Integrand size = 26, antiderivative size = 508 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {5 \left (8 A c e \left (16 c^2 d^2-20 b c d e+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 d^2 e+88 b^2 c d e^2-b^3 e^3\right )-2 c e \left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c d e+b^2 e^2\right )\right ) x\right ) \sqrt {b x+c x^2}}{64 c e^6}-\frac {5 (B d (24 c d-13 b e)-2 A e (8 c d-3 b e)+e (6 B c d-b B e-4 A c e) x) \left (b x+c x^2\right )^{3/2}}{24 e^4 (d+e x)}+\frac {(3 B d-2 A e+B e x) \left (b x+c x^2\right )^{5/2}}{4 e^2 (d+e x)^2}-\frac {5 \left (8 A c e \left (32 c^3 d^3-48 b c^2 d^2 e+18 b^2 c d e^2-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 d^3 e+288 b^2 c^2 d^2 e^2-24 b^3 c d e^3-b^4 e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{3/2} e^7}+\frac {5 \sqrt {d} \sqrt {c d-b e} \left (A e \left (16 c^2 d^2-16 b c d e+3 b^2 e^2\right )-B d \left (24 c^2 d^2-28 b c d e+7 b^2 e^2\right )\right ) \text {arctanh}\left (\frac {b d+(2 c d-b e) x}{2 \sqrt {d} \sqrt {c d-b e} \sqrt {b x+c x^2}}\right )}{8 e^7} \]
-5/24*(B*d*(-13*b*e+24*c*d)-2*A*e*(-3*b*e+8*c*d)+e*(-4*A*c*e-B*b*e+6*B*c*d )*x)*(c*x^2+b*x)^(3/2)/e^4/(e*x+d)+1/4*(B*e*x-2*A*e+3*B*d)*(c*x^2+b*x)^(5/ 2)/e^2/(e*x+d)^2-5/64*(8*A*c*e*(-b^3*e^3+18*b^2*c*d*e^2-48*b*c^2*d^2*e+32* c^3*d^3)-B*(-b^4*e^4-24*b^3*c*d*e^3+288*b^2*c^2*d^2*e^2-640*b*c^3*d^3*e+38 4*c^4*d^4))*arctanh(x*c^(1/2)/(c*x^2+b*x)^(1/2))/c^(3/2)/e^7+5/8*(A*e*(3*b ^2*e^2-16*b*c*d*e+16*c^2*d^2)-B*d*(7*b^2*e^2-28*b*c*d*e+24*c^2*d^2))*arcta nh(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))*d^ (1/2)*(-b*e+c*d)^(1/2)/e^7+5/64*(8*A*c*e*(5*b^2*e^2-20*b*c*d*e+16*c^2*d^2) -B*(-b^3*e^3+88*b^2*c*d*e^2-272*b*c^2*d^2*e+192*c^3*d^3)-2*c*e*(16*A*c*e*( -b*e+2*c*d)-B*(b^2*e^2-32*b*c*d*e+48*c^2*d^2))*x)*(c*x^2+b*x)^(1/2)/c/e^6
Time = 14.49 (sec) , antiderivative size = 971, normalized size of antiderivative = 1.91 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\frac {(x (b+c x))^{5/2} \left (\frac {(-B d+A e) x^{7/2} (b+c x)}{(d+e x)^2}+\frac {(B d (10 c d-7 b e)+3 A e (-2 c d+b e)) x^{7/2} (b+c x)}{2 d (c d-b e) (d+e x)}+\frac {-e \left (-8 A c^2 d^2+5 b^2 e (7 B d-3 A e)+4 b c d (-14 B d+11 A e)\right ) \left (15 \left (-256 c^5 d^5+640 b c^4 d^4 e-480 b^2 c^3 d^3 e^2+80 b^3 c^2 d^2 e^3+10 b^4 c d e^4+3 b^5 e^5\right ) (b+c x) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {b} \sqrt {c} \sqrt {1+\frac {c x}{b}} \left (e \sqrt {x} (b+c x) \left (-45 b^4 e^4+30 b^3 c e^3 (-5 d+e x)+4 b^2 c^2 e^2 \left (660 d^2-295 d e x+186 e^2 x^2\right )+16 b c^3 e \left (-270 d^3+130 d^2 e x-85 d e^2 x^2+63 e^3 x^3\right )+32 c^4 \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )+3840 c^2 d^{5/2} (c d-b e)^{5/2} \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )+3 (B d (10 c d-7 b e)+3 A e (-2 c d+b e)) \left (-15 \left (-1024 c^6 d^6+2560 b c^5 d^5 e-1920 b^2 c^4 d^4 e^2+320 b^3 c^3 d^3 e^3+40 b^4 c^2 d^2 e^4+12 b^5 c d e^5+5 b^6 e^6\right ) (b+c x) \text {arcsinh}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )+\sqrt {b} \sqrt {c} \sqrt {1+\frac {c x}{b}} \left (e \sqrt {x} (b+c x) \left (75 b^5 e^5+10 b^4 c e^4 (18 d-5 e x)+40 b^3 c^2 e^3 \left (15 d^2-3 d e x+e^2 x^2\right )+16 b^2 c^3 e^2 \left (-660 d^3+295 d^2 e x-186 d e^2 x^2+135 e^3 x^3\right )+64 b c^4 e \left (270 d^4-130 d^3 e x+85 d^2 e^2 x^2-63 d e^3 x^3+50 e^4 x^4\right )-128 c^5 \left (60 d^5-30 d^4 e x+20 d^3 e^2 x^2-15 d^2 e^3 x^3+12 d e^4 x^4-10 e^5 x^5\right )\right )-15360 c^3 d^{7/2} (c d-b e)^{5/2} \sqrt {b+c x} \text {arctanh}\left (\frac {\sqrt {c d-b e} \sqrt {x}}{\sqrt {d} \sqrt {b+c x}}\right )\right )\right )}{7680 \sqrt {b} c^{5/2} d e^7 (-c d+b e) (b+c x)^3 \sqrt {1+\frac {c x}{b}}}\right )}{2 d (-c d+b e) x^{5/2}} \]
((x*(b + c*x))^(5/2)*(((-(B*d) + A*e)*x^(7/2)*(b + c*x))/(d + e*x)^2 + ((B *d*(10*c*d - 7*b*e) + 3*A*e*(-2*c*d + b*e))*x^(7/2)*(b + c*x))/(2*d*(c*d - b*e)*(d + e*x)) + (-(e*(-8*A*c^2*d^2 + 5*b^2*e*(7*B*d - 3*A*e) + 4*b*c*d* (-14*B*d + 11*A*e))*(15*(-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 + c*x)*ArcSinh[( Sqrt[c]*Sqrt[x])/Sqrt[b]] + Sqrt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*(e*Sqrt[x]*( b + c*x)*(-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)) + 3840*c^2*d^(5/2)*(c*d - b*e)^(5/2)*Sqrt[b + c*x]* ArcTanh[(Sqrt[c*d - b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])]))) + 3*(B*d*(10 *c*d - 7*b*e) + 3*A*e*(-2*c*d + b*e))*(-15*(-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)*(b + c*x)*ArcSinh[(Sqrt[c]*Sqrt[x])/Sqrt[b]] + Sq rt[b]*Sqrt[c]*Sqrt[1 + (c*x)/b]*(e*Sqrt[x]*(b + c*x)*(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 - 63*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^3*d^(7/2)*(c*d - b*e)^(5/2)*Sqrt[b + c*x]...
Time = 1.04 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.05, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {1230, 27, 1230, 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.
| \(\displaystyle \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \int \frac {2 (b (3 B d-2 A e)+(6 B c d-b B e-4 A c e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^2}dx}{16 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \int \frac {(b (3 B d-2 A e)+(6 B c d-b B e-4 A c e) x) \left (c x^2+b x\right )^{3/2}}{(d+e x)^2}dx}{8 e^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\int \frac {\left (b (B d (24 c d-13 b e)-2 A e (8 c d-3 b e))-\left (16 A c e (2 c d-b e)-B \left (48 c^2 d^2-32 b c e d+b^2 e^2\right )\right ) x\right ) \sqrt {c x^2+b x}}{d+e x}dx}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\int \frac {b d \left (8 A c e \left (16 c^2 d^2-20 b c e d+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 e d^2+88 b^2 c e^2 d-b^3 e^3\right )\right )+\left (8 A c e \left (32 c^3 d^3-48 b c^2 e d^2+18 b^2 c e^2 d-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 e d^3+288 b^2 c^2 e^2 d^2-24 b^3 c e^3 d-b^4 e^4\right )\right ) x}{2 (d+e x) \sqrt {c x^2+b x}}dx}{4 c e^2}}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\int \frac {b d \left (8 A c e \left (16 c^2 d^2-20 b c e d+5 b^2 e^2\right )-B \left (192 c^3 d^3-272 b c^2 e d^2+88 b^2 c e^2 d-b^3 e^3\right )\right )+\left (8 A c e \left (32 c^3 d^3-48 b c^2 e d^2+18 b^2 c e^2 d-b^3 e^3\right )-B \left (384 c^4 d^4-640 b c^3 e d^3+288 b^2 c^2 e^2 d^2-24 b^3 c e^3 d-b^4 e^4\right )\right ) x}{(d+e x) \sqrt {c x^2+b x}}dx}{8 c e^2}}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {\left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right ) \int \frac {1}{\sqrt {c x^2+b x}}dx}{e}-\frac {16 c d (c d-b e) \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 1091 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right ) \int \frac {1}{1-\frac {c x^2}{c x^2+b x}}d\frac {x}{\sqrt {c x^2+b x}}}{e}-\frac {16 c d (c d-b e) \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{\sqrt {c} e}-\frac {16 c d (c d-b e) \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \int \frac {1}{(d+e x) \sqrt {c x^2+b x}}dx}{e}}{8 c e^2}}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {32 c d (c d-b e) \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \int \frac {1}{4 d (c d-b e)-\frac {(b d+(2 c d-b e) x)^2}{c x^2+b x}}d\left (-\frac {b d+(2 c d-b e) x}{\sqrt {c x^2+b x}}\right )}{e}+\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{\sqrt {c} e}}{8 c e^2}}{2 e^2}\right )}{8 e^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\left (b x+c x^2\right )^{5/2} (-2 A e+3 B d+B e x)}{4 e^2 (d+e x)^2}-\frac {5 \left (\frac {\left (b x+c x^2\right )^{3/2} (e x (-4 A c e-b B e+6 B c d)-2 A e (8 c d-3 b e)+B d (24 c d-13 b e))}{3 e^2 (d+e x)}-\frac {\frac {\sqrt {b x+c x^2} \left (-2 c e x \left (16 A c e (2 c d-b e)-B \left (b^2 e^2-32 b c d e+48 c^2 d^2\right )\right )+8 A c e \left (5 b^2 e^2-20 b c d e+16 c^2 d^2\right )-2 B \left (-\frac {b^3 e^3}{2}+44 b^2 c d e^2-136 b c^2 d^2 e+96 c^3 d^3\right )\right )}{4 c e^2}-\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \left (8 A c e \left (-b^3 e^3+18 b^2 c d e^2-48 b c^2 d^2 e+32 c^3 d^3\right )-B \left (-b^4 e^4-24 b^3 c d e^3+288 b^2 c^2 d^2 e^2-640 b c^3 d^3 e+384 c^4 d^4\right )\right )}{\sqrt {c} e}-\frac {16 c \sqrt {d} \sqrt {c d-b e} \left (A e \left (3 b^2 e^2-16 b c d e+16 c^2 d^2\right )-B d \left (7 b^2 e^2-28 b c d e+24 c^2 d^2\right )\right ) \text {arctanh}\left (\frac {x (2 c d-b e)+b d}{2 \sqrt {d} \sqrt {b x+c x^2} \sqrt {c d-b e}}\right )}{e}}{8 c e^2}}{2 e^2}\right )}{8 e^2}\) |
((3*B*d - 2*A*e + B*e*x)*(b*x + c*x^2)^(5/2))/(4*e^2*(d + e*x)^2) - (5*((( B*d*(24*c*d - 13*b*e) - 2*A*e*(8*c*d - 3*b*e) + e*(6*B*c*d - b*B*e - 4*A*c *e)*x)*(b*x + c*x^2)^(3/2))/(3*e^2*(d + e*x)) - (((8*A*c*e*(16*c^2*d^2 - 2 0*b*c*d*e + 5*b^2*e^2) - 2*B*(96*c^3*d^3 - 136*b*c^2*d^2*e + 44*b^2*c*d*e^ 2 - (b^3*e^3)/2) - 2*c*e*(16*A*c*e*(2*c*d - b*e) - B*(48*c^2*d^2 - 32*b*c* d*e + b^2*e^2))*x)*Sqrt[b*x + c*x^2])/(4*c*e^2) - ((2*(8*A*c*e*(32*c^3*d^3 - 48*b*c^2*d^2*e + 18*b^2*c*d*e^2 - b^3*e^3) - B*(384*c^4*d^4 - 640*b*c^3 *d^3*e + 288*b^2*c^2*d^2*e^2 - 24*b^3*c*d*e^3 - b^4*e^4))*ArcTanh[(Sqrt[c] *x)/Sqrt[b*x + c*x^2]])/(Sqrt[c]*e) - (16*c*Sqrt[d]*Sqrt[c*d - b*e]*(A*e*( 16*c^2*d^2 - 16*b*c*d*e + 3*b^2*e^2) - B*d*(24*c^2*d^2 - 28*b*c*d*e + 7*b^ 2*e^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))/(2*e^2)))/(8*e^2)
3.12.88.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(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 = 0.81 (sec) , antiderivative size = 498, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {\frac {15 \left (\frac {32 \left (A e -\frac {13 B d}{8}\right ) d^{2} e b \,c^{\frac {7}{2}}}{3}-\frac {19 \left (A e -\frac {35 B d}{19}\right ) d \,e^{2} b^{2} c^{\frac {5}{2}}}{3}+b^{3} e^{3} \left (A e -\frac {7 B d}{3}\right ) c^{\frac {3}{2}}-\frac {16 c^{\frac {9}{2}} \left (A e -\frac {3 B d}{2}\right ) d^{3}}{3}\right ) d \left (e x +d \right )^{2} \arctan \left (\frac {\sqrt {x \left (c x +b \right )}\, d}{x \sqrt {d \left (b e -c d \right )}}\right )}{4}+\frac {25 \sqrt {d \left (b e -c d \right )}\, \left (\frac {\left (b^{3} \left (A c -\frac {B b}{8}\right ) e^{4}+\left (-18 A \,b^{2} c^{2}-3 B \,b^{3} c \right ) d \,e^{3}+48 c^{2} d^{2} \left (A c +\frac {3 B b}{4}\right ) b \,e^{2}-32 c^{3} \left (A c +\frac {5 B b}{2}\right ) d^{3} e +48 B \,c^{4} d^{4}\right ) \left (e x +d \right )^{2} \operatorname {arctanh}\left (\frac {\sqrt {x \left (c x +b \right )}}{x \sqrt {c}}\right )}{5}+\left (\left (\frac {8 \left (\frac {3 B x}{4}+A \right ) x^{4} e^{5}}{75}-\frac {4 x^{3} d \left (\frac {3 B x}{5}+A \right ) e^{4}}{15}+\frac {16 x^{2} d^{2} \left (\frac {3 B x}{8}+A \right ) e^{3}}{15}+\frac {24 x \,d^{3} \left (-\frac {B x}{3}+A \right ) e^{2}}{5}+\frac {16 \left (-\frac {9 B x}{4}+A \right ) d^{4} e}{5}-\frac {24 B \,d^{5}}{5}\right ) c^{\frac {7}{2}}+\left (\left (\frac {26 x^{3} \left (\frac {17 B x}{26}+A \right ) e^{4}}{75}-\frac {22 x^{2} d \left (\frac {2 B x}{5}+A \right ) e^{3}}{15}-\frac {92 x \left (-\frac {37 B x}{92}+A \right ) d^{2} e^{2}}{15}-4 \left (-\frac {13 B x}{5}+A \right ) d^{3} e +\frac {34 B \,d^{4}}{5}\right ) c^{\frac {5}{2}}+\left (\left (\frac {11 \left (\frac {59 B x}{132}+A \right ) x^{2} e^{3}}{25}+\frac {8 x d \left (-\frac {139 B x}{240}+A \right ) e^{2}}{5}+d^{2} \left (-\frac {209 B x}{60}+A \right ) e -\frac {11 B \,d^{3}}{5}\right ) c^{\frac {3}{2}}+\frac {B b e \sqrt {c}\, \left (e x +d \right )^{2}}{40}\right ) e b \right ) e b \right ) \sqrt {x \left (c x +b \right )}\, e \right )}{8}}{\left (e x +d \right )^{2} e^{7} \sqrt {d \left (b e -c d \right )}\, c^{\frac {3}{2}}}\) | \(498\) |
| risch | \(\text {Expression too large to display}\) | \(1447\) |
| default | \(\text {Expression too large to display}\) | \(3958\) |
25/8/(d*(b*e-c*d))^(1/2)*(6/5*(32/3*(A*e-13/8*B*d)*d^2*e*b*c^(7/2)-19/3*(A *e-35/19*B*d)*d*e^2*b^2*c^(5/2)+b^3*e^3*(A*e-7/3*B*d)*c^(3/2)-16/3*c^(9/2) *(A*e-3/2*B*d)*d^3)*d*(e*x+d)^2*arctan((x*(c*x+b))^(1/2)/x*d/(d*(b*e-c*d)) ^(1/2))+(d*(b*e-c*d))^(1/2)*(1/5*(b^3*(A*c-1/8*B*b)*e^4+(-18*A*b^2*c^2-3*B *b^3*c)*d*e^3+48*c^2*d^2*(A*c+3/4*B*b)*b*e^2-32*c^3*(A*c+5/2*B*b)*d^3*e+48 *B*c^4*d^4)*(e*x+d)^2*arctanh((x*(c*x+b))^(1/2)/x/c^(1/2))+((8/75*(3/4*B*x +A)*x^4*e^5-4/15*x^3*d*(3/5*B*x+A)*e^4+16/15*x^2*d^2*(3/8*B*x+A)*e^3+24/5* x*d^3*(-1/3*B*x+A)*e^2+16/5*(-9/4*B*x+A)*d^4*e-24/5*B*d^5)*c^(7/2)+((26/75 *x^3*(17/26*B*x+A)*e^4-22/15*x^2*d*(2/5*B*x+A)*e^3-92/15*x*(-37/92*B*x+A)* d^2*e^2-4*(-13/5*B*x+A)*d^3*e+34/5*B*d^4)*c^(5/2)+((11/25*(59/132*B*x+A)*x ^2*e^3+8/5*x*d*(-139/240*B*x+A)*e^2+d^2*(-209/60*B*x+A)*e-11/5*B*d^3)*c^(3 /2)+1/40*B*b*e*c^(1/2)*(e*x+d)^2)*e*b)*e*b)*(x*(c*x+b))^(1/2)*e))/c^(3/2)/ (e*x+d)^2/e^7
Leaf count of result is larger than twice the leaf count of optimal. 1015 vs. \(2 (474) = 948\).
Time = 6.81 (sec) , antiderivative size = 4075, normalized size of antiderivative = 8.02 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
[-1/384*(15*(384*B*c^4*d^6 - 128*(5*B*b*c^3 + 2*A*c^4)*d^5*e + 96*(3*B*b^2 *c^2 + 4*A*b*c^3)*d^4*e^2 - 24*(B*b^3*c + 6*A*b^2*c^2)*d^3*e^3 - (B*b^4 - 8*A*b^3*c)*d^2*e^4 + (384*B*c^4*d^4*e^2 - 128*(5*B*b*c^3 + 2*A*c^4)*d^3*e^ 3 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^2*e^4 - 24*(B*b^3*c + 6*A*b^2*c^2)*d*e^ 5 - (B*b^4 - 8*A*b^3*c)*e^6)*x^2 + 2*(384*B*c^4*d^5*e - 128*(5*B*b*c^3 + 2 *A*c^4)*d^4*e^2 + 96*(3*B*b^2*c^2 + 4*A*b*c^3)*d^3*e^3 - 24*(B*b^3*c + 6*A *b^2*c^2)*d^2*e^4 - (B*b^4 - 8*A*b^3*c)*d*e^5)*x)*sqrt(c)*log(2*c*x + b - 2*sqrt(c*x^2 + b*x)*sqrt(c)) - 240*(24*B*c^4*d^5 - 3*A*b^2*c^2*d^2*e^3 - 4 *(7*B*b*c^3 + 4*A*c^4)*d^4*e + (7*B*b^2*c^2 + 16*A*b*c^3)*d^3*e^2 + (24*B* c^4*d^3*e^2 - 3*A*b^2*c^2*e^5 - 4*(7*B*b*c^3 + 4*A*c^4)*d^2*e^3 + (7*B*b^2 *c^2 + 16*A*b*c^3)*d*e^4)*x^2 + 2*(24*B*c^4*d^4*e - 3*A*b^2*c^2*d*e^4 - 4* (7*B*b*c^3 + 4*A*c^4)*d^3*e^2 + (7*B*b^2*c^2 + 16*A*b*c^3)*d^2*e^3)*x)*sqr t(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*(48*B*c^4*e^6*x^5 - 2880*B*c^4*d^5*e + 240*(17 *B*b*c^3 + 8*A*c^4)*d^4*e^2 - 120*(11*B*b^2*c^2 + 20*A*b*c^3)*d^3*e^3 + 15 *(B*b^3*c + 40*A*b^2*c^2)*d^2*e^4 - 8*(12*B*c^4*d*e^5 - (17*B*b*c^3 + 8*A* c^4)*e^6)*x^4 + 2*(120*B*c^4*d^2*e^4 - 16*(11*B*b*c^3 + 5*A*c^4)*d*e^5 + ( 59*B*b^2*c^2 + 104*A*b*c^3)*e^6)*x^3 - (960*B*c^4*d^3*e^3 - 40*(37*B*b*c^3 + 16*A*c^4)*d^2*e^4 + 4*(139*B*b^2*c^2 + 220*A*b*c^3)*d*e^5 - 3*(5*B*b^3* c + 88*A*b^2*c^2)*e^6)*x^2 - 10*(432*B*c^4*d^4*e^2 - 48*(13*B*b*c^3 + 6...
\[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{\left (d + e x\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(b*e-c*d>0)', see `assume?` for m ore detail
Leaf count of result is larger than twice the leaf count of optimal. 1448 vs. \(2 (474) = 948\).
Time = 0.38 (sec) , antiderivative size = 1448, normalized size of antiderivative = 2.85 \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\text {Too large to display} \]
1/192*sqrt(c*x^2 + b*x)*(2*(4*(6*B*c^2*x/e^3 - (24*B*c^5*d*e^21 - 17*B*b*c ^4*e^22 - 8*A*c^5*e^22)/(c^3*e^25))*x + (288*B*c^5*d^2*e^20 - 312*B*b*c^4* d*e^21 - 144*A*c^5*d*e^21 + 59*B*b^2*c^3*e^22 + 104*A*b*c^4*e^22)/(c^3*e^2 5))*x - 3*(640*B*c^5*d^3*e^19 - 864*B*b*c^4*d^2*e^20 - 384*A*c^5*d^2*e^20 + 264*B*b^2*c^3*d*e^21 + 432*A*b*c^4*d*e^21 - 5*B*b^3*c^2*e^22 - 88*A*b^2* c^3*e^22)/(c^3*e^25)) - 5/4*(24*B*c^3*d^5 - 52*B*b*c^2*d^4*e - 16*A*c^3*d^ 4*e + 35*B*b^2*c*d^3*e^2 + 32*A*b*c^2*d^3*e^2 - 7*B*b^3*d^2*e^3 - 19*A*b^2 *c*d^2*e^3 + 3*A*b^3*d*e^4)*arctan(-((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + s qrt(c)*d)/sqrt(-c*d^2 + b*d*e))/(sqrt(-c*d^2 + b*d*e)*e^7) - 5/128*(384*B* c^4*d^4 - 640*B*b*c^3*d^3*e - 256*A*c^4*d^3*e + 288*B*b^2*c^2*d^2*e^2 + 38 4*A*b*c^3*d^2*e^2 - 24*B*b^3*c*d*e^3 - 144*A*b^2*c^2*d*e^3 - B*b^4*e^4 + 8 *A*b^3*c*e^4)*log(abs(2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) + b))/(c^( 3/2)*e^7) - 1/4*(48*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*c^3*d^5*e - 100*(s qrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b*c^2*d^4*e^2 - 40*(sqrt(c)*x - sqrt(c*x ^2 + b*x))^3*A*c^3*d^4*e^2 + 65*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*B*b^2*c* d^3*e^3 + 80*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b*c^2*d^3*e^3 - 13*(sqrt( c)*x - sqrt(c*x^2 + b*x))^3*B*b^3*d^2*e^4 - 49*(sqrt(c)*x - sqrt(c*x^2 + b *x))^3*A*b^2*c*d^2*e^4 + 9*(sqrt(c)*x - sqrt(c*x^2 + b*x))^3*A*b^3*d*e^5 + 88*(sqrt(c)*x - sqrt(c*x^2 + b*x))^2*B*c^(7/2)*d^6 - 156*(sqrt(c)*x - sqr t(c*x^2 + b*x))^2*B*b*c^(5/2)*d^5*e - 72*(sqrt(c)*x - sqrt(c*x^2 + b*x)...
Timed out. \[ \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{(d+e x)^3} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \]