Integrand size = 62, antiderivative size = 1128 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\frac {2 b d \left (9 a^3 C d f h-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {c+d x}}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}-\frac {2 b^2 \left (9 a^3 C d f h-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {a+b x}}-\frac {2 b \sqrt {d g-c h} \sqrt {f g-e h} \left (9 a^3 C d f h-b^3 (2 B d e g-c (3 C e g-2 B f g-2 B e h))+a b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h))-a^2 b (6 B d f h+5 C (d f g+d e h+c f h))\right ) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right )}{3 (b c-a d)^2 (b e-a f)^2 (b g-a h)^2 \sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}-\frac {2 \left (3 a^3 C d^2 f h-b^3 \left (2 B d^2 e g-B c^2 f h-c d (3 C e g-B f g-B e h)\right )-3 a^2 b d (B d f h+C (d f g+d e h-c f h))+a b^2 \left (3 B d^2 (f g+e h)+C \left (d^2 e g-c d f g-c d e h-2 c^2 f h\right )\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{3 (b c-a d)^2 (b e-a f) (b g-a h)^{3/2} \sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}} \]
2/3*b*d*(9*a^3*C*d*f*h-b^3*(2*B*d*e*g-c*(-2*B*e*h-2*B*f*g+3*C*e*g))+a*b^2* (C*(c*e*h+c*f*g+d*e*g)+4*B*(c*f*h+d*e*h+d*f*g))-a^2*b*(6*B*d*f*h+5*C*(c*f* h+d*e*h+d*f*g)))*(b*x+a)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(- a*f+b*e)^2/(-a*h+b*g)^2/(d*x+c)^(1/2)-2/3*b^2*(B*b-2*C*a)*(d*x+c)^(1/2)*(f *x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)/(-a*f+b*e)/(-a*h+b*g)/(b*x+a)^(3/2)-2 /3*b^2*(9*a^3*C*d*f*h-b^3*(2*B*d*e*g-c*(-2*B*e*h-2*B*f*g+3*C*e*g))+a*b^2*( C*(c*e*h+c*f*g+d*e*g)+4*B*(c*f*h+d*e*h+d*f*g))-a^2*b*(6*B*d*f*h+5*C*(c*f*h +d*e*h+d*f*g)))*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a *f+b*e)^2/(-a*h+b*g)^2/(b*x+a)^(1/2)-2/3*(3*a^3*C*d^2*f*h-b^3*(2*B*d^2*e*g -B*c^2*f*h-c*d*(-B*e*h-B*f*g+3*C*e*g))-3*a^2*b*d*(B*d*f*h+C*(-c*f*h+d*e*h+ d*f*g))+a*b^2*(3*B*d^2*(e*h+f*g)+C*(-2*c^2*f*h-c*d*e*h-c*d*f*g+d^2*e*g)))* EllipticF((-a*h+b*g)^(1/2)*(f*x+e)^(1/2)/(-e*h+f*g)^(1/2)/(b*x+a)^(1/2),(- (-a*d+b*c)*(-e*h+f*g)/(-c*f+d*e)/(-a*h+b*g))^(1/2))*((-a*f+b*e)*(d*x+c)/(- c*f+d*e)/(b*x+a))^(1/2)*(h*x+g)^(1/2)/(-a*d+b*c)^2/(-a*f+b*e)/(-a*h+b*g)^( 3/2)/(-e*h+f*g)^(1/2)/(d*x+c)^(1/2)/(-(-a*f+b*e)*(h*x+g)/(-e*h+f*g)/(b*x+a ))^(1/2)-2/3*b*(9*a^3*C*d*f*h-b^3*(2*B*d*e*g-c*(-2*B*e*h-2*B*f*g+3*C*e*g)) +a*b^2*(C*(c*e*h+c*f*g+d*e*g)+4*B*(c*f*h+d*e*h+d*f*g))-a^2*b*(6*B*d*f*h+5* C*(c*f*h+d*e*h+d*f*g)))*EllipticE((-c*h+d*g)^(1/2)*(f*x+e)^(1/2)/(-e*h+f*g )^(1/2)/(d*x+c)^(1/2),((-a*d+b*c)*(-e*h+f*g)/(-a*f+b*e)/(-c*h+d*g))^(1/2)) *(-c*h+d*g)^(1/2)*(-e*h+f*g)^(1/2)*(b*x+a)^(1/2)*(-(-c*f+d*e)*(h*x+g)/(...
Leaf count is larger than twice the leaf count of optimal. \(10836\) vs. \(2(1128)=2256\).
Time = 40.01 (sec) , antiderivative size = 10836, normalized size of antiderivative = 9.61 \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Result too large to show} \]
Integrate[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]),x]
Time = 4.57 (sec) , antiderivative size = 1105, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.145, Rules used = {2004, 2102, 2102, 2105, 27, 188, 194, 321, 327}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {a^2 (-C)+a b B+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx\) |
\(\Big \downarrow \) 2004 |
\(\displaystyle \int \frac {\frac {a b B-a^2 C}{a}+b C x}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx\) |
\(\Big \downarrow \) 2102 |
\(\displaystyle \frac {\int \frac {C (3 b c e g-a (d e g+c f g+c e h)) b^2+(b B-2 a C) (3 a d f h-b (d f g+d e h+c f h)) x b-(b B-a C) \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right )}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 2102 |
\(\displaystyle \frac {\frac {\int \frac {2 d f h \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) x^2 b^2+(b B-2 a C) (b c e g-a (d e g+c f g+c e h)) (3 a d f h-b (d f g+d e h+c f h)) b^2+(a d f h+b (d f g+d e h+c f h)) \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) x b-a (a d f h-b (d f g+d e h+c f h)) \left (b^2 C (3 b c e g-a (d e g+c f g+c e h))-(b B-a C) \left (3 d f h a^2-3 b (d f g+d e h+c f h) a+2 b^2 (d e g+c f g+c e h)\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} \left (9 a^3 C d f h-a^2 b (6 B d f h+5 C (c f h+d e h+d f g))+a b^2 (4 B (c f h+d e h+d f g)+C (c e h+c f g+d e g))+b^3 (-2 B c (e h+f g)-2 B d e g+3 c C e g)\right )}{\sqrt {a+b x} (b c-a d) (b e-a f) (b g-a h)}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x} (b B-2 a C)}{3 (a+b x)^{3/2} (b c-a d) (b e-a f) (b g-a h)}\) |
\(\Big \downarrow \) 2105 |
\(\displaystyle \frac {\frac {b (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {c+d x}}+\frac {\int -\frac {2 b d f (b e-a f) h (b g-a h) \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B (f g+e h)) c+2 B d^2 e g\right )\right )}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{2 b d f h}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {b (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {c+d x}}-(b e-a f) (b g-a h) \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B (f g+e h)) c+2 B d^2 e g\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}dx}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 188 |
\(\displaystyle \frac {\frac {b (d e-c f) (d g-c h) \int \frac {\sqrt {a+b x}}{(c+d x)^{3/2} \sqrt {e+f x} \sqrt {g+h x}}dx \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) (b g-a h) \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B (f g+e h)) c+2 B d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 194 |
\(\displaystyle \frac {\frac {-\frac {2 b (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) (b g-a h) \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B (f g+e h)) c+2 B d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \int \frac {1}{\sqrt {\frac {(b c-a d) (e+f x)}{(d e-c f) (a+b x)}+1} \sqrt {1-\frac {(b g-a h) (e+f x)}{(f g-e h) (a+b x)}}}d\frac {\sqrt {e+f x}}{\sqrt {a+b x}}}{(f g-e h) \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle \frac {\frac {-\frac {2 b (d g-c h) \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} \int \frac {\sqrt {1-\frac {(b c-a d) (e+f x)}{(b e-a f) (c+d x)}}}{\sqrt {1-\frac {(d g-c h) (e+f x)}{(f g-e h) (c+d x)}}}d\frac {\sqrt {e+f x}}{\sqrt {c+d x}} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) \sqrt {b g-a h} \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B (f g+e h)) c+2 B d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\frac {-\frac {2 b \sqrt {d g-c h} \sqrt {f g-e h} \sqrt {a+b x} \sqrt {-\frac {(d e-c f) (g+h x)}{(f g-e h) (c+d x)}} E\left (\arcsin \left (\frac {\sqrt {d g-c h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {c+d x}}\right )|\frac {(b c-a d) (f g-e h)}{(b e-a f) (d g-c h)}\right ) \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {\frac {(d e-c f) (a+b x)}{(b e-a f) (c+d x)}} \sqrt {g+h x}}+\frac {2 b d \sqrt {a+b x} \sqrt {e+f x} \sqrt {g+h x} \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right )}{\sqrt {c+d x}}-\frac {2 (b e-a f) \sqrt {b g-a h} \left (3 C d^2 f h a^3-3 b d (B d f h+C (d f g+d e h-c f h)) a^2+b^2 \left (3 B (f g+e h) d^2+C \left (-2 f h c^2-d (f g+e h) c+d^2 e g\right )\right ) a-b^3 \left (-B f h c^2-d (3 C e g-B (f g+e h)) c+2 B d^2 e g\right )\right ) \sqrt {\frac {(b e-a f) (c+d x)}{(d e-c f) (a+b x)}} \sqrt {g+h x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b g-a h} \sqrt {e+f x}}{\sqrt {f g-e h} \sqrt {a+b x}}\right ),-\frac {(b c-a d) (f g-e h)}{(d e-c f) (b g-a h)}\right )}{\sqrt {f g-e h} \sqrt {c+d x} \sqrt {-\frac {(b e-a f) (g+h x)}{(f g-e h) (a+b x)}}}}{(b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 \left (9 C d f h a^3-b (6 B d f h+5 C (d f g+d e h+c f h)) a^2+b^2 (C (d e g+c f g+c e h)+4 B (d f g+d e h+c f h)) a+b^3 (3 c C e g-2 B d e g-2 B c (f g+e h))\right ) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{(b c-a d) (b e-a f) (b g-a h) \sqrt {a+b x}}}{3 (b c-a d) (b e-a f) (b g-a h)}-\frac {2 b^2 (b B-2 a C) \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}}{3 (b c-a d) (b e-a f) (b g-a h) (a+b x)^{3/2}}\) |
Int[(a*b*B - a^2*C + b^2*B*x + b^2*C*x^2)/((a + b*x)^(7/2)*Sqrt[c + d*x]*S qrt[e + f*x]*Sqrt[g + h*x]),x]
(-2*b^2*(b*B - 2*a*C)*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/(3*(b*c - a*d)*(b*e - a*f)*(b*g - a*h)*(a + b*x)^(3/2)) + ((-2*b^2*(9*a^3*C*d*f*h + b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/((b*c - a*d)* (b*e - a*f)*(b*g - a*h)*Sqrt[a + b*x]) + ((2*b*d*(9*a^3*C*d*f*h + b^3*(3*c *C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d*e*h + c*f*h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[e + f*x]*Sqrt[g + h*x])/Sqrt[c + d*x] - (2*b*S qrt[d*g - c*h]*Sqrt[f*g - e*h]*(9*a^3*C*d*f*h + b^3*(3*c*C*e*g - 2*B*d*e*g - 2*B*c*(f*g + e*h)) + a*b^2*(C*(d*e*g + c*f*g + c*e*h) + 4*B*(d*f*g + d* e*h + c*f*h)) - a^2*b*(6*B*d*f*h + 5*C*(d*f*g + d*e*h + c*f*h)))*Sqrt[a + b*x]*Sqrt[-(((d*e - c*f)*(g + h*x))/((f*g - e*h)*(c + d*x)))]*EllipticE[Ar cSin[(Sqrt[d*g - c*h]*Sqrt[e + f*x])/(Sqrt[f*g - e*h]*Sqrt[c + d*x])], ((b *c - a*d)*(f*g - e*h))/((b*e - a*f)*(d*g - c*h))])/(Sqrt[((d*e - c*f)*(a + b*x))/((b*e - a*f)*(c + d*x))]*Sqrt[g + h*x]) - (2*(b*e - a*f)*Sqrt[b*g - a*h]*(3*a^3*C*d^2*f*h - 3*a^2*b*d*(B*d*f*h + C*(d*f*g + d*e*h - c*f*h)) - b^3*(2*B*d^2*e*g - B*c^2*f*h - c*d*(3*C*e*g - B*(f*g + e*h))) + a*b^2*(3* B*d^2*(f*g + e*h) + C*(d^2*e*g - 2*c^2*f*h - c*d*(f*g + e*h))))*Sqrt[((b*e - a*f)*(c + d*x))/((d*e - c*f)*(a + b*x))]*Sqrt[g + h*x]*EllipticF[Arc...
3.1.25.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[2*Sqrt[g + h*x]*(Sqrt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))]/((f*g - e*h)*Sqrt[c + d*x]*Sqrt[( -(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))])) Subst[Int[1/(Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]*Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))]), x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[Sqrt[(c_.) + (d_.)*(x_)]/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(e_.) + (f_.) *(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2*Sqrt[c + d*x]*(Sqrt[(-(b*e - a*f))*((g + h*x)/((f*g - e*h)*(a + b*x)))]/((b*e - a*f)*Sqrt[g + h*x]*Sq rt[(b*e - a*f)*((c + d*x)/((d*e - c*f)*(a + b*x)))])) Subst[Int[Sqrt[1 + (b*c - a*d)*(x^2/(d*e - c*f))]/Sqrt[1 - (b*g - a*h)*(x^2/(f*g - e*h))], x], x, Sqrt[e + f*x]/Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[(u_)*((d_) + (e_.)*(x_))^(q_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.) , x_Symbol] :> Int[u*(d + e*x)^(p + q)*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, b , c, d, e, q}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p]
Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_)))/(Sqrt[(c_.) + (d_.)*(x _)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[( A*b^2 - a*b*B)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x] /((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] - Simp[1/(2*(m + 1)*(b* c - a*d)*(b*e - a*f)*(b*g - a*h)) Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*S qrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m + 1) - 2*a*b*(m + 1)*(d *f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g + c*e*h)) - b*B*(a*(d* e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B)* x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B}, x] && IntegerQ[2*m ] && LtQ[m, -1]
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_. ) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbo l] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c + d*x ])), x] + (Simp[1/(2*b*d*f*h) Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*A*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d* f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Simp[C*(d*e - c*f)*((d*g - c*h)/(2*b*d*f*h)) Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[ e + f*x]*Sqrt[g + h*x]), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C} , x]
Leaf count of result is larger than twice the leaf count of optimal. \(3424\) vs. \(2(1056)=2112\).
Time = 10.27 (sec) , antiderivative size = 3425, normalized size of antiderivative = 3.04
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(3425\) |
default | \(\text {Expression too large to display}\) | \(110289\) |
int((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x+e)^(1 /2)/(h*x+g)^(1/2),x,method=_RETURNVERBOSE)
((b*x+a)*(d*x+c)*(f*x+e)*(h*x+g))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e )^(1/2)/(h*x+g)^(1/2)*(2/3/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+ a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g)*(B*b-2*C*a)*(b*d*f*h*x^4+a* d*f*h*x^3+b*c*f*h*x^3+b*d*e*h*x^3+b*d*f*g*x^3+a*c*f*h*x^2+a*d*e*h*x^2+a*d* f*g*x^2+b*c*e*h*x^2+b*c*f*g*x^2+b*d*e*g*x^2+a*c*e*h*x+a*c*f*g*x+a*d*e*g*x+ b*c*e*g*x+a*c*e*g)^(1/2)/(x+a/b)^2+2/3*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^ 2+b*d*f*g*x^2+b*c*e*h*x+b*c*f*g*x+b*d*e*g*x+b*c*e*g)*b/(a^3*d*f*h-a^2*b*c* f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f*g+a*b^2*d*e*g-b^3*c*e*g) ^2*(6*B*a^2*b*d*f*h-4*B*a*b^2*c*f*h-4*B*a*b^2*d*e*h-4*B*a*b^2*d*f*g+2*B*b^ 3*c*e*h+2*B*b^3*c*f*g+2*B*b^3*d*e*g-9*C*a^3*d*f*h+5*C*a^2*b*c*f*h+5*C*a^2* b*d*e*h+5*C*a^2*b*d*f*g-C*a*b^2*c*e*h-C*a*b^2*c*f*g-C*a*b^2*d*e*g-3*C*b^3* c*e*g)/((x+a/b)*(b*d*f*h*x^3+b*c*f*h*x^2+b*d*e*h*x^2+b*d*f*g*x^2+b*c*e*h*x +b*c*f*g*x+b*d*e*g*x+b*c*e*g))^(1/2)+2*(-1/3*(3*B*a*b*d*f*h-B*b^2*c*f*h-B* b^2*d*e*h-B*b^2*d*f*g-6*C*a^2*d*f*h+2*C*a*b*c*f*h+2*C*a*b*d*e*h+2*C*a*b*d* f*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b*d*f*g+a*b^2*c*e*h+a*b^2*c*f* g+a*b^2*d*e*g-b^3*c*e*g)+1/3*(a^2*d*f*h-a*b*c*f*h-a*b*d*e*h-a*b*d*f*g+b^2* c*e*h+b^2*c*f*g+b^2*d*e*g)*(6*B*a^2*b*d*f*h-4*B*a*b^2*c*f*h-4*B*a*b^2*d*e* h-4*B*a*b^2*d*f*g+2*B*b^3*c*e*h+2*B*b^3*c*f*g+2*B*b^3*d*e*g-9*C*a^3*d*f*h+ 5*C*a^2*b*c*f*h+5*C*a^2*b*d*e*h+5*C*a^2*b*d*f*g-C*a*b^2*c*e*h-C*a*b^2*c*f* g-C*a*b^2*d*e*g-3*C*b^3*c*e*g)/(a^3*d*f*h-a^2*b*c*f*h-a^2*b*d*e*h-a^2*b...
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="fricas")
integral((C*b*x - C*a + B*b)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e)*sqr t(h*x + g)/(b^3*d*f*h*x^6 + a^3*c*e*g + (b^3*d*f*g + (b^3*d*e + (b^3*c + 3 *a*b^2*d)*f)*h)*x^5 + ((b^3*d*e + (b^3*c + 3*a*b^2*d)*f)*g + ((b^3*c + 3*a *b^2*d)*e + 3*(a*b^2*c + a^2*b*d)*f)*h)*x^4 + (((b^3*c + 3*a*b^2*d)*e + 3* (a*b^2*c + a^2*b*d)*f)*g + (3*(a*b^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)* f)*h)*x^3 + ((3*(a*b^2*c + a^2*b*d)*e + (3*a^2*b*c + a^3*d)*f)*g + (a^3*c* f + (3*a^2*b*c + a^3*d)*e)*h)*x^2 + (a^3*c*e*h + (a^3*c*f + (3*a^2*b*c + a ^3*d)*e)*g)*x), x)
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\text {Timed out} \]
integrate((C*b**2*x**2+B*b**2*x+B*a*b-C*a**2)/(b*x+a)**(7/2)/(d*x+c)**(1/2 )/(f*x+e)**(1/2)/(h*x+g)**(1/2),x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="maxima")
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
\[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int { \frac {C b^{2} x^{2} + B b^{2} x - C a^{2} + B a b}{{\left (b x + a\right )}^{\frac {7}{2}} \sqrt {d x + c} \sqrt {f x + e} \sqrt {h x + g}} \,d x } \]
integrate((C*b^2*x^2+B*b^2*x+B*a*b-C*a^2)/(b*x+a)^(7/2)/(d*x+c)^(1/2)/(f*x +e)^(1/2)/(h*x+g)^(1/2),x, algorithm="giac")
integrate((C*b^2*x^2 + B*b^2*x - C*a^2 + B*a*b)/((b*x + a)^(7/2)*sqrt(d*x + c)*sqrt(f*x + e)*sqrt(h*x + g)), x)
Timed out. \[ \int \frac {a b B-a^2 C+b^2 B x+b^2 C x^2}{(a+b x)^{7/2} \sqrt {c+d x} \sqrt {e+f x} \sqrt {g+h x}} \, dx=\int \frac {-C\,a^2+B\,a\,b+C\,b^2\,x^2+B\,b^2\,x}{\sqrt {e+f\,x}\,\sqrt {g+h\,x}\,{\left (a+b\,x\right )}^{7/2}\,\sqrt {c+d\,x}} \,d x \]
int((C*b^2*x^2 - C*a^2 + B*a*b + B*b^2*x)/((e + f*x)^(1/2)*(g + h*x)^(1/2) *(a + b*x)^(7/2)*(c + d*x)^(1/2)),x)