Integrand size = 38, antiderivative size = 964 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\frac {2 \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 (b c-a d)^2 (b e-a f) \sqrt {a+b x}}+\frac {2 \left (6 a^3 C d f+a b^2 (10 c C e+3 B d e+3 B c f-4 A d f)-b^3 (5 B c e-2 A (d e+c f))-a^2 b (B d f+8 C (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{15 b^2 (b c-a d) (b e-a f)^2 (a+b x)^{3/2}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}+\frac {2 \sqrt {d} \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (2 A d^2 e^2-c d e (5 B e+2 A f)-c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 (d e-c f) \left (24 a^3 C d^2 f-a^2 b d (23 C d e+41 c C f+4 B d f)-b^3 \left (15 c^2 C e-2 A d^2 e+c d (5 B e+A f)\right )+a b^2 \left (15 c^2 C f+d^2 (3 B e-A f)+c (40 C d e+6 B d f)\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 \sqrt {d} (-b c+a d)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \]
-2/5*(A*b^2-a*(B*b-C*a))*(d*x+c)^(3/2)*(f*x+e)^(3/2)/b/(-a*d+b*c)/(-a*f+b* e)/(b*x+a)^(5/2)+2/15*(6*a^3*C*d*f+a*b^2*(-4*A*d*f+3*B*c*f+3*B*d*e+10*C*c* e)-b^3*(5*B*c*e-2*A*(c*f+d*e))-a^2*b*(B*d*f+8*C*(c*f+d*e)))*(f*x+e)^(3/2)* (d*x+c)^(1/2)/b^2/(-a*d+b*c)/(-a*f+b*e)^2/(b*x+a)^(3/2)+2/15*(24*a^3*C*d^2 *f-a^2*b*d*(4*B*d*f+41*C*c*f+23*C*d*e)-b^3*(15*c^2*C*e-2*A*d^2*e+c*d*(A*f+ 5*B*e))+a*b^2*(15*c^2*C*f+d^2*(-A*f+3*B*e)+c*(6*B*d*f+40*C*d*e)))*(d*x+c)^ (1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)^2/(-a*f+b*e)/(b*x+a)^(1/2)+2/15*(48*a^4 *C*d^2*f^2-8*a^3*b*d*f*(B*d*f+11*C*(c*f+d*e))-b^4*(2*A*d^2*e^2-c*d*e*(2*A* f+5*B*e)-c^2*(-2*A*f^2+5*B*e*f+30*C*e^2))-a*b^3*(d^2*e*(-2*A*f+3*B*e)+c^2* f*(3*B*f+70*C*e)+2*c*d*(-A*f^2+11*B*e*f+35*C*e^2))+a^2*b^2*(2*C*(19*c^2*f^ 2+81*c*d*e*f+19*d^2*e^2)-d*f*(2*A*d*f-13*B*(c*f+d*e))))*EllipticE(d^(1/2)* (b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))*d^(1/2)*( b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^4/(a*d-b*c)^(3/2)/(-a*f+b*e)^2 /(d*x+c)^(1/2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)+2/15*(-c*f+d*e)*(24*a^3*C*d^2* f-a^2*b*d*(4*B*d*f+41*C*c*f+23*C*d*e)-b^3*(15*c^2*C*e-2*A*d^2*e+c*d*(A*f+5 *B*e))+a*b^2*(15*c^2*C*f+d^2*(-A*f+3*B*e)+c*(6*B*d*f+40*C*d*e)))*EllipticF (d^(1/2)*(b*x+a)^(1/2)/(a*d-b*c)^(1/2),((-a*d+b*c)*f/d/(-a*f+b*e))^(1/2))* (b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b^4/(a*d-b*c)^(3 /2)/(-a*f+b*e)/d^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 33.80 (sec) , antiderivative size = 1444, normalized size of antiderivative = 1.50 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (-\frac {2 \left (A b^2-a b B+a^2 C\right )}{5 b^3 (a+b x)^3}-\frac {2 \left (5 b^3 B c e-10 a b^2 c C e+A b^3 d e-6 a b^2 B d e+11 a^2 b C d e+A b^3 c f-6 a b^2 B c f+11 a^2 b c C f-2 a A b^2 d f+7 a^2 b B d f-12 a^3 C d f\right )}{15 b^3 (b c-a d) (b e-a f) (a+b x)^2}-\frac {2 \left (15 b^4 c^2 C e^2+5 b^4 B c d e^2-40 a b^3 c C d e^2-2 A b^4 d^2 e^2-3 a b^3 B d^2 e^2+23 a^2 b^2 C d^2 e^2+5 b^4 B c^2 e f-40 a b^3 c^2 C e f+2 A b^4 c d e f-22 a b^3 B c d e f+102 a^2 b^2 c C d e f+2 a A b^3 d^2 e f+13 a^2 b^2 B d^2 e f-58 a^3 b C d^2 e f-2 A b^4 c^2 f^2-3 a b^3 B c^2 f^2+23 a^2 b^2 c^2 C f^2+2 a A b^3 c d f^2+13 a^2 b^2 B c d f^2-58 a^3 b c C d f^2-2 a^2 A b^2 d^2 f^2-8 a^3 b B d^2 f^2+33 a^4 C d^2 f^2\right )}{15 b^3 (b c-a d)^2 (b e-a f)^2 (a+b x)}\right )+\frac {2 (a+b x)^{3/2} \left (\sqrt {-a+\frac {b c}{d}} \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (d e+c f))+b^4 \left (-2 A d^2 e^2+c d e (5 B e+2 A f)+c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )-a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )+a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )+d f (-2 A d f+13 B (d e+c f))\right )\right ) \left (d+\frac {b c}{a+b x}-\frac {a d}{a+b x}\right ) \left (f+\frac {b e}{a+b x}-\frac {a f}{a+b x}\right )+\frac {i (-b c+a d) f \left (-48 a^4 C d^2 f^2+8 a^3 b d f (B d f+11 C (d e+c f))-b^4 \left (-2 A d^2 e^2+c d e (5 B e+2 A f)+c^2 \left (30 C e^2+5 B e f-2 A f^2\right )\right )+a b^3 \left (d^2 e (3 B e-2 A f)+c^2 f (70 C e+3 B f)+2 c d \left (35 C e^2+11 B e f-A f^2\right )\right )-a^2 b^2 \left (2 C \left (19 d^2 e^2+81 c d e f+19 c^2 f^2\right )+d f (-2 A d f+13 B (d e+c f))\right )\right ) \sqrt {1-\frac {a}{a+b x}+\frac {b c}{d (a+b x)}} \sqrt {1-\frac {a}{a+b x}+\frac {b e}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {a+b x}}-\frac {i b (-b c+a d) (d e-c f) \left (-24 a^3 C d f^2+a^2 b f (41 C d e+23 c C f+4 B d f)+b^3 \left (15 c C e^2+A d e f+c f (5 B e-2 A f)\right )-a b^2 (5 C e (3 d e+8 c f)+f (6 B d e+3 B c f-A d f))\right ) \sqrt {1-\frac {a}{a+b x}+\frac {b c}{d (a+b x)}} \sqrt {1-\frac {a}{a+b x}+\frac {b e}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {a+b x}}\right )}{15 b^5 \sqrt {-a+\frac {b c}{d}} (b c-a d)^2 (b e-a f)^2 \sqrt {c+\frac {(a+b x) \left (d-\frac {a d}{a+b x}\right )}{b}} \sqrt {e+\frac {(a+b x) \left (f-\frac {a f}{a+b x}\right )}{b}}} \]
Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*((-2*(A*b^2 - a*b*B + a^2*C))/(5 *b^3*(a + b*x)^3) - (2*(5*b^3*B*c*e - 10*a*b^2*c*C*e + A*b^3*d*e - 6*a*b^2 *B*d*e + 11*a^2*b*C*d*e + A*b^3*c*f - 6*a*b^2*B*c*f + 11*a^2*b*c*C*f - 2*a *A*b^2*d*f + 7*a^2*b*B*d*f - 12*a^3*C*d*f))/(15*b^3*(b*c - a*d)*(b*e - a*f )*(a + b*x)^2) - (2*(15*b^4*c^2*C*e^2 + 5*b^4*B*c*d*e^2 - 40*a*b^3*c*C*d*e ^2 - 2*A*b^4*d^2*e^2 - 3*a*b^3*B*d^2*e^2 + 23*a^2*b^2*C*d^2*e^2 + 5*b^4*B* c^2*e*f - 40*a*b^3*c^2*C*e*f + 2*A*b^4*c*d*e*f - 22*a*b^3*B*c*d*e*f + 102* a^2*b^2*c*C*d*e*f + 2*a*A*b^3*d^2*e*f + 13*a^2*b^2*B*d^2*e*f - 58*a^3*b*C* d^2*e*f - 2*A*b^4*c^2*f^2 - 3*a*b^3*B*c^2*f^2 + 23*a^2*b^2*c^2*C*f^2 + 2*a *A*b^3*c*d*f^2 + 13*a^2*b^2*B*c*d*f^2 - 58*a^3*b*c*C*d*f^2 - 2*a^2*A*b^2*d ^2*f^2 - 8*a^3*b*B*d^2*f^2 + 33*a^4*C*d^2*f^2))/(15*b^3*(b*c - a*d)^2*(b*e - a*f)^2*(a + b*x))) + (2*(a + b*x)^(3/2)*(Sqrt[-a + (b*c)/d]*(48*a^4*C*d ^2*f^2 - 8*a^3*b*d*f*(B*d*f + 11*C*(d*e + c*f)) + b^4*(-2*A*d^2*e^2 + c*d* e*(5*B*e + 2*A*f) + c^2*(30*C*e^2 + 5*B*e*f - 2*A*f^2)) - a*b^3*(d^2*e*(3* B*e - 2*A*f) + c^2*f*(70*C*e + 3*B*f) + 2*c*d*(35*C*e^2 + 11*B*e*f - A*f^2 )) + a^2*b^2*(2*C*(19*d^2*e^2 + 81*c*d*e*f + 19*c^2*f^2) + d*f*(-2*A*d*f + 13*B*(d*e + c*f))))*(d + (b*c)/(a + b*x) - (a*d)/(a + b*x))*(f + (b*e)/(a + b*x) - (a*f)/(a + b*x)) + (I*(-(b*c) + a*d)*f*(-48*a^4*C*d^2*f^2 + 8*a^ 3*b*d*f*(B*d*f + 11*C*(d*e + c*f)) - b^4*(-2*A*d^2*e^2 + c*d*e*(5*B*e + 2* A*f) + c^2*(30*C*e^2 + 5*B*e*f - 2*A*f^2)) + a*b^3*(d^2*e*(3*B*e - 2*A*...
Time = 2.17 (sec) , antiderivative size = 996, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2117, 27, 167, 27, 167, 27, 176, 124, 123, 131, 131, 130}
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.
| \(\displaystyle \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 2117 |
| \(\displaystyle -\frac {2 \int -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (3 C (d e+c f) a^2-b (5 c C e+3 B d e+3 B c f-5 A d f) a+b^2 (5 B c e-2 A (d e+c f))-b \left (-\frac {6 C d f a^2}{b}+B d f a+5 C (d e+c f) a-b (5 c C e+A d f)\right ) x\right )}{2 b (a+b x)^{5/2}}dx}{5 (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (3 C (d e+c f) a^2-b (5 c C e+3 B d e+3 B c f-5 A d f) a+b^2 (5 B c e-2 A (d e+c f))-b \left (-\frac {6 C d f a^2}{b}+B d f a+5 C (d e+c f) a-b (5 c C e+A d f)\right ) x\right )}{(a+b x)^{5/2}}dx}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {\frac {2 \int -\frac {\sqrt {e+f x} \left (6 C d f (d e+3 c f) a^3-b \left (B d f (d e+3 c f)+C \left (8 d^2 e^2+41 c d f e+15 c^2 f^2\right )\right ) a^2+b^2 \left (30 C e f c^2+d \left (25 C e^2+6 B f e+3 A f^2\right ) c+d^2 e (3 B e-4 A f)\right ) a-b^3 e \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )+d \left (24 C d f^2 a^3-b f (41 C d e+23 c C f+4 B d f) a^2+b^2 (5 C e (3 d e+8 c f)+f (6 B d e+3 B c f-A d f)) a-b^3 \left (15 c C e^2+A d f e+c f (5 B e-2 A f)\right )\right ) x\right )}{2 (a+b x)^{3/2} \sqrt {c+d x}}dx}{3 b (b e-a f)}+\frac {2 \sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (B d f+8 C (c f+d e))+a b^2 (-4 A d f+3 B c f+3 B d e+10 c C e)-b^3 (5 B c e-2 A (c f+d e))\right )}{3 b (a+b x)^{3/2} (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (B d f+8 C (c f+d e))+a b^2 (-4 A d f+3 B c f+3 B d e+10 c C e)-b^3 (5 B c e-2 A (c f+d e))\right )}{3 b (a+b x)^{3/2} (b e-a f)}-\frac {\int \frac {\sqrt {e+f x} \left (6 C d f (d e+3 c f) a^3-b \left (B d f (d e+3 c f)+C \left (8 d^2 e^2+41 c d f e+15 c^2 f^2\right )\right ) a^2+b^2 \left (30 C e f c^2+d \left (25 C e^2+6 B f e+3 A f^2\right ) c+d^2 e (3 B e-4 A f)\right ) a-b^3 e \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )+d \left (24 C d f^2 a^3-b f (41 C d e+23 c C f+4 B d f) a^2+b^2 (5 C e (3 d e+8 c f)+f (6 B d e+3 B c f-A d f)) a-b^3 \left (15 c C e^2+A d f e+c f (5 B e-2 A f)\right )\right ) x\right )}{(a+b x)^{3/2} \sqrt {c+d x}}dx}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {\frac {2 \int -\frac {24 C d^2 f^2 (d e+c f) a^4-b d f \left (4 B d f (d e+c f)+C \left (41 d^2 e^2+94 c d f e+41 c^2 f^2\right )\right ) a^3+b^2 \left (C \left (15 d^3 e^3+104 c d^2 f e^2+104 c^2 d f^2 e+15 c^3 f^3\right )-d f \left (A d f (d e+c f)-2 B \left (3 d^2 e^2+7 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^3 \left (30 C e f^2 c^3+d f \left (80 C e^2+14 B f e+A f^2\right ) c^2+2 d^2 e \left (15 C e^2+7 B f e-3 A f^2\right ) c+A d^3 e^2 f\right ) a+b^4 c e \left (15 C e f c^2+d \left (15 C e^2+10 B f e-A f^2\right ) c-A d^2 e f\right )+d f \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{b (b c-a d)}-\frac {2 (b e-a f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {c+d x} \sqrt {e+f x}}{b (b c-a d) \sqrt {a+b x}}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {-\frac {2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right )}{b (b c-a d) \sqrt {a+b x}}-\frac {\int \frac {24 C d^2 f^2 (d e+c f) a^4-b d f \left (4 B d f (d e+c f)+C \left (41 d^2 e^2+94 c d f e+41 c^2 f^2\right )\right ) a^3+b^2 \left (C \left (15 d^3 e^3+104 c d^2 f e^2+104 c^2 d f^2 e+15 c^3 f^3\right )-d f \left (A d f (d e+c f)-2 B \left (3 d^2 e^2+7 c d f e+3 c^2 f^2\right )\right )\right ) a^2-b^3 \left (30 C e f^2 c^3+d f \left (80 C e^2+14 B f e+A f^2\right ) c^2+2 d^2 e \left (15 C e^2+7 B f e-3 A f^2\right ) c+A d^3 e^2 f\right ) a+b^4 c e \left (15 C e f c^2+d \left (15 C e^2+10 B f e-A f^2\right ) c-A d^2 e f\right )+d f \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{b (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\frac {2 \sqrt {c+d x} (e+f x)^{3/2} \left (6 a^3 C d f-a^2 b (B d f+8 C (c f+d e))+a b^2 (-4 A d f+3 B c f+3 B d e+10 c C e)-b^3 (5 B c e-2 A (c f+d e))\right )}{3 b (a+b x)^{3/2} (b e-a f)}-\frac {-\frac {(b e-a f) (d e-c f) \left (24 a^3 C d^2 f-a^2 b d (4 B d f+41 c C f+23 C d e)+a b^2 \left (d^2 (3 B e-A f)+c (6 B d f+40 C d e)+15 c^2 C f\right )-b^3 \left (c d (A f+5 B e)-2 A d^2 e+15 c^2 C e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+d \left (48 a^4 C d^2 f^2-8 a^3 b d f (B d f+11 C (c f+d e))+a^2 b^2 \left (2 C \left (19 c^2 f^2+81 c d e f+19 d^2 e^2\right )-d f (2 A d f-13 B (c f+d e))\right )-a b^3 \left (2 c d \left (-A f^2+11 B e f+35 C e^2\right )+d^2 e (3 B e-2 A f)+c^2 f (3 B f+70 C e)\right )-b^4 \left (-\left (c^2 \left (-2 A f^2+5 B e f+30 C e^2\right )\right )-c d e (2 A f+5 B e)+2 A d^2 e^2\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{b (b c-a d)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} (b e-a f) \left (24 a^3 C d^2 f-a^2 b d (4 B d f+41 c C f+23 C d e)+a b^2 \left (d^2 (3 B e-A f)+c (6 B d f+40 C d e)+15 c^2 C f\right )-b^3 \left (c d (A f+5 B e)-2 A d^2 e+15 c^2 C e\right )\right )}{b \sqrt {a+b x} (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{5 b (a+b x)^{5/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {-\frac {2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right )}{b (b c-a d) \sqrt {a+b x}}-\frac {(b e-a f) (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+\frac {d \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{\sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{b (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {-\frac {2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right )}{b (b c-a d) \sqrt {a+b x}}-\frac {\frac {2 \sqrt {d} \sqrt {a d-b c} \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+(b e-a f) (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{b (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {-\frac {2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right )}{b (b c-a d) \sqrt {a+b x}}-\frac {\frac {2 \sqrt {d} \sqrt {a d-b c} \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{\sqrt {c+d x}}}{b (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {-\frac {2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right )}{b (b c-a d) \sqrt {a+b x}}-\frac {\frac {2 \sqrt {d} \sqrt {a d-b c} \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {(b e-a f) (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{\sqrt {c+d x} \sqrt {e+f x}}}{b (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {\frac {2 \left (6 C d f a^3-b (B d f+8 C (d e+c f)) a^2+b^2 (10 c C e+3 B d e+3 B c f-4 A d f) a-b^3 (5 B c e-2 A (d e+c f))\right ) \sqrt {c+d x} (e+f x)^{3/2}}{3 b (b e-a f) (a+b x)^{3/2}}-\frac {-\frac {2 (b e-a f) \sqrt {c+d x} \sqrt {e+f x} \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right )}{b (b c-a d) \sqrt {a+b x}}-\frac {\frac {2 \sqrt {d} \sqrt {a d-b c} \left (48 C d^2 f^2 a^4-8 b d f (B d f+11 C (d e+c f)) a^3+b^2 \left (2 C \left (19 d^2 e^2+81 c d f e+19 c^2 f^2\right )-d f (2 A d f-13 B (d e+c f))\right ) a^2-b^3 \left (f (70 C e+3 B f) c^2+2 d \left (35 C e^2+11 B f e-A f^2\right ) c+d^2 e (3 B e-2 A f)\right ) a-b^4 \left (-\left (\left (30 C e^2+5 B f e-2 A f^2\right ) c^2\right )-d e (5 B e+2 A f) c+2 A d^2 e^2\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \left (24 C d^2 f a^3-b d (23 C d e+41 c C f+4 B d f) a^2+b^2 \left (15 C f c^2+(40 C d e+6 B d f) c+d^2 (3 B e-A f)\right ) a-b^3 \left (15 C e c^2+d (5 B e+A f) c-2 A d^2 e\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}}}{b (b c-a d)}}{3 b (b e-a f)}}{5 b (b c-a d) (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{5 b (b c-a d) (b e-a f) (a+b x)^{5/2}}\) |
(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(5*b*(b*c - a *d)*(b*e - a*f)*(a + b*x)^(5/2)) + ((2*(6*a^3*C*d*f + a*b^2*(10*c*C*e + 3* B*d*e + 3*B*c*f - 4*A*d*f) - b^3*(5*B*c*e - 2*A*(d*e + c*f)) - a^2*b*(B*d* f + 8*C*(d*e + c*f)))*Sqrt[c + d*x]*(e + f*x)^(3/2))/(3*b*(b*e - a*f)*(a + b*x)^(3/2)) - ((-2*(b*e - a*f)*(24*a^3*C*d^2*f - a^2*b*d*(23*C*d*e + 41*c *C*f + 4*B*d*f) - b^3*(15*c^2*C*e - 2*A*d^2*e + c*d*(5*B*e + A*f)) + a*b^2 *(15*c^2*C*f + d^2*(3*B*e - A*f) + c*(40*C*d*e + 6*B*d*f)))*Sqrt[c + d*x]* Sqrt[e + f*x])/(b*(b*c - a*d)*Sqrt[a + b*x]) - ((2*Sqrt[d]*Sqrt[-(b*c) + a *d]*(48*a^4*C*d^2*f^2 - 8*a^3*b*d*f*(B*d*f + 11*C*(d*e + c*f)) - b^4*(2*A* d^2*e^2 - c*d*e*(5*B*e + 2*A*f) - c^2*(30*C*e^2 + 5*B*e*f - 2*A*f^2)) - a* b^3*(d^2*e*(3*B*e - 2*A*f) + c^2*f*(70*C*e + 3*B*f) + 2*c*d*(35*C*e^2 + 11 *B*e*f - A*f^2)) + a^2*b^2*(2*C*(19*d^2*e^2 + 81*c*d*e*f + 19*c^2*f^2) - d *f*(2*A*d*f - 13*B*(d*e + c*f))))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[e + f*x]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f )]) + (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*(d*e - c*f)*(24*a^3*C*d^2*f - a^2* b*d*(23*C*d*e + 41*c*C*f + 4*B*d*f) - b^3*(15*c^2*C*e - 2*A*d^2*e + c*d*(5 *B*e + A*f)) + a*b^2*(15*c^2*C*f + d^2*(3*B*e - A*f) + c*(40*C*d*e + 6*B*d *f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*Elli pticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*...
3.1.65.3.1 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[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ [b/(b*e - a*f), 0] && SimplerQ[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f *x] && (PosQ[-(b*c - a*d)/d] || NegQ[-(b*e - a*f)/f])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x] Int[1/(Sq rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[(b*c - a*d)/b, 0] && Simpler Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_ .)*(x_))^(p_.), x_Symbol] :> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px, a + b*x, x]}, Simp[b*R*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Si mp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n* (e + f*x)^p*ExpandToSum[(m + 1)*(b*c - a*d)*(b*e - a*f)*Qx + a*d*f*R*(m + 1 ) - b*R*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*R*(m + n + p + 3)*x, x] , x], x]] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && PolyQ[Px, x] && LtQ[m, - 1] && IntegersQ[2*m, 2*n, 2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(2291\) vs. \(2(902)=1804\).
Time = 4.64 (sec) , antiderivative size = 2292, normalized size of antiderivative = 2.38
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2292\) |
| default | \(\text {Expression too large to display}\) | \(34614\) |
((b*x+a)*(d*x+c)*(f*x+e))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* (-2/5*(A*b^2-B*a*b+C*a^2)/b^6*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c *f*x+a*d*e*x+b*c*e*x+a*c*e)^(1/2)/(x+a/b)^3+2/15*(2*A*a*b^2*d*f-A*b^3*c*f- A*b^3*d*e-7*B*a^2*b*d*f+6*B*a*b^2*c*f+6*B*a*b^2*d*e-5*B*b^3*c*e+12*C*a^3*d *f-11*C*a^2*b*c*f-11*C*a^2*b*d*e+10*C*a*b^2*c*e)/b^5/(a^2*d*f-a*b*c*f-a*b* d*e+b^2*c*e)*(b*d*f*x^3+a*d*f*x^2+b*c*f*x^2+b*d*e*x^2+a*c*f*x+a*d*e*x+b*c* e*x+a*c*e)^(1/2)/(x+a/b)^2+2/15*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e)/(a^2*d*f -a*b*c*f-a*b*d*e+b^2*c*e)^2/b^4*(2*A*a^2*b^2*d^2*f^2-2*A*a*b^3*c*d*f^2-2*A *a*b^3*d^2*e*f+2*A*b^4*c^2*f^2-2*A*b^4*c*d*e*f+2*A*b^4*d^2*e^2+8*B*a^3*b*d ^2*f^2-13*B*a^2*b^2*c*d*f^2-13*B*a^2*b^2*d^2*e*f+3*B*a*b^3*c^2*f^2+22*B*a* b^3*c*d*e*f+3*B*a*b^3*d^2*e^2-5*B*b^4*c^2*e*f-5*B*b^4*c*d*e^2-33*C*a^4*d^2 *f^2+58*C*a^3*b*c*d*f^2+58*C*a^3*b*d^2*e*f-23*C*a^2*b^2*c^2*f^2-102*C*a^2* b^2*c*d*e*f-23*C*a^2*b^2*d^2*e^2+40*C*a*b^3*c^2*e*f+40*C*a*b^3*c*d*e^2-15* C*b^4*c^2*e^2)/((x+a/b)*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e))^(1/2)+2*((B*b*d *f-3*C*a*d*f+C*b*c*f+C*b*d*e)/b^4+1/15*d*f*(2*A*a*b^2*d*f-A*b^3*c*f-A*b^3* d*e-7*B*a^2*b*d*f+6*B*a*b^2*c*f+6*B*a*b^2*d*e-5*B*b^3*c*e+12*C*a^3*d*f-11* C*a^2*b*c*f-11*C*a^2*b*d*e+10*C*a*b^2*c*e)/b^4/(a^2*d*f-a*b*c*f-a*b*d*e+b^ 2*c*e)-1/15/b^4*(a*d*f-b*c*f-b*d*e)*(2*A*a^2*b^2*d^2*f^2-2*A*a*b^3*c*d*f^2 -2*A*a*b^3*d^2*e*f+2*A*b^4*c^2*f^2-2*A*b^4*c*d*e*f+2*A*b^4*d^2*e^2+8*B*a^3 *b*d^2*f^2-13*B*a^2*b^2*c*d*f^2-13*B*a^2*b^2*d^2*e*f+3*B*a*b^3*c^2*f^2+...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.86 (sec) , antiderivative size = 4721, normalized size of antiderivative = 4.90 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\text {Too large to display} \]
-2/45*(3*((15*C*a^4*b^4*d^3 + (8*C*a^2*b^6 + 2*B*a*b^7 + 3*A*b^8)*c^2*d - 5*(5*C*a^3*b^5 + A*a*b^7)*c*d^2)*e^2*f - (5*(5*C*a^3*b^5 + A*a*b^7)*c^2*d - 10*(7*C*a^4*b^4 - B*a^3*b^5 + A*a^2*b^6)*c*d^2 + (41*C*a^5*b^3 - 6*B*a^4 *b^4 + A*a^3*b^5)*d^3)*e*f^2 + (15*C*a^4*b^4*c^2*d - (41*C*a^5*b^3 - 6*B*a ^4*b^4 + A*a^3*b^5)*c*d^2 + (24*C*a^6*b^2 - 4*B*a^5*b^3 - A*a^4*b^4)*d^3)* f^3 + ((15*C*b^8*c^2*d - 5*(8*C*a*b^7 - B*b^8)*c*d^2 + (23*C*a^2*b^6 - 3*B *a*b^7 - 2*A*b^8)*d^3)*e^2*f - (5*(8*C*a*b^7 - B*b^8)*c^2*d - 2*(51*C*a^2* b^6 - 11*B*a*b^7 + A*b^8)*c*d^2 + (58*C*a^3*b^5 - 13*B*a^2*b^6 - 2*A*a*b^7 )*d^3)*e*f^2 + ((23*C*a^2*b^6 - 3*B*a*b^7 - 2*A*b^8)*c^2*d - (58*C*a^3*b^5 - 13*B*a^2*b^6 - 2*A*a*b^7)*c*d^2 + (33*C*a^4*b^4 - 8*B*a^3*b^5 - 2*A*a^2 *b^6)*d^3)*f^3)*x^2 + ((5*(4*C*a*b^7 + B*b^8)*c^2*d - (59*C*a^2*b^6 + B*a* b^7 - A*b^8)*c*d^2 + 5*(7*C*a^3*b^5 - A*a*b^7)*d^3)*e^2*f - ((59*C*a^2*b^6 + B*a*b^7 - A*b^8)*c^2*d - 20*(8*C*a^3*b^5 - B*a^2*b^6)*c*d^2 + (93*C*a^4 *b^4 - 13*B*a^3*b^5 - 7*A*a^2*b^6)*d^3)*e*f^2 + (5*(7*C*a^3*b^5 - A*a*b^7) *c^2*d - (93*C*a^4*b^4 - 13*B*a^3*b^5 - 7*A*a^2*b^6)*c*d^2 + 3*(18*C*a^5*b ^3 - 3*B*a^4*b^4 - 2*A*a^3*b^5)*d^3)*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*s qrt(f*x + e) - ((15*C*a^3*b^5*c^2*d - 5*(4*C*a^4*b^4 + B*a^3*b^5)*c*d^2 + (7*C*a^5*b^3 + 3*B*a^4*b^4 + 2*A*a^3*b^5)*d^3)*e^3 + (15*C*a^3*b^5*c^3 - 1 0*(13*C*a^4*b^4 - 2*B*a^3*b^5)*c^2*d + (182*C*a^5*b^3 - 22*B*a^4*b^4 - 3*A *a^3*b^5)*c*d^2 - (73*C*a^6*b^2 - 8*B*a^5*b^3 + 3*A*a^4*b^4)*d^3)*e^2*f...
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\int \frac {\sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )}{\left (a + b x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{7/2}} \, dx=\int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{7/2}} \,d x \]