Integrand size = 38, antiderivative size = 687 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=\frac {2 \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{3 b^3 (b c-a d) (b e-a f)}-\frac {2 (b B-2 a C) \sqrt {c+d x} (e+f x)^{3/2}}{b^2 (b e-a f) \sqrt {a+b x}}-\frac {2 \left (A b^2-a (b B-a C)\right ) (c+d x)^{3/2} (e+f x)^{3/2}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {2 \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d e f+c^2 f^2\right )\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^4 \sqrt {d} \sqrt {-b c+a d} f (b e-a f) \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}+\frac {2 (d e-c f) \left (8 a^2 C d f+b^2 (c C e+3 B c f+A d f)-a b (C d e+7 c C f+4 B d f)\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^4 \sqrt {d} \sqrt {-b c+a d} f \sqrt {c+d x} \sqrt {e+f x}} \]
-2/3*(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)^(3/2)-2*(B*b-2*C*a)*(f*x+e)^(3/2)*(d*x+c)^(1/2)/b^2/(-a*f+b*e)/ (b*x+a)^(1/2)+2/3*(8*a^2*C*d*f+b^2*(A*d*f+3*B*c*f+C*c*e)-a*b*(4*B*d*f+7*C* c*f+C*d*e))*(b*x+a)^(1/2)*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^3/(-a*d+b*c)/(-a*f +b*e)+2/3*(16*a^3*C*d^2*f^2-8*a^2*b*d*f*(B*d*f+2*C*(c*f+d*e))-b^3*(c^2*C*e *f+A*d^2*e*f+c*d*(A*f^2+6*B*e*f+C*e^2))+a*b^2*(d*f*(2*A*d*f+7*B*c*f+7*B*d* e)+C*(c^2*f^2+16*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))*(b*(d*x+c)/(-a*d+b*c))^(1/2)* (f*x+e)^(1/2)/b^4/f/(-a*f+b*e)/d^(1/2)/(a*d-b*c)^(1/2)/(d*x+c)^(1/2)/(b*(f *x+e)/(-a*f+b*e))^(1/2)+2/3*(-c*f+d*e)*(8*a^2*C*d*f+b^2*(A*d*f+3*B*c*f+C*c *e)-a*b*(4*B*d*f+7*C*c*f+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/f/d^(1/2)/(a*d-b*c)^(1/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)
Result contains complex when optimal does not.
Time = 28.99 (sec) , antiderivative size = 815, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} d f (c+d x) (e+f x) \left (\left (A b^2+a (-b B+a C)\right ) (b c-a d) (b e-a f)+\left (-8 a^3 C d f+b^3 (3 B c e+A d e+A c f)-2 a b^2 (3 c C e+2 B d e+2 B c f+A d f)+a^2 b (5 B d f+7 C (d e+c f))\right ) (a+b x)-C (b c-a d) (b e-a f) (a+b x)^2\right )+(a+b x) \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d e f+c^2 f^2\right )\right )\right ) (c+d x) (e+f x)+i (b c-a d) f \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (d e+c f))-b^3 \left (c^2 C e f+A d^2 e f+c d \left (C e^2+6 B e f+A f^2\right )\right )+a b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d e f+c^2 f^2\right )\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b (b c-a d) f (d e-c f) \left (8 a^2 C d f+b^2 (c C e+3 B d e+A d f)-a b (7 C d e+c C f+4 B d f)\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^5 \sqrt {-a+\frac {b c}{d}} d (b c-a d) f (b e-a f) (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \]
(-2*(b^2*Sqrt[-a + (b*c)/d]*d*f*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C))*(b*c - a*d)*(b*e - a*f) + (-8*a^3*C*d*f + b^3*(3*B*c*e + A*d*e + A*c *f) - 2*a*b^2*(3*c*C*e + 2*B*d*e + 2*B*c*f + A*d*f) + a^2*b*(5*B*d*f + 7*C *(d*e + c*f)))*(a + b*x) - C*(b*c - a*d)*(b*e - a*f)*(a + b*x)^2) + (a + b *x)*(b^2*Sqrt[-a + (b*c)/d]*(16*a^3*C*d^2*f^2 - 8*a^2*b*d*f*(B*d*f + 2*C*( d*e + c*f)) - b^3*(c^2*C*e*f + A*d^2*e*f + c*d*(C*e^2 + 6*B*e*f + A*f^2)) + a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*c*d*e*f + c^2 *f^2)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f*(16*a^3*C*d^2*f^2 - 8*a^2*b* d*f*(B*d*f + 2*C*(d*e + c*f)) - b^3*(c^2*C*e*f + A*d^2*e*f + c*d*(C*e^2 + 6*B*e*f + A*f^2)) + a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d^2*e^2 + 16*c*d*e*f + c^2*f^2)))*(a + b*x)^(3/2)*Sqrt[(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*(b*c - a*d)*f*(d*e - c*f)*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*d*e + A*d*f) - a*b*(7*C*d*e + c*C* f + 4*B*d*f))*(a + b*x)^(3/2)*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(b*(e + f*x))/(f*(a + b*x))]*EllipticF[I*ArcSinh[Sqrt[-a + (b*c)/d]/Sqrt[a + b* x]], (b*d*e - a*d*f)/(b*c*f - a*d*f)])))/(3*b^5*Sqrt[-a + (b*c)/d]*d*(b*c - a*d)*f*(b*e - a*f)*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])
Time = 1.44 (sec) , antiderivative size = 697, normalized size of antiderivative = 1.01, 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, 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.
| \(\displaystyle \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 2117 |
| \(\displaystyle -\frac {2 \int -\frac {3 \sqrt {c+d x} \sqrt {e+f x} \left (C (d e+c f) a^2-b (c C e+B d e+B c f-A d f) a+b^2 B c e+b \left (\frac {2 C d f a^2}{b}-(C d e+c C f+B d f) a+b (c C e+A d f)\right ) x\right )}{2 b (a+b x)^{3/2}}dx}{3 (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (C (d e+c f) a^2-b (c C e+B d e+B c f-A d f) a+b^2 B c e+b \left (\frac {2 C d f a^2}{b}-(C d e+c C f+B d f) a+b (c C e+A d f)\right ) x\right )}{(a+b x)^{3/2}}dx}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {\frac {2 \int \frac {(b e-a f) \sqrt {e+f x} \left (2 C d (d e+3 c f) a^2-b \left (5 C f c^2+3 d (C e+B f) c+B d^2 e\right ) a+b^2 c (c C e+B d e+2 B c f+A d f)+d \left (8 C d f a^2-b (C d e+7 c C f+4 B d f) a+b^2 (c C e+3 B c f+A d f)\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x}}dx}{b (b e-a f)}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\sqrt {e+f x} \left (2 C d (d e+3 c f) a^2-b \left (5 C f c^2+3 d (C e+B f) c+B d^2 e\right ) a+b^2 c (c C e+B d e+2 B c f+A d f)+d \left (8 C d f a^2-b (C d e+7 c C f+4 B d f) a+b^2 (c C e+3 B c f+A d f)\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}}dx}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {d \left ((b c e+a d e+a c f) \left (8 C d f a^2-b (C d e+7 c C f+4 B d f) a+b^2 (c C e+3 B c f+A d f)\right )-3 b e \left (2 C d (d e+3 c f) a^2-b \left (5 C f c^2+3 d (C e+B f) c+B d^2 e\right ) a+b^2 c (c C e+B d e+2 B c f+A d f)\right )+\left (16 C d^2 f^2 a^3-8 b d f (B d f+2 C (d e+c f)) a^2+b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d f e+c^2 f^2\right )\right ) a-b^3 \left (C e f c^2+d \left (C e^2+6 B f e+A f^2\right ) c+A d^2 e f\right )\right ) x\right )}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b d}+\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\int \frac {(b c e+a d e+a c f) \left (8 C d f a^2-b (C d e+7 c C f+4 B d f) a+b^2 (c C e+3 B c f+A d f)\right )-3 b e \left (2 C d (d e+3 c f) a^2-b \left (5 C f c^2+3 d (C e+B f) c+B d^2 e\right ) a+b^2 c (c C e+B d e+2 B c f+A d f)\right )+\left (16 C d^2 f^2 a^3-8 b d f (B d f+2 C (d e+c f)) a^2+b^2 \left (d f (7 B d e+7 B c f+2 A d f)+C \left (d^2 e^2+16 c d f e+c^2 f^2\right )\right ) a-b^3 \left (C e f c^2+d \left (C e^2+6 B f e+A f^2\right ) c+A d^2 e f\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\frac {(b e-a f) (d e-c f) \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{f}}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 124 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\frac {(b e-a f) (d e-c f) \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {\sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\frac {(b e-a f) (d e-c f) \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{f}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\frac {(b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{f \sqrt {c+d x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 131 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\frac {(b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{f \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
| \(\Big \downarrow \) 130 |
| \(\displaystyle \frac {\frac {\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right )}{3 b}-\frac {\frac {2 \sqrt {a d-b c} (b e-a f) (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (8 a^2 C d f-a b (4 B d f+7 c C f+C d e)+b^2 (A d f+3 B c f+c C e)\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (16 a^3 C d^2 f^2-8 a^2 b d f (B d f+2 C (c f+d e))+a b^2 \left (d f (2 A d f+7 B c f+7 B d e)+C \left (c^2 f^2+16 c d e f+d^2 e^2\right )\right )-b^3 \left (c d \left (A f^2+6 B e f+C e^2\right )+A d^2 e f+c^2 C e f\right )\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} f \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{3 b}}{b}-\frac {2 \sqrt {c+d x} (e+f x)^{3/2} (b B-2 a C) (b c-a d)}{b \sqrt {a+b x}}}{b (b c-a d) (b e-a f)}-\frac {2 (c+d x)^{3/2} (e+f x)^{3/2} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
(-2*(A*b^2 - a*(b*B - a*C))*(c + d*x)^(3/2)*(e + f*x)^(3/2))/(3*b*(b*c - a *d)*(b*e - a*f)*(a + b*x)^(3/2)) + ((-2*(b*B - 2*a*C)*(b*c - a*d)*Sqrt[c + d*x]*(e + f*x)^(3/2))/(b*Sqrt[a + b*x]) + ((2*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*c*f + A*d*f) - a*b*(C*d*e + 7*c*C*f + 4*B*d*f))*Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b) - ((2*Sqrt[-(b*c) + a*d]*(16*a^3*C*d^2*f^2 - 8 *a^2*b*d*f*(B*d*f + 2*C*(d*e + c*f)) - b^3*(c^2*C*e*f + A*d^2*e*f + c*d*(C *e^2 + 6*B*e*f + A*f^2)) + a*b^2*(d*f*(7*B*d*e + 7*B*c*f + 2*A*d*f) + C*(d ^2*e^2 + 16*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]*(b*e - a*f)*(d*e - c*f)*(8*a^2*C*d*f + b^2*(c*C*e + 3*B*c*f + A*d*f) - a*b*(C*d*e + 7*c*C*f + 4*B*d*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))/b)/(b*(b*c - a*d)*( b*e - a*f))
3.1.64.3.1 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[((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]
Leaf count of result is larger than twice the leaf count of optimal. \(1377\) vs. \(2(627)=1254\).
Time = 5.45 (sec) , antiderivative size = 1378, normalized size of antiderivative = 2.01
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1378\) |
| default | \(\text {Expression too large to display}\) | \(15769\) |
((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)* (-2/3*(A*b^2-B*a*b+C*a^2)/b^5*(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+2/3*(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)/b^4*(2*A*a*b^2*d*f-A*b^3*c*f-A*b ^3*d*e-5*B*a^2*b*d*f+4*B*a*b^2*c*f+4*B*a*b^2*d*e-3*B*b^3*c*e+8*C*a^3*d*f-7 *C*a^2*b*c*f-7*C*a^2*b*d*e+6*C*a*b^2*c*e)/((x+a/b)*(b*d*f*x^2+b*c*f*x+b*d* e*x+b*c*e))^(1/2)+2/3*C/b^3*(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*b^2*d*f-2*B*a*b*d*f+B*b^2*c*f+B*b^2* d*e+3*C*a^2*d*f-2*C*a*b*c*f-2*C*a*b*d*e+C*b^2*c*e)/b^4-1/3*(A*b^2-B*a*b+C* a^2)/b^4*d*f-1/3/b^4*(a*d*f-b*c*f-b*d*e)*(2*A*a*b^2*d*f-A*b^3*c*f-A*b^3*d* e-5*B*a^2*b*d*f+4*B*a*b^2*c*f+4*B*a*b^2*d*e-3*B*b^3*c*e+8*C*a^3*d*f-7*C*a^ 2*b*c*f-7*C*a^2*b*d*e+6*C*a*b^2*c*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)-1/3 *(b*c*f+b*d*e)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^4*(2*A*a*b^2*d*f-A*b^3* c*f-A*b^3*d*e-5*B*a^2*b*d*f+4*B*a*b^2*c*f+4*B*a*b^2*d*e-3*B*b^3*c*e+8*C*a^ 3*d*f-7*C*a^2*b*c*f-7*C*a^2*b*d*e+6*C*a*b^2*c*e)-2/3*C/b^3*(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*(B*b*d*f-2*C*a*d*f+C*b*c*f+C*b*d*e )-1/3/b^3*d*f*(2*A*a*b^2*d*f-A*b^3*c*f-A*b^3*d*e-5*B*a^2*b*d*f+4*B*a*b^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.27 (sec) , antiderivative size = 2588, normalized size of antiderivative = 3.77 \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=\text {Too large to display} \]
2/9*(3*(((6*C*a^2*b^4 - 2*B*a*b^5 - A*b^6)*c*d^2 - (7*C*a^3*b^3 - 3*B*a^2* b^4)*d^3)*e*f^2 - ((7*C*a^3*b^3 - 3*B*a^2*b^4)*c*d^2 - (8*C*a^4*b^2 - 4*B* a^3*b^3 + A*a^2*b^4)*d^3)*f^3 + ((C*b^6*c*d^2 - C*a*b^5*d^3)*e*f^2 - (C*a* b^5*c*d^2 - C*a^2*b^4*d^3)*f^3)*x^2 + (((8*C*a*b^5 - 3*B*b^6)*c*d^2 - (9*C *a^2*b^4 - 4*B*a*b^5 + A*b^6)*d^3)*e*f^2 - ((9*C*a^2*b^4 - 4*B*a*b^5 + A*b ^6)*c*d^2 - (10*C*a^3*b^3 - 5*B*a^2*b^4 + 2*A*a*b^5)*d^3)*f^3)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) - ((C*a^2*b^4*c*d^2 - C*a^3*b^3*d^3)*e^3 - (4*C*a^2*b^4*c^2*d - (11*C*a^3*b^3 - 3*B*a^2*b^4)*c*d^2 + (6*C*a^4*b^2 - 2*B*a^3*b^3 - A*a^2*b^4)*d^3)*e^2*f + (C*a^2*b^4*c^3 + (11*C*a^3*b^3 - 3 *B*a^2*b^4)*c^2*d - 2*(19*C*a^4*b^2 - 8*B*a^3*b^3 + 2*A*a^2*b^4)*c*d^2 + ( 24*C*a^5*b - 11*B*a^4*b^2 + 2*A*a^3*b^3)*d^3)*e*f^2 - (C*a^3*b^3*c^3 + (6* C*a^4*b^2 - 2*B*a^3*b^3 - A*a^2*b^4)*c^2*d - (24*C*a^5*b - 11*B*a^4*b^2 + 2*A*a^3*b^3)*c*d^2 + 2*(8*C*a^6 - 4*B*a^5*b + A*a^4*b^2)*d^3)*f^3 + ((C*b^ 6*c*d^2 - C*a*b^5*d^3)*e^3 - (4*C*b^6*c^2*d - (11*C*a*b^5 - 3*B*b^6)*c*d^2 + (6*C*a^2*b^4 - 2*B*a*b^5 - A*b^6)*d^3)*e^2*f + (C*b^6*c^3 + (11*C*a*b^5 - 3*B*b^6)*c^2*d - 2*(19*C*a^2*b^4 - 8*B*a*b^5 + 2*A*b^6)*c*d^2 + (24*C*a ^3*b^3 - 11*B*a^2*b^4 + 2*A*a*b^5)*d^3)*e*f^2 - (C*a*b^5*c^3 + (6*C*a^2*b^ 4 - 2*B*a*b^5 - A*b^6)*c^2*d - (24*C*a^3*b^3 - 11*B*a^2*b^4 + 2*A*a*b^5)*c *d^2 + 2*(8*C*a^4*b^2 - 4*B*a^3*b^3 + A*a^2*b^4)*d^3)*f^3)*x^2 + 2*((C*a*b ^5*c*d^2 - C*a^2*b^4*d^3)*e^3 - (4*C*a*b^5*c^2*d - (11*C*a^2*b^4 - 3*B*...
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=\int \frac {\sqrt {c + d x} \sqrt {e + f x} \left (A + B x + C x^{2}\right )}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c} \sqrt {f x + e}}{{\left (b x + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c+d x} \sqrt {e+f x} \left (A+B x+C x^2\right )}{(a+b x)^{5/2}} \, dx=\int \frac {\sqrt {e+f\,x}\,\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{{\left (a+b\,x\right )}^{5/2}} \,d x \]