Integrand size = 36, antiderivative size = 1348 \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}-\frac {(2 a C d f-b (4 B d f-3 C (d e+c f))) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)+(4 a d f-7 b (d e+c f)) (2 a C d f-b (4 B d f-3 C (d e+c f)))) x\right )}{960 b d^4 f^4}-\frac {(d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 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 )}{512 d^{11/2} f^{11/2}} \]
[Out]
Time = 1.56 (sec) , antiderivative size = 1345, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1629, 158, 152, 52, 65, 223, 212} \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {C (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^3}{6 b d f}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (c+d x)^{3/2} (e+f x)^{3/2} (a+b x)^2}{20 b d^2 f^2}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (\left (7 C \left (15 d^3 e^3+17 c d^2 f e^2+17 c^2 d f^2 e+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )\right )\right ) b^3-8 a d f \left (C \left (35 d^2 e^2+38 c d f e+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right ) b^2-8 a^2 d^2 f^2 (16 B d f-7 C (d e+c f)) b+6 d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x b+64 a^3 C d^3 f^3\right )}{960 b d^4 f^4}-\frac {(d e-c f)^2 \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{512 d^{11/2} f^{11/2}}+\frac {\left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(d e-c f) \left (\left (C \left (21 d^4 e^4+28 c d^3 f e^3+30 c^2 d^2 f^2 e^2+28 c^3 d f^3 e+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )\right )\right ) b^2-8 a d f \left (C \left (7 d^3 e^3+9 c d^2 f e^2+9 c^2 d f^2 e+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )\right )\right ) b+8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d f e+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5} \]
[In]
[Out]
Rule 52
Rule 65
Rule 152
Rule 158
Rule 212
Rule 223
Rule 1629
Rubi steps \begin{align*} \text {integral}& = \frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}+\frac {\int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3}{2} b (2 b c C e+a C d e+a c C f-4 A b d f)+\frac {3}{2} b (4 b B d f-2 a C d f-3 b C (d e+c f)) x\right ) \, dx}{6 b^2 d f} \\ & = \frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}+\frac {\int (a+b x) \sqrt {c+d x} \sqrt {e+f x} \left (-\frac {3}{4} b (10 a d f (2 b c C e+a C d e+a c C f-4 A b d f)+(4 b c e+3 a (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f)))-\frac {3}{4} b (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right ) \, dx}{30 b^2 d^2 f^2} \\ & = \frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \int \sqrt {c+d x} \sqrt {e+f x} \, dx}{128 d^4 f^4} \\ & = \frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}+\frac {\left ((d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {e+f x}} \, dx}{512 d^5 f^4} \\ & = \frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {\left ((d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}} \, dx}{1024 d^5 f^5} \\ & = \frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {\left ((d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {e-\frac {c f}{d}+\frac {f x^2}{d}}} \, dx,x,\sqrt {c+d x}\right )}{512 d^6 f^5} \\ & = \frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {\left ((d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {f x^2}{d}} \, dx,x,\frac {\sqrt {c+d x}}{\sqrt {e+f x}}\right )}{512 d^6 f^5} \\ & = \frac {(d e-c f) \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{512 d^5 f^5}+\frac {\left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) (c+d x)^{3/2} \sqrt {e+f x}}{256 d^5 f^4}+\frac {(4 b B d f-2 a C d f-3 b C (d e+c f)) (a+b x)^2 (c+d x)^{3/2} (e+f x)^{3/2}}{20 b d^2 f^2}+\frac {C (a+b x)^3 (c+d x)^{3/2} (e+f x)^{3/2}}{6 b d f}-\frac {(c+d x)^{3/2} (e+f x)^{3/2} \left (64 a^3 C d^3 f^3-8 a^2 b d^2 f^2 (16 B d f-7 C (d e+c f))-8 a b^2 d f \left (C \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )+10 d f (8 A d f-5 B (d e+c f))\right )+b^3 \left (7 C \left (15 d^3 e^3+17 c d^2 e^2 f+17 c^2 d e f^2+15 c^3 f^3\right )+4 d f \left (50 A d f (d e+c f)-B \left (35 d^2 e^2+38 c d e f+35 c^2 f^2\right )\right )\right )+6 b d f (10 b d f (2 b c C e+a C d e+a c C f-4 A b d f)-(4 a d f-7 b (d e+c f)) (4 b B d f-2 a C d f-3 b C (d e+c f))) x\right )}{960 b d^4 f^4}-\frac {(d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{512 d^{11/2} f^{11/2}} \\ \end{align*}
Time = 4.43 (sec) , antiderivative size = 1253, normalized size of antiderivative = 0.93 \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\frac {\sqrt {c+d x} \sqrt {e+f x} \left (40 a^2 d^2 f^2 \left (C \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )+8 d f \left (6 A d f (c f+d (e+2 f x))+B \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )\right )\right )+8 a b d f \left (C \left (-105 c^4 f^4+10 c^3 d f^3 (4 e+7 f x)-2 c^2 d^2 f^2 \left (-17 e^2+11 e f x+28 f^2 x^2\right )+2 c d^3 f \left (20 e^3-11 e^2 f x+8 e f^2 x^2+24 f^3 x^3\right )+d^4 \left (-105 e^4+70 e^3 f x-56 e^2 f^2 x^2+48 e f^3 x^3+384 f^4 x^4\right )\right )+10 d f \left (8 A d f \left (-3 c^2 f^2+2 c d f (e+f x)+d^2 \left (-3 e^2+2 e f x+8 f^2 x^2\right )\right )+B \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )\right )\right )+b^2 \left (C \left (315 c^5 f^5-105 c^4 d f^4 (e+2 f x)+2 c^3 d^2 f^3 \left (-41 e^2+28 e f x+84 f^2 x^2\right )-2 c^2 d^3 f^2 \left (41 e^3-26 e^2 f x+20 e f^2 x^2+72 f^3 x^3\right )+c d^4 f \left (-105 e^4+56 e^3 f x-40 e^2 f^2 x^2+32 e f^3 x^3+128 f^4 x^4\right )+d^5 \left (315 e^5-210 e^4 f x+168 e^3 f^2 x^2-144 e^2 f^3 x^3+128 e f^4 x^4+1280 f^5 x^5\right )\right )+4 d f \left (10 A d f \left (15 c^3 f^3-c^2 d f^2 (7 e+10 f x)+c d^2 f \left (-7 e^2+4 e f x+8 f^2 x^2\right )+d^3 \left (15 e^3-10 e^2 f x+8 e f^2 x^2+48 f^3 x^3\right )\right )+B \left (-105 c^4 f^4+10 c^3 d f^3 (4 e+7 f x)-2 c^2 d^2 f^2 \left (-17 e^2+11 e f x+28 f^2 x^2\right )+2 c d^3 f \left (20 e^3-11 e^2 f x+8 e f^2 x^2+24 f^3 x^3\right )+d^4 \left (-105 e^4+70 e^3 f x-56 e^2 f^2 x^2+48 e f^3 x^3+384 f^4 x^4\right )\right )\right )\right )\right )}{7680 d^5 f^5}-\frac {(d e-c f)^2 \left (8 a^2 d^2 f^2 \left (C \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )+8 d f (2 A d f-B (d e+c f))\right )-8 a b d f \left (C \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 c^3 f^3\right )+2 d f \left (8 A d f (d e+c f)-B \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )\right )\right )+b^2 \left (C \left (21 d^4 e^4+28 c d^3 e^3 f+30 c^2 d^2 e^2 f^2+28 c^3 d e f^3+21 c^4 f^4\right )+4 d f \left (2 A d f \left (5 d^2 e^2+6 c d e f+5 c^2 f^2\right )-B \left (7 d^3 e^3+9 c d^2 e^2 f+9 c^2 d e f^2+7 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 )}{512 d^{11/2} f^{11/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(5733\) vs. \(2(1304)=2608\).
Time = 1.68 (sec) , antiderivative size = 5734, normalized size of antiderivative = 4.25
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Time = 1.58 (sec) , antiderivative size = 3096, normalized size of antiderivative = 2.30 \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \]
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\[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\int \left (a + b x\right )^{2} \sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )\, dx \]
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Exception generated. \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 4656 vs. \(2 (1304) = 2608\).
Time = 0.86 (sec) , antiderivative size = 4656, normalized size of antiderivative = 3.45 \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Too large to display} \]
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Timed out. \[ \int (a+b x)^2 \sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right ) \, dx=\text {Hanged} \]
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