Integrand size = 36, antiderivative size = 718 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {(2 a C d f-b (8 B d f-7 C (d e+c f))) (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x}}{24 b d^2 f^2}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (d e+c f))-16 a b^2 d f \left (C \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )+6 d f (4 A d f-3 B (d e+c f))\right )+b^3 \left (5 C \left (21 d^3 e^3+19 c d^2 e^2 f+19 c^2 d e f^2+21 c^3 f^3\right )+8 d f \left (18 A d f (d e+c f)-B \left (15 d^2 e^2+14 c d e f+15 c^2 f^2\right )\right )\right )+2 b d f (6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)+(4 a d f-5 b (d e+c f)) (2 a C d f-b (8 B d f-7 C (d e+c f)))) x\right )}{192 b d^4 f^4}+\frac {\left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{64 d^{9/2} f^{9/2}} \]
1/64*(16*a^2*d^2*f^2*(C*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)+4*d*f*(2*A*d*f-B*( c*f+d*e)))-16*a*b*d*f*(C*(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^2*e^2*f+5*d^3*e^3) +2*d*f*(4*A*d*f*(c*f+d*e)-B*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)))+b^2*(C*(35*c ^4*f^4+20*c^3*d*e*f^3+18*c^2*d^2*e^2*f^2+20*c*d^3*e^3*f+35*d^4*e^4)+8*d*f* (2*A*d*f*(3*c^2*f^2+2*c*d*e*f+3*d^2*e^2)-B*(5*c^3*f^3+3*c^2*d*e*f^2+3*c*d^ 2*e^2*f+5*d^3*e^3))))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e)^(1/2)) /d^(9/2)/f^(9/2)-1/24*(2*a*C*d*f-b*(8*B*d*f-7*C*(c*f+d*e)))*(b*x+a)^2*(d*x +c)^(1/2)*(f*x+e)^(1/2)/b/d^2/f^2+1/4*C*(b*x+a)^3*(d*x+c)^(1/2)*(f*x+e)^(1 /2)/b/d/f-1/192*(32*a^3*C*d^3*f^3-8*a^2*b*d^2*f^2*(16*B*d*f-11*C*(c*f+d*e) )-16*a*b^2*d*f*(C*(15*c^2*f^2+14*c*d*e*f+15*d^2*e^2)+6*d*f*(4*A*d*f-3*B*(c *f+d*e)))+b^3*(5*C*(21*c^3*f^3+19*c^2*d*e*f^2+19*c*d^2*e^2*f+21*d^3*e^3)+8 *d*f*(18*A*d*f*(c*f+d*e)-B*(15*c^2*f^2+14*c*d*e*f+15*d^2*e^2)))+2*b*d*f*(6 *b*d*f*(-8*A*b*d*f+C*a*c*f+C*a*d*e+6*C*b*c*e)+(4*a*d*f-5*b*(c*f+d*e))*(2*a *C*d*f-b*(8*B*d*f-7*C*(c*f+d*e))))*x)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/d^4/f^ 4
Time = 2.22 (sec) , antiderivative size = 632, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {\sqrt {d} \sqrt {f} \sqrt {c+d x} \sqrt {e+f x} \left (48 a^2 d^2 f^2 (4 B d f+C (-3 d e-3 c f+2 d f x))+16 a b d f \left (6 d f (4 A d f+B (-3 d e-3 c f+2 d f x))+C \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )+b^2 \left (-C \left (105 c^3 f^3+5 c^2 d f^2 (19 e-14 f x)+c d^2 f \left (95 e^2-68 e f x+56 f^2 x^2\right )+d^3 \left (105 e^3-70 e^2 f x+56 e f^2 x^2-48 f^3 x^3\right )\right )+8 d f \left (6 A d f (-3 d e-3 c f+2 d f x)+B \left (15 c^2 f^2+2 c d f (7 e-5 f x)+d^2 \left (15 e^2-10 e f x+8 f^2 x^2\right )\right )\right )\right )\right )+3 \left (16 a^2 d^2 f^2 \left (C \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )+4 d f (2 A d f-B (d e+c f))\right )-16 a b d f \left (C \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )+2 d f \left (4 A d f (d e+c f)-B \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )\right )\right )+b^2 \left (C \left (35 d^4 e^4+20 c d^3 e^3 f+18 c^2 d^2 e^2 f^2+20 c^3 d e f^3+35 c^4 f^4\right )+8 d f \left (2 A d f \left (3 d^2 e^2+2 c d e f+3 c^2 f^2\right )-B \left (5 d^3 e^3+3 c d^2 e^2 f+3 c^2 d e f^2+5 c^3 f^3\right )\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {f} \sqrt {c+d x}}\right )}{192 d^{9/2} f^{9/2}} \]
(Sqrt[d]*Sqrt[f]*Sqrt[c + d*x]*Sqrt[e + f*x]*(48*a^2*d^2*f^2*(4*B*d*f + C* (-3*d*e - 3*c*f + 2*d*f*x)) + 16*a*b*d*f*(6*d*f*(4*A*d*f + B*(-3*d*e - 3*c *f + 2*d*f*x)) + C*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10* e*f*x + 8*f^2*x^2))) + b^2*(-(C*(105*c^3*f^3 + 5*c^2*d*f^2*(19*e - 14*f*x) + c*d^2*f*(95*e^2 - 68*e*f*x + 56*f^2*x^2) + d^3*(105*e^3 - 70*e^2*f*x + 56*e*f^2*x^2 - 48*f^3*x^3))) + 8*d*f*(6*A*d*f*(-3*d*e - 3*c*f + 2*d*f*x) + B*(15*c^2*f^2 + 2*c*d*f*(7*e - 5*f*x) + d^2*(15*e^2 - 10*e*f*x + 8*f^2*x^ 2))))) + 3*(16*a^2*d^2*f^2*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f* (2*A*d*f - B*(d*e + c*f))) - 16*a*b*d*f*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3* c^2*d*e*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c *d*e*f + 3*c^2*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e ^2*f^2 + 20*c^3*d*e*f^3 + 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2 + 2*c*d* e*f + 3*c^2*f^2) - B*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^ 3))))*ArcTanh[(Sqrt[d]*Sqrt[e + f*x])/(Sqrt[f]*Sqrt[c + d*x])])/(192*d^(9/ 2)*f^(9/2))
Time = 1.20 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.02, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2118, 27, 170, 27, 164, 66, 221}
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.
| \(\displaystyle \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
| \(\Big \downarrow \) 2118 |
| \(\displaystyle \frac {\int -\frac {b (a+b x)^2 (6 b c C e+a C d e+a c C f-8 A b d f-(8 b B d f-2 a C d f-7 b C (d e+c f)) x)}{2 \sqrt {c+d x} \sqrt {e+f x}}dx}{4 b^2 d f}+\frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\int \frac {(a+b x)^2 (6 b c C e+a C d e+a c C f-8 A b d f-(8 b B d f-2 a C d f-7 b C (d e+c f)) x)}{\sqrt {c+d x} \sqrt {e+f x}}dx}{8 b d f}\) |
| \(\Big \downarrow \) 170 |
| \(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\int \frac {(a+b x) (6 a d f (6 b c C e+a C d e+a c C f-8 A b d f)+(4 b c e+a d e+a c f) (8 b B d f-2 a C d f-7 b C (d e+c f))+(6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x)}{2 \sqrt {c+d x} \sqrt {e+f x}}dx}{3 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\int \frac {(a+b x) (6 a d f (6 b c C e+a C d e+a c C f-8 A b d f)+(4 b c e+a d e+a c f) (8 b B d f-2 a C d f-7 b C (d e+c f))+(6 b d f (6 b c C e+a C d e+a c C f-8 A b d f)-(4 a d f-5 b (d e+c f)) (8 b B d f-2 a C d f-7 b C (d e+c f))) x)}{\sqrt {c+d x} \sqrt {e+f x}}dx}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{4 d^2 f^2}-\frac {3 b \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{8 d^2 f^2}}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\) |
| \(\Big \downarrow \) 66 |
| \(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{4 d^2 f^2}-\frac {3 b \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{4 d^2 f^2}}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {C (a+b x)^3 \sqrt {c+d x} \sqrt {e+f x}}{4 b d f}-\frac {\frac {\frac {\sqrt {c+d x} \sqrt {e+f x} \left (32 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-11 C (c f+d e))-16 a b^2 d f \left (6 d f (4 A d f-3 B (c f+d e))+C \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+2 b d f x (6 b d f (a c C f+a C d e-8 A b d f+6 b c C e)-(4 a d f-5 b (c f+d e)) (-2 a C d f+8 b B d f-7 b C (c f+d e)))+b^3 \left (8 d f \left (18 A d f (c f+d e)-B \left (15 c^2 f^2+14 c d e f+15 d^2 e^2\right )\right )+5 C \left (21 c^3 f^3+19 c^2 d e f^2+19 c d^2 e^2 f+21 d^3 e^3\right )\right )\right )}{4 d^2 f^2}-\frac {3 b \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (16 a^2 d^2 f^2 \left (4 d f (2 A d f-B (c f+d e))+C \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )-16 a b d f \left (2 d f \left (4 A d f (c f+d e)-B \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )\right )+C \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+b^2 \left (8 d f \left (2 A d f \left (3 c^2 f^2+2 c d e f+3 d^2 e^2\right )-B \left (5 c^3 f^3+3 c^2 d e f^2+3 c d^2 e^2 f+5 d^3 e^3\right )\right )+C \left (35 c^4 f^4+20 c^3 d e f^3+18 c^2 d^2 e^2 f^2+20 c d^3 e^3 f+35 d^4 e^4\right )\right )\right )}{4 d^{5/2} f^{5/2}}}{6 d f}-\frac {(a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} (-2 a C d f+8 b B d f-7 b C (c f+d e))}{3 d f}}{8 b d f}\) |
(C*(a + b*x)^3*Sqrt[c + d*x]*Sqrt[e + f*x])/(4*b*d*f) - (-1/3*((8*b*B*d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f))*(a + b*x)^2*Sqrt[c + d*x]*Sqrt[e + f*x])/ (d*f) + ((Sqrt[c + d*x]*Sqrt[e + f*x]*(32*a^3*C*d^3*f^3 - 8*a^2*b*d^2*f^2* (16*B*d*f - 11*C*(d*e + c*f)) - 16*a*b^2*d*f*(C*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2) + 6*d*f*(4*A*d*f - 3*B*(d*e + c*f))) + b^3*(5*C*(21*d^3*e^3 + 19*c*d^2*e^2*f + 19*c^2*d*e*f^2 + 21*c^3*f^3) + 8*d*f*(18*A*d*f*(d*e + c* f) - B*(15*d^2*e^2 + 14*c*d*e*f + 15*c^2*f^2))) + 2*b*d*f*(6*b*d*f*(6*b*c* C*e + a*C*d*e + a*c*C*f - 8*A*b*d*f) - (4*a*d*f - 5*b*(d*e + c*f))*(8*b*B* d*f - 2*a*C*d*f - 7*b*C*(d*e + c*f)))*x))/(4*d^2*f^2) - (3*b*(16*a^2*d^2*f ^2*(C*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) + 4*d*f*(2*A*d*f - B*(d*e + c*f) )) - 16*a*b*d*f*(C*(5*d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3) + 2*d*f*(4*A*d*f*(d*e + c*f) - B*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2))) + b^2*(C*(35*d^4*e^4 + 20*c*d^3*e^3*f + 18*c^2*d^2*e^2*f^2 + 20*c^3*d*e*f^3 + 35*c^4*f^4) + 8*d*f*(2*A*d*f*(3*d^2*e^2 + 2*c*d*e*f + 3*c^2*f^2) - B*(5* d^3*e^3 + 3*c*d^2*e^2*f + 3*c^2*d*e*f^2 + 5*c^3*f^3))))*ArcTanh[(Sqrt[f]*S qrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(4*d^(5/2)*f^(5/2)))/(6*d*f))/(8*b *d*f)
3.1.54.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 2 Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre eQ[{a, b, c, d}, x] && !GtQ[c - a*(d/b), 0]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f _.)*(x_))^(p_.), x_Symbol] :> With[{q = Expon[Px, x], k = Coeff[Px, x, Expo n[Px, x]]}, Simp[k*(a + b*x)^(m + q - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*b^(q - 1)*(m + n + p + q + 1))), x] + Simp[1/(d*f*b^q*(m + n + p + q + 1)) Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*ExpandToSum[d*f*b^q*(m + n + p + q + 1)*Px - d*f*k*(m + n + p + q + 1)*(a + b*x)^q + k*(a + b*x)^(q - 2)*(a^2*d*f*(m + n + p + q + 1) - b*(b*c*e*(m + q - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*(m + q) + n + p) - b*(d*e*(m + q + n) + c*f*(m + q + p)))*x), x], x], x] /; NeQ[m + n + p + q + 1, 0]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[Px, x]
Leaf count of result is larger than twice the leaf count of optimal. \(2527\) vs. \(2(686)=1372\).
Time = 1.68 (sec) , antiderivative size = 2528, normalized size of antiderivative = 3.52
1/384*(-112*C*b^2*d^3*e*f^2*x^2*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)-72*B*l n(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2)) *b^2*c^2*d^2*e*f^3-72*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1 /2)+c*f+d*e)/(d*f)^(1/2))*b^2*c*d^3*e^2*f^2+60*C*ln(1/2*(2*d*f*x+2*((d*x+c )*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c*d^3*e^3*f+54*C*ln (1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))* b^2*c^2*d^2*e^2*f^2-320*C*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*a*b*c*d^2*f^ 3*x-320*C*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*a*b*d^3*e*f^2*x+136*C*((d*x+ c)*(f*x+e))^(1/2)*(d*f)^(1/2)*b^2*c*d^2*e*f^2*x+144*A*ln(1/2*(2*d*f*x+2*(( d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*d^4*e^2*f^2-19 2*B*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^( 1/2))*a^2*c*d^3*f^4+144*A*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^ (1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*c^2*d^2*f^4-288*A*(d*f)^(1/2)*((d*x+c)*(f* x+e))^(1/2)*b^2*c*d^2*f^3+448*C*(d*f)^(1/2)*((d*x+c)*(f*x+e))^(1/2)*a*b*c* d^2*e*f^2+192*C*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)*a^2*d^3*f^3*x+96*C*ln( 1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a ^2*c*d^3*e*f^3+105*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e))^(1/2)*(d*f)^(1/2) +c*f+d*e)/(d*f)^(1/2))*b^2*c^4*f^4+105*C*ln(1/2*(2*d*f*x+2*((d*x+c)*(f*x+e ))^(1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^2*d^4*e^4-288*A*(d*f)^(1/2)*( (d*x+c)*(f*x+e))^(1/2)*b^2*d^3*e*f^2+384*B*(d*f)^(1/2)*((d*x+c)*(f*x+e)...
Time = 1.12 (sec) , antiderivative size = 1436, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \]
[1/768*(3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*(2*C*a*b + B*b^2)*d^4)*e ^3*f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 + 8*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^2 + 8*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35* C*b^2*c^4 + 128*A*a^2*d^4 - 40*(2*C*a*b + B*b^2)*c^3*d + 48*(C*a^2 + 2*B*a *b + A*b^2)*c^2*d^2 - 64*(B*a^2 + 2*A*a*b)*c*d^3)*f^4)*sqrt(d*f)*log(8*d^2 *f^2*x^2 + d^2*e^2 + 6*c*d*e*f + c^2*f^2 + 4*(2*d*f*x + d*e + c*f)*sqrt(d* f)*sqrt(d*x + c)*sqrt(f*x + e) + 8*(d^2*e*f + c*d*f^2)*x) + 4*(48*C*b^2*d^ 4*f^4*x^3 - 105*C*b^2*d^4*e^3*f - 5*(19*C*b^2*c*d^3 - 24*(2*C*a*b + B*b^2) *d^4)*e^2*f^2 - (95*C*b^2*c^2*d^2 - 112*(2*C*a*b + B*b^2)*c*d^3 + 144*(C*a ^2 + 2*B*a*b + A*b^2)*d^4)*e*f^3 - 3*(35*C*b^2*c^3*d - 40*(2*C*a*b + B*b^2 )*c^2*d^2 + 48*(C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 64*(B*a^2 + 2*A*a*b)*d^4) *f^4 - 8*(7*C*b^2*d^4*e*f^3 + (7*C*b^2*c*d^3 - 8*(2*C*a*b + B*b^2)*d^4)*f^ 4)*x^2 + 2*(35*C*b^2*d^4*e^2*f^2 + 2*(17*C*b^2*c*d^3 - 20*(2*C*a*b + B*b^2 )*d^4)*e*f^3 + (35*C*b^2*c^2*d^2 - 40*(2*C*a*b + B*b^2)*c*d^3 + 48*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*f^4)*x)*sqrt(d*x + c)*sqrt(f*x + e))/(d^5*f^5), -1 /384*(3*(35*C*b^2*d^4*e^4 + 20*(C*b^2*c*d^3 - 2*(2*C*a*b + B*b^2)*d^4)*e^3 *f + 6*(3*C*b^2*c^2*d^2 - 4*(2*C*a*b + B*b^2)*c*d^3 + 8*(C*a^2 + 2*B*a*b + A*b^2)*d^4)*e^2*f^2 + 4*(5*C*b^2*c^3*d - 6*(2*C*a*b + B*b^2)*c^2*d^2 + 8* (C*a^2 + 2*B*a*b + A*b^2)*c*d^3 - 16*(B*a^2 + 2*A*a*b)*d^4)*e*f^3 + (35...
\[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {\left (a + b x\right )^{2} \left (A + B x + C x^{2}\right )}{\sqrt {c + d x} \sqrt {e + f x}}\, dx \]
Exception generated. \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m ore detail
Time = 0.36 (sec) , antiderivative size = 946, normalized size of antiderivative = 1.32 \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\frac {{\left (\sqrt {d^{2} e + {\left (d x + c\right )} d f - c d f} {\left (2 \, {\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {6 \, {\left (d x + c\right )} C b^{2}}{d^{5} f} - \frac {7 \, C b^{2} d^{20} e f^{5} + 25 \, C b^{2} c d^{19} f^{6} - 16 \, C a b d^{20} f^{6} - 8 \, B b^{2} d^{20} f^{6}}{d^{24} f^{7}}\right )} + \frac {35 \, C b^{2} d^{21} e^{2} f^{4} + 90 \, C b^{2} c d^{20} e f^{5} - 80 \, C a b d^{21} e f^{5} - 40 \, B b^{2} d^{21} e f^{5} + 163 \, C b^{2} c^{2} d^{19} f^{6} - 208 \, C a b c d^{20} f^{6} - 104 \, B b^{2} c d^{20} f^{6} + 48 \, C a^{2} d^{21} f^{6} + 96 \, B a b d^{21} f^{6} + 48 \, A b^{2} d^{21} f^{6}}{d^{24} f^{7}}\right )} - \frac {3 \, {\left (35 \, C b^{2} d^{22} e^{3} f^{3} + 55 \, C b^{2} c d^{21} e^{2} f^{4} - 80 \, C a b d^{22} e^{2} f^{4} - 40 \, B b^{2} d^{22} e^{2} f^{4} + 73 \, C b^{2} c^{2} d^{20} e f^{5} - 128 \, C a b c d^{21} e f^{5} - 64 \, B b^{2} c d^{21} e f^{5} + 48 \, C a^{2} d^{22} e f^{5} + 96 \, B a b d^{22} e f^{5} + 48 \, A b^{2} d^{22} e f^{5} + 93 \, C b^{2} c^{3} d^{19} f^{6} - 176 \, C a b c^{2} d^{20} f^{6} - 88 \, B b^{2} c^{2} d^{20} f^{6} + 80 \, C a^{2} c d^{21} f^{6} + 160 \, B a b c d^{21} f^{6} + 80 \, A b^{2} c d^{21} f^{6} - 64 \, B a^{2} d^{22} f^{6} - 128 \, A a b d^{22} f^{6}\right )}}{d^{24} f^{7}}\right )} \sqrt {d x + c} - \frac {3 \, {\left (35 \, C b^{2} d^{4} e^{4} + 20 \, C b^{2} c d^{3} e^{3} f - 80 \, C a b d^{4} e^{3} f - 40 \, B b^{2} d^{4} e^{3} f + 18 \, C b^{2} c^{2} d^{2} e^{2} f^{2} - 48 \, C a b c d^{3} e^{2} f^{2} - 24 \, B b^{2} c d^{3} e^{2} f^{2} + 48 \, C a^{2} d^{4} e^{2} f^{2} + 96 \, B a b d^{4} e^{2} f^{2} + 48 \, A b^{2} d^{4} e^{2} f^{2} + 20 \, C b^{2} c^{3} d e f^{3} - 48 \, C a b c^{2} d^{2} e f^{3} - 24 \, B b^{2} c^{2} d^{2} e f^{3} + 32 \, C a^{2} c d^{3} e f^{3} + 64 \, B a b c d^{3} e f^{3} + 32 \, A b^{2} c d^{3} e f^{3} - 64 \, B a^{2} d^{4} e f^{3} - 128 \, A a b d^{4} e f^{3} + 35 \, C b^{2} c^{4} f^{4} - 80 \, C a b c^{3} d f^{4} - 40 \, B b^{2} c^{3} d f^{4} + 48 \, C a^{2} c^{2} d^{2} f^{4} + 96 \, B a b c^{2} d^{2} f^{4} + 48 \, A b^{2} c^{2} d^{2} f^{4} - 64 \, B a^{2} c d^{3} f^{4} - 128 \, A a b c d^{3} f^{4} + 128 \, A a^{2} d^{4} f^{4}\right )} \log \left ({\left | -\sqrt {d f} \sqrt {d x + c} + \sqrt {d^{2} e + {\left (d x + c\right )} d f - c d f} \right |}\right )}{\sqrt {d f} d^{4} f^{4}}\right )} d}{192 \, {\left | d \right |}} \]
1/192*(sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*(2*(d*x + c)*(4*(d*x + c)*(6*(d *x + c)*C*b^2/(d^5*f) - (7*C*b^2*d^20*e*f^5 + 25*C*b^2*c*d^19*f^6 - 16*C*a *b*d^20*f^6 - 8*B*b^2*d^20*f^6)/(d^24*f^7)) + (35*C*b^2*d^21*e^2*f^4 + 90* C*b^2*c*d^20*e*f^5 - 80*C*a*b*d^21*e*f^5 - 40*B*b^2*d^21*e*f^5 + 163*C*b^2 *c^2*d^19*f^6 - 208*C*a*b*c*d^20*f^6 - 104*B*b^2*c*d^20*f^6 + 48*C*a^2*d^2 1*f^6 + 96*B*a*b*d^21*f^6 + 48*A*b^2*d^21*f^6)/(d^24*f^7)) - 3*(35*C*b^2*d ^22*e^3*f^3 + 55*C*b^2*c*d^21*e^2*f^4 - 80*C*a*b*d^22*e^2*f^4 - 40*B*b^2*d ^22*e^2*f^4 + 73*C*b^2*c^2*d^20*e*f^5 - 128*C*a*b*c*d^21*e*f^5 - 64*B*b^2* c*d^21*e*f^5 + 48*C*a^2*d^22*e*f^5 + 96*B*a*b*d^22*e*f^5 + 48*A*b^2*d^22*e *f^5 + 93*C*b^2*c^3*d^19*f^6 - 176*C*a*b*c^2*d^20*f^6 - 88*B*b^2*c^2*d^20* f^6 + 80*C*a^2*c*d^21*f^6 + 160*B*a*b*c*d^21*f^6 + 80*A*b^2*c*d^21*f^6 - 6 4*B*a^2*d^22*f^6 - 128*A*a*b*d^22*f^6)/(d^24*f^7))*sqrt(d*x + c) - 3*(35*C *b^2*d^4*e^4 + 20*C*b^2*c*d^3*e^3*f - 80*C*a*b*d^4*e^3*f - 40*B*b^2*d^4*e^ 3*f + 18*C*b^2*c^2*d^2*e^2*f^2 - 48*C*a*b*c*d^3*e^2*f^2 - 24*B*b^2*c*d^3*e ^2*f^2 + 48*C*a^2*d^4*e^2*f^2 + 96*B*a*b*d^4*e^2*f^2 + 48*A*b^2*d^4*e^2*f^ 2 + 20*C*b^2*c^3*d*e*f^3 - 48*C*a*b*c^2*d^2*e*f^3 - 24*B*b^2*c^2*d^2*e*f^3 + 32*C*a^2*c*d^3*e*f^3 + 64*B*a*b*c*d^3*e*f^3 + 32*A*b^2*c*d^3*e*f^3 - 64 *B*a^2*d^4*e*f^3 - 128*A*a*b*d^4*e*f^3 + 35*C*b^2*c^4*f^4 - 80*C*a*b*c^3*d *f^4 - 40*B*b^2*c^3*d*f^4 + 48*C*a^2*c^2*d^2*f^4 + 96*B*a*b*c^2*d^2*f^4 + 48*A*b^2*c^2*d^2*f^4 - 64*B*a^2*c*d^3*f^4 - 128*A*a*b*c*d^3*f^4 + 128*A...
Timed out. \[ \int \frac {(a+b x)^2 \left (A+B x+C x^2\right )}{\sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Hanged} \]