b^2 c^2 e^2 d^2-10 b^3 c e^3 d-3 b^4 e^4\right )-B \left (512 c^5 d^5-1152 b c^4 e d^4+704 b^2 c^3 e^2 d^3-40 b^3 c^2 e^3 d^2-12 b^4 c e^4 d-5 b^5 e^5\right )\right )+\left (4 A c e \left (256 c^5 d^5-640 b c^4 e d^4+480 b^2 c^3 e^2 d^3-80 b^3 c^2 e^3 d^2-10 b^4 c e^4 d-3 b^5 e^5\right )-B \left (1024 c^6 d^6-2560 b c^5 e d^5+1920 b^2 c^4 e^2 d^4-320 b^3 c^3 e^3 d^3-40 b^4 c^2 e^4 d^2-12 b^5 c e^5 d-5 b^6 e^6\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\frac {3 \left (\frac {\left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}+\frac {1024 c^3 d^3 (B d-A e) (c d-b e)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}\right )}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\frac {3 \left (\frac {2 \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}+\frac {1024 c^3 d^3 (B d-A e) (c d-b e)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}\right )}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\frac {3 \left (\frac {1024 c^3 d^3 (B d-A e) (c d-b e)^3 \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{\sqrt {c} e}\right )}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\frac {3 \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{\sqrt {c} e}-\frac {2048 c^3 d^3 (B d-A e) (c d-b e)^3 \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}\right )}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\left (b x+c x^2\right )^{3/2} \left (-2 c e x \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+4 A c e \left (3 b^2 e^2-22 b c d e+16 c^2 d^2\right )-B \left (5 b^3 e^3+12 b^2 c d e^2-88 b c^2 d^2 e+64 c^3 d^3\right )\right )}{8 c e^2}-\frac {\frac {3 \left (\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (4 A c e \left (-3 b^5 e^5-10 b^4 c d e^4-80 b^3 c^2 d^2 e^3+480 b^2 c^3 d^3 e^2-640 b c^4 d^4 e+256 c^5 d^5\right )-B \left (-5 b^6 e^6-12 b^5 c d e^5-40 b^4 c^2 d^2 e^4-320 b^3 c^3 d^3 e^3+1920 b^2 c^4 d^4 e^2-2560 b c^5 d^5 e+1024 c^6 d^6\right )\right )}{\sqrt {c} e}+\frac {1024 c^3 d^{5/2} (B d-A e) (c d-b e)^{5/2} \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}\right )}{8 c e^2}-\frac {\sqrt {b x+c x^2} \left (3 \left (4 A c e \left (-3 b^4 e^4-10 b^3 c d e^3+176 b^2 c^2 d^2 e^2-288 b c^3 d^3 e+128 c^4 d^4\right )-B \left (-5 b^5 e^5-12 b^4 c d e^4-40 b^3 c^2 d^2 e^3+704 b^2 c^3 d^3 e^2-1152 b c^4 d^4 e+512 c^5 d^5\right )\right )-2 c e x \left (2 \left (-\frac {3 b^2 e^2}{2}-4 b c d e+8 c^2 d^2\right ) \left (12 A c e (2 c d-b e)-B \left (-5 b^2 e^2-12 b c d e+24 c^2 d^2\right )\right )+8 b c d e (2 c d-b e) (-12 A c e-5 b B e+12 B c d)\right )\right )}{4 c e^2}}{16 c e^2}}{24 c e^2}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c e-5 b B e+12 B c d-10 B c e x)}{60 c e^2}\) |
-1/60*((12*B*c*d - 5*b*B*e - 12*A*c*e - 10*B*c*e*x)*(b*x + c*x^2)^(5/2))/( c*e^2) + (((4*A*c*e*(16*c^2*d^2 - 22*b*c*d*e + 3*b^2*e^2) - B*(64*c^3*d^3 - 88*b*c^2*d^2*e + 12*b^2*c*d*e^2 + 5*b^3*e^3) - 2*c*e*(12*A*c*e*(2*c*d - b*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2))*x)*(b*x + c*x^2)^(3/2))/(8 *c*e^2) - (-1/4*((3*(4*A*c*e*(128*c^4*d^4 - 288*b*c^3*d^3*e + 176*b^2*c^2* d^2*e^2 - 10*b^3*c*d*e^3 - 3*b^4*e^4) - B*(512*c^5*d^5 - 1152*b*c^4*d^4*e + 704*b^2*c^3*d^3*e^2 - 40*b^3*c^2*d^2*e^3 - 12*b^4*c*d*e^4 - 5*b^5*e^5)) - 2*c*e*(8*b*c*d*e*(2*c*d - b*e)*(12*B*c*d - 5*b*B*e - 12*A*c*e) + 2*(8*c^ 2*d^2 - 4*b*c*d*e - (3*b^2*e^2)/2)*(12*A*c*e*(2*c*d - b*e) - B*(24*c^2*d^2 - 12*b*c*d*e - 5*b^2*e^2)))*x)*Sqrt[b*x + c*x^2])/(c*e^2) + (3*((2*(4*A*c *e*(256*c^5*d^5 - 640*b*c^4*d^4*e + 480*b^2*c^3*d^3*e^2 - 80*b^3*c^2*d^2*e ^3 - 10*b^4*c*d*e^4 - 3*b^5*e^5) - B*(1024*c^6*d^6 - 2560*b*c^5*d^5*e + 19 20*b^2*c^4*d^4*e^2 - 320*b^3*c^3*d^3*e^3 - 40*b^4*c^2*d^2*e^4 - 12*b^5*c*d *e^5 - 5*b^6*e^6))*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e) + ( 1024*c^3*d^(5/2)*(B*d - A*e)*(c*d - b*e)^(5/2)*ArcTanh[(b*d + (2*c*d - b*e )*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/e))/(8*c*e^2))/(16*c* e^2))/(24*c*e^2)
3.12.86.3.1 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/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[Int[1/(1 - c*x^2), x], x, x/Sqrt[b*x + c*x^2]], x] /; FreeQ[{b, c}, x]
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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[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[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.34 (sec) , antiderivative size = 927, normalized size of antiderivative = 1.32
| method | result | size |
| risch | \(-\frac {\left (-1280 B \,c^{5} e^{5} x^{5}-1536 A \,c^{5} e^{5} x^{4}-3200 B b \,c^{4} e^{5} x^{4}+1536 B \,c^{5} d \,e^{4} x^{4}-4032 A b \,c^{4} e^{5} x^{3}+1920 A \,c^{5} d \,e^{4} x^{3}-2160 B \,b^{2} c^{3} e^{5} x^{3}+4032 B b \,c^{4} d \,e^{4} x^{3}-1920 B \,c^{5} d^{2} e^{3} x^{3}-2976 A \,b^{2} c^{3} e^{5} x^{2}+5440 A b \,c^{4} d \,e^{4} x^{2}-2560 A \,c^{5} d^{2} e^{3} x^{2}-40 B \,b^{3} c^{2} e^{5} x^{2}+2976 B \,b^{2} c^{3} d \,e^{4} x^{2}-5440 B b \,c^{4} d^{2} e^{3} x^{2}+2560 B \,c^{5} d^{3} e^{2} x^{2}-120 A \,b^{3} c^{2} e^{5} x +4720 A \,b^{2} c^{3} d \,e^{4} x -8320 A b \,c^{4} d^{2} e^{3} x +3840 A \,c^{5} d^{3} e^{2} x +50 B \,b^{4} c \,e^{5} x +120 B \,b^{3} c^{2} d \,e^{4} x -4720 B \,b^{2} c^{3} d^{2} e^{3} x +8320 B b \,c^{4} d^{3} e^{2} x -3840 B \,c^{5} d^{4} e x +180 A \,b^{4} c \,e^{5}+600 A \,b^{3} c^{2} d \,e^{4}-10560 A \,b^{2} c^{3} d^{2} e^{3}+17280 A b \,c^{4} d^{3} e^{2}-7680 A \,c^{5} d^{4} e -75 B \,b^{5} e^{5}-180 B \,b^{4} c d \,e^{4}-600 B \,b^{3} c^{2} d^{2} e^{3}+10560 B \,b^{2} c^{3} d^{3} e^{2}-17280 B b \,c^{4} d^{4} e +7680 B \,c^{5} d^{5}\right ) x \left (c x +b \right )}{7680 c^{3} e^{6} \sqrt {x \left (c x +b \right )}}+\frac {\frac {1024 d^{3} \left (A \,b^{3} e^{4}-3 A \,b^{2} c d \,e^{3}+3 A b \,c^{2} d^{2} e^{2}-A \,c^{3} d^{3} e -B \,b^{3} d \,e^{3}+3 B \,b^{2} c \,d^{2} e^{2}-3 B b \,c^{2} d^{3} e +B \,c^{3} d^{4}\right ) c^{3} \ln \left (\frac {-\frac {2 d \left (b e -c d \right )}{e^{2}}+\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c +\frac {\left (b e -2 c d \right ) \left (x +\frac {d}{e}\right )}{e}-\frac {d \left (b e -c d \right )}{e^{2}}}}{x +\frac {d}{e}}\right )}{e^{2} \sqrt {-\frac {d \left (b e -c d \right )}{e^{2}}}}+\frac {\left (12 A \,b^{5} c \,e^{6}+40 A \,b^{4} c^{2} d \,e^{5}+320 A \,b^{3} c^{3} d^{2} e^{4}-1920 A \,b^{2} c^{4} d^{3} e^{3}+2560 A b \,c^{5} d^{4} e^{2}-1024 A \,c^{6} d^{5} e -5 B \,b^{6} e^{6}-12 B \,b^{5} c d \,e^{5}-40 B \,b^{4} c^{2} d^{2} e^{4}-320 B \,b^{3} c^{3} d^{3} e^{3}+1920 B \,b^{2} c^{4} d^{4} e^{2}-2560 B b \,c^{5} d^{5} e +1024 B \,c^{6} d^{6}\right ) \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{e \sqrt {c}}}{1024 e^{6} c^{3}}\) | \(927\) |
| default | \(\text {Expression too large to display}\) | \(1057\) |
-1/7680/c^3*(-1280*B*c^5*e^5*x^5-1536*A*c^5*e^5*x^4-3200*B*b*c^4*e^5*x^4+1 536*B*c^5*d*e^4*x^4-4032*A*b*c^4*e^5*x^3+1920*A*c^5*d*e^4*x^3-2160*B*b^2*c ^3*e^5*x^3+4032*B*b*c^4*d*e^4*x^3-1920*B*c^5*d^2*e^3*x^3-2976*A*b^2*c^3*e^ 5*x^2+5440*A*b*c^4*d*e^4*x^2-2560*A*c^5*d^2*e^3*x^2-40*B*b^3*c^2*e^5*x^2+2 976*B*b^2*c^3*d*e^4*x^2-5440*B*b*c^4*d^2*e^3*x^2+2560*B*c^5*d^3*e^2*x^2-12 0*A*b^3*c^2*e^5*x+4720*A*b^2*c^3*d*e^4*x-8320*A*b*c^4*d^2*e^3*x+3840*A*c^5 *d^3*e^2*x+50*B*b^4*c*e^5*x+120*B*b^3*c^2*d*e^4*x-4720*B*b^2*c^3*d^2*e^3*x +8320*B*b*c^4*d^3*e^2*x-3840*B*c^5*d^4*e*x+180*A*b^4*c*e^5+600*A*b^3*c^2*d *e^4-10560*A*b^2*c^3*d^2*e^3+17280*A*b*c^4*d^3*e^2-7680*A*c^5*d^4*e-75*B*b ^5*e^5-180*B*b^4*c*d*e^4-600*B*b^3*c^2*d^2*e^3+10560*B*b^2*c^3*d^3*e^2-172 80*B*b*c^4*d^4*e+7680*B*c^5*d^5)*x*(c*x+b)/e^6/(x*(c*x+b))^(1/2)+1/1024/e^ 6/c^3*(1024*d^3*(A*b^3*e^4-3*A*b^2*c*d*e^3+3*A*b*c^2*d^2*e^2-A*c^3*d^3*e-B *b^3*d*e^3+3*B*b^2*c*d^2*e^2-3*B*b*c^2*d^3*e+B*c^3*d^4)*c^3/e^2/(-d*(b*e-c *d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d )/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x +d/e))+(12*A*b^5*c*e^6+40*A*b^4*c^2*d*e^5+320*A*b^3*c^3*d^2*e^4-1920*A*b^2 *c^4*d^3*e^3+2560*A*b*c^5*d^4*e^2-1024*A*c^6*d^5*e-5*B*b^6*e^6-12*B*b^5*c* d*e^5-40*B*b^4*c^2*d^2*e^4-320*B*b^3*c^3*d^3*e^3+1920*B*b^2*c^4*d^4*e^2-25 60*B*b*c^5*d^5*e+1024*B*c^6*d^6)/e*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x)^(1/2 ))/c^(1/2))
Time = 165.05 (sec) , antiderivative size = 3059, normalized size of antiderivative = 4.35 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Too large to display} \]
[1/15360*(15*(1024*B*c^6*d^6 - 512*(5*B*b*c^5 + 2*A*c^6)*d^5*e + 640*(3*B* b^2*c^4 + 4*A*b*c^5)*d^4*e^2 - 320*(B*b^3*c^3 + 6*A*b^2*c^4)*d^3*e^3 - 40* (B*b^4*c^2 - 8*A*b^3*c^3)*d^2*e^4 - 4*(3*B*b^5*c - 10*A*b^4*c^2)*d*e^5 - ( 5*B*b^6 - 12*A*b^5*c)*e^6)*sqrt(c)*log(2*c*x + b + 2*sqrt(c*x^2 + b*x)*sqr t(c)) + 15360*(B*c^6*d^5 - A*b^2*c^4*d^2*e^3 - (2*B*b*c^5 + A*c^6)*d^4*e + (B*b^2*c^4 + 2*A*b*c^5)*d^3*e^2)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x - 2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(1280*B*c ^6*e^6*x^5 - 7680*B*c^6*d^5*e + 1920*(9*B*b*c^5 + 4*A*c^6)*d^4*e^2 - 960*( 11*B*b^2*c^4 + 18*A*b*c^5)*d^3*e^3 + 120*(5*B*b^3*c^3 + 88*A*b^2*c^4)*d^2* e^4 + 60*(3*B*b^4*c^2 - 10*A*b^3*c^3)*d*e^5 + 15*(5*B*b^5*c - 12*A*b^4*c^2 )*e^6 - 128*(12*B*c^6*d*e^5 - (25*B*b*c^5 + 12*A*c^6)*e^6)*x^4 + 48*(40*B* c^6*d^2*e^4 - 4*(21*B*b*c^5 + 10*A*c^6)*d*e^5 + 3*(15*B*b^2*c^4 + 28*A*b*c ^5)*e^6)*x^3 - 8*(320*B*c^6*d^3*e^3 - 40*(17*B*b*c^5 + 8*A*c^6)*d^2*e^4 + 4*(93*B*b^2*c^4 + 170*A*b*c^5)*d*e^5 - (5*B*b^3*c^3 + 372*A*b^2*c^4)*e^6)* x^2 + 10*(384*B*c^6*d^4*e^2 - 64*(13*B*b*c^5 + 6*A*c^6)*d^3*e^3 + 8*(59*B* b^2*c^4 + 104*A*b*c^5)*d^2*e^4 - 4*(3*B*b^3*c^3 + 118*A*b^2*c^4)*d*e^5 - ( 5*B*b^4*c^2 - 12*A*b^3*c^3)*e^6)*x)*sqrt(c*x^2 + b*x))/(c^4*e^7), -1/15360 *(30720*(B*c^6*d^5 - A*b^2*c^4*d^2*e^3 - (2*B*b*c^5 + A*c^6)*d^4*e + (B*b^ 2*c^4 + 2*A*b*c^5)*d^3*e^2)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d *e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) - 15*(1024*B*c^6*d^6 - 512*(5*B*...
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{d + e x}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{d+e x} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{d+e\,x} \,d x \]