Integrand size = 38, antiderivative size = 706 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\frac {2 \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^3 d f (b e-a f)}+\frac {2 \left (6 a^2 C d f+b^2 (c C e+5 A d f)-a b (C d e+c C f+5 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2}}{5 b^2 (b c-a d) f (b e-a f)}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{b (b c-a d) (b e-a f) \sqrt {a+b x}}+\frac {2 \sqrt {-b c+a d} \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (d e-c f) \left (24 a^2 C d f^2-a b f (7 C d e+c C f+20 B d f)+b^2 (5 d f (B e+3 A f)-C e (2 d e-c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^4 d^{3/2} f^2 \sqrt {c+d x} \sqrt {e+f x}} \]
-2*(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)^(1/2)+2/5*(6*a^2*C*d*f+b^2*(5*A*d*f+C*c*e)-a*b*(5*B*d*f+C*c*f+C*d *e))*(f*x+e)^(3/2)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/b^2/(-a*d+b*c)/f/(-a*f+b*e) +2/15*(24*a^2*C*d*f^2-a*b*f*(20*B*d*f+C*c*f+7*C*d*e)+b^2*(5*d*f*(3*A*f+B*e )-C*e*(-c*f+2*d*e)))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/d/f/(-a *f+b*e)+2/15*(48*a^2*C*d^2*f^2-8*a*b*d*f*(5*B*d*f+C*c*f+C*d*e)+b^2*(5*d*f* (6*A*d*f+B*c*f+B*d*e)-2*C*(c^2*f^2-c*d*e*f+d^2*e^2)))*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))*(a*d-b*c)^( 1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)/b^4/d^(3/2)/f^2/(d*x+c)^(1 /2)/(b*(f*x+e)/(-a*f+b*e))^(1/2)-2/15*(-c*f+d*e)*(24*a^2*C*d*f^2-a*b*f*(20 *B*d*f+C*c*f+7*C*d*e)+b^2*(5*d*f*(3*A*f+B*e)-C*e*(-c*f+2*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))*( a*d-b*c)^(1/2)*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e)/(-a*f+b*e))^(1/2)/b ^4/d^(3/2)/f^2/(d*x+c)^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 25.90 (sec) , antiderivative size = 633, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=-\frac {2 \left (-b^2 \sqrt {-a+\frac {b c}{d}} \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) (c+d x) (e+f x)+b^2 \sqrt {-a+\frac {b c}{d}} d f (c+d x) (e+f x) \left (15 \left (A b^2+a (-b B+a C)\right ) d f-(-9 a C d f+b (C d e+c C f+5 B d f)) (a+b x)-3 b C d f x (a+b x)\right )-i (b c-a d) f \left (48 a^2 C d^2 f^2-8 a b d f (C d e+c C f+5 B d f)+b^2 \left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d e f+c^2 f^2\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b f (d e-c f) \left (24 a^2 C d^2 f-a b d (C d e+7 c C f+20 B d f)+b^2 \left (-2 c^2 C f+15 A d^2 f+c d (C e+5 B f)\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )}{15 b^5 \sqrt {-a+\frac {b c}{d}} d^2 f^2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \]
(-2*(-(b^2*Sqrt[-a + (b*c)/d]*(48*a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e* f + c^2*f^2)))*(c + d*x)*(e + f*x)) + b^2*Sqrt[-a + (b*c)/d]*d*f*(c + d*x) *(e + f*x)*(15*(A*b^2 + a*(-(b*B) + a*C))*d*f - (-9*a*C*d*f + b*(C*d*e + c *C*f + 5*B*d*f))*(a + b*x) - 3*b*C*d*f*x*(a + b*x)) - I*(b*c - a*d)*f*(48* a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*(a + b*x)^(3/2)*Sqr t[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticE [I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a*d *f)] - I*b*f*(d*e - c*f)*(24*a^2*C*d^2*f - a*b*d*(C*d*e + 7*c*C*f + 20*B*d *f) + b^2*(-2*c^2*C*f + 15*A*d^2*f + c*d*(C*e + 5*B*f)))*(a + b*x)^(3/2)*S qrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*Ellipti cF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b*x]], (b*d*e - a*d*f)/(b*c*f - a *d*f)]))/(15*b^5*Sqrt[-a + (b*c)/d]*d^2*f^2*Sqrt[a + b*x]*Sqrt[c + d*x]*Sq rt[e + f*x])
Time = 1.42 (sec) , antiderivative size = 729, 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, 171, 27, 171, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2117 |
| \(\displaystyle -\frac {2 \int -\frac {\sqrt {c+d x} \sqrt {e+f x} \left (3 C (d e+c f) a^2-b (c C e+3 B d e+3 B c f-A d f) a+b^2 (B c e+2 A (d e+c f))+b \left (\frac {6 C d f a^2}{b}-(C d e+c C f+5 B d f) a+b (c C e+5 A d f)\right ) x\right )}{2 b \sqrt {a+b x}}dx}{(b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (3 C (d e+c f) a^2-b (c C e+3 B d e+3 B c f-A d f) a+b^2 (B c e+2 A (d e+c f))+b \left (\frac {6 C d f a^2}{b}-(C d e+c C f+5 B d f) a+b (c C e+5 A d f)\right ) x\right )}{\sqrt {a+b x}}dx}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\frac {2 \int \frac {(b c-a d) \sqrt {e+f x} \left (6 C f (d e+3 c f) a^2-b (5 B f (d e+3 c f)+C e (d e+7 c f)) a-b^2 \left (c C e^2-5 A d f e-5 c f (B e+2 A f)\right )+\left ((5 d f (B e+3 A f)-C e (2 d e-c f)) b^2-a f (7 C d e+c C f+20 B d f) b+24 a^2 C d f^2\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {(b c-a d) \int \frac {\sqrt {e+f x} \left (6 C f (d e+3 c f) a^2-b (5 B f (d e+3 c f)+C e (d e+7 c f)) a-b^2 \left (c C e^2-5 A d f e-5 c f (B e+2 A f)\right )+\left ((5 d f (B e+3 A f)-C e (2 d e-c f)) b^2-a f (7 C d e+c C f+20 B d f) b+24 a^2 C d f^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {2 \int \frac {(b e-a f) \left (24 C d f (d e+c f) a^2-b \left (20 B d f (d e+c f)+C \left (d^2 e^2+14 c d f e+c^2 f^2\right )\right ) a-b^2 \left (C e f c^2+d \left (C e^2-5 f (2 B e+3 A f)\right ) c-15 A d^2 e f\right )+\left (\left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a d f (C d e+c C f+5 B d f) b+48 a^2 C d^2 f^2\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \int \frac {24 C d f (d e+c f) a^2-b \left (20 B d f (d e+c f)+C \left (d^2 e^2+14 c d f e+c^2 f^2\right )\right ) a-b^2 \left (C e f c^2+d \left (C e^2-5 f (2 B e+3 A f)\right ) c-15 A d^2 e f\right )+\left (\left (5 d f (B d e+B c f+6 A d f)-2 C \left (d^2 e^2-c d f e+c^2 f^2\right )\right ) b^2-8 a d f (C d e+c C f+5 B d f) b+48 a^2 C d^2 f^2\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {\left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}-\frac {(d e-c f) \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {\sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(d e-c f) \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(d e-c f) \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}\right )}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}\right )}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {(d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}\right )}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {\frac {(b c-a d) \left (\frac {(b e-a f) \left (\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (48 a^2 C d^2 f^2-8 a b d f (5 B d f+c C f+C d e)+b^2 \left (5 d f (6 A d f+B c f+B d e)-2 C \left (c^2 f^2-c d e f+d^2 e^2\right )\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a d-b c} (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}\right )}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (24 a^2 C d f^2-a b f (20 B d f+c C f+7 C d e)+b^2 (5 d f (3 A f+B e)-C e (2 d e-c f))\right )}{3 b d}\right )}{5 b f}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} (e+f x)^{3/2} \left (\frac {6 a^2 C d f}{b}-a (5 B d f+c C f+C d e)+b (5 A d f+c C e)\right )}{5 f}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{b \sqrt {a+b x} (b c-a d) (b e-a f)}\) |
(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(b*(b*c - a*d )*(b*e - a*f)*Sqrt[a + b*x]) + ((2*((6*a^2*C*d*f)/b + b*(c*C*e + 5*A*d*f) - a*(C*d*e + c*C*f + 5*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*(e + f*x)^(3/2) )/(5*f) + ((b*c - a*d)*((2*(24*a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B *d*f) + b^2*(5*d*f*(B*e + 3*A*f) - C*e*(2*d*e - c*f)))*Sqrt[a + b*x]*Sqrt[ c + d*x]*Sqrt[e + f*x])/(3*b*d) + ((b*e - a*f)*((2*Sqrt[-(b*c) + a*d]*(48* a^2*C*d^2*f^2 - 8*a*b*d*f*(C*d*e + c*C*f + 5*B*d*f) + b^2*(5*d*f*(B*d*e + B*c*f + 6*A*d*f) - 2*C*(d^2*e^2 - c*d*e*f + c^2*f^2)))*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[d]*f*Sqrt[c + d*x ]*Sqrt[(b*(e + f*x))/(b*e - a*f)]) - (2*Sqrt[-(b*c) + a*d]*(d*e - c*f)*(24 *a^2*C*d*f^2 - a*b*f*(7*C*d*e + c*C*f + 20*B*d*f) + b^2*(5*d*f*(B*e + 3*A* f) - C*e*(2*d*e - c*f)))*Sqrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x) )/(b*e - a*f)]*EllipticF[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d] ], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*f*Sqrt[c + d*x]*Sqrt[e + f *x])))/(3*b*d)))/(5*b*f))/(b*(b*c - a*d)*(b*e - a*f))
3.1.63.3.1 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[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] && 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]
Time = 2.28 (sec) , antiderivative size = 1163, normalized size of antiderivative = 1.65
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1163\) |
| default | \(\text {Expression too large to display}\) | \(5787\) |
((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*(b*d*f*x^2+b*c*f*x+b*d*e*x+b*c*e)*(A*b^2-B*a*b+C*a^2)/b^4/((x+a/b)*(b* d*f*x^2+b*c*f*x+b*d*e*x+b*c*e))^(1/2)+2/5*C/b^2*x*(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)+2/3*(1/b^2*(B*b*d*f- C*a*d*f+C*b*c*f+C*b*d*e)-2/5*C/b^2*(2*a*d*f+2*b*c*f+2*b*d*e))/b/d/f*(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)+2* (-(A*a*b^2*d*f-A*b^3*c*f-A*b^3*d*e-B*a^2*b*d*f+B*a*b^2*c*f+B*a*b^2*d*e-B*b ^3*c*e+C*a^3*d*f-C*a^2*b*c*f-C*a^2*b*d*e+C*a*b^2*c*e)/b^4+(A*b^2-B*a*b+C*a ^2)/b^4*(a*d*f-b*c*f-b*d*e)+(b*c*f+b*d*e)*(A*b^2-B*a*b+C*a^2)/b^4-2/5*C/b^ 2*a*c*e-2/3*(1/b^2*(B*b*d*f-C*a*d*f+C*b*c*f+C*b*d*e)-2/5*C/b^2*(2*a*d*f+2* b*c*f+2*b*d*e))/b/d/f*(1/2*a*c*f+1/2*a*d*e+1/2*b*c*e))*(e/f-c/d)*((x+e/f)/ (e/f-c/d))^(1/2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(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) *EllipticF(((x+e/f)/(e/f-c/d))^(1/2),((-e/f+c/d)/(-e/f+a/b))^(1/2))+2*(1/b ^3*(A*b^2*d*f-B*a*b*d*f+B*b^2*c*f+B*b^2*d*e+C*a^2*d*f-C*a*b*c*f-C*a*b*d*e+ C*b^2*c*e)+(A*b^2-B*a*b+C*a^2)/b^3*d*f-2/5*C/b^2*(3/2*a*c*f+3/2*a*d*e+3/2* b*c*e)-2/3*(1/b^2*(B*b*d*f-C*a*d*f+C*b*c*f+C*b*d*e)-2/5*C/b^2*(2*a*d*f+2*b *c*f+2*b*d*e))/b/d/f*(a*d*f+b*c*f+b*d*e))*(e/f-c/d)*((x+e/f)/(e/f-c/d))^(1 /2)*((x+a/b)/(-e/f+a/b))^(1/2)*((x+c/d)/(-e/f+c/d))^(1/2)/(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)*((-e/f+a/...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 1463, normalized size of antiderivative = 2.07 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\text {Too large to display} \]
2/45*(3*(3*C*b^4*d^3*f^3*x^2 + C*a*b^3*d^3*e*f^2 + (C*a*b^3*c*d^2 - (24*C* a^2*b^2 - 20*B*a*b^3 + 15*A*b^4)*d^3)*f^3 + (C*b^4*d^3*e*f^2 + (C*b^4*c*d^ 2 - (6*C*a*b^3 - 5*B*b^4)*d^3)*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f* x + e) + (2*C*a*b^3*d^3*e^3 - (3*C*a*b^3*c*d^2 - (7*C*a^2*b^2 - 5*B*a*b^3) *d^3)*e^2*f - (3*C*a*b^3*c^2*d + 4*(7*C*a^2*b^2 - 5*B*a*b^3)*c*d^2 - (32*C *a^3*b - 25*B*a^2*b^2 + 15*A*a*b^3)*d^3)*e*f^2 + (2*C*a*b^3*c^3 + (7*C*a^2 *b^2 - 5*B*a*b^3)*c^2*d + (32*C*a^3*b - 25*B*a^2*b^2 + 15*A*a*b^3)*c*d^2 - 2*(24*C*a^4 - 20*B*a^3*b + 15*A*a^2*b^2)*d^3)*f^3 + (2*C*b^4*d^3*e^3 - (3 *C*b^4*c*d^2 - (7*C*a*b^3 - 5*B*b^4)*d^3)*e^2*f - (3*C*b^4*c^2*d + 4*(7*C* a*b^3 - 5*B*b^4)*c*d^2 - (32*C*a^2*b^2 - 25*B*a*b^3 + 15*A*b^4)*d^3)*e*f^2 + (2*C*b^4*c^3 + (7*C*a*b^3 - 5*B*b^4)*c^2*d + (32*C*a^2*b^2 - 25*B*a*b^3 + 15*A*b^4)*c*d^2 - 2*(24*C*a^3*b - 20*B*a^2*b^2 + 15*A*a*b^3)*d^3)*f^3)* x)*sqrt(b*d*f)*weierstrassPInverse(4/3*(b^2*d^2*e^2 - (b^2*c*d + a*b*d^2)* e*f + (b^2*c^2 - a*b*c*d + a^2*d^2)*f^2)/(b^2*d^2*f^2), -4/27*(2*b^3*d^3*e ^3 - 3*(b^3*c*d^2 + a*b^2*d^3)*e^2*f - 3*(b^3*c^2*d - 4*a*b^2*c*d^2 + a^2* b*d^3)*e*f^2 + (2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3)*f^3 )/(b^3*d^3*f^3), 1/3*(3*b*d*f*x + b*d*e + (b*c + a*d)*f)/(b*d*f)) + 3*(2*C *a*b^3*d^3*e^2*f - (2*C*a*b^3*c*d^2 - (8*C*a^2*b^2 - 5*B*a*b^3)*d^3)*e*f^2 + (2*C*a*b^3*c^2*d + (8*C*a^2*b^2 - 5*B*a*b^3)*c*d^2 - 2*(24*C*a^3*b - 20 *B*a^2*b^2 + 15*A*a*b^3)*d^3)*f^3 + (2*C*b^4*d^3*e^2*f - (2*C*b^4*c*d^2...
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\int \frac {\sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )}{\left (a + b x\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{3/2}} \, dx=\int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{3/2}} \,d x \]