Integrand size = 38, antiderivative size = 642 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d) (b e-a f) (a+b x)^{3/2}}+\frac {2 \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt {c+d x} \sqrt {e+f x}}{3 b (b c-a d)^2 (b e-a f)^2 \sqrt {a+b x}}-\frac {2 \sqrt {d} \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)-b^3 (3 B c e-2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 (-b c+a d)^{3/2} (b e-a f)^2 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \left (a^2 C d (d e-c f)-b^2 \left (3 c^2 C e-3 B c d e+2 A d^2 e+A c d f\right )+a b \left (3 \left (c^2 C+A d^2\right ) f-B d (d e+2 c f)\right )\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{3 b^2 \sqrt {d} (-b c+a d)^{3/2} (b e-a f) \sqrt {c+d x} \sqrt {e+f x}} \] Output:
-2/3*(A*b^2-a*(B*b-C*a))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b/(-a*d+b*c)/(-a*f+b* e)/(b*x+a)^(3/2)+2/3*(2*a^3*C*d*f+a*b^2*(-4*A*d*f+B*c*f+B*d*e+6*C*c*e)-b^3 *(3*B*c*e-2*A*(c*f+d*e))+a^2*b*(B*d*f-4*C*(c*f+d*e)))*(d*x+c)^(1/2)*(f*x+e )^(1/2)/b/(-a*d+b*c)^2/(-a*f+b*e)^2/(b*x+a)^(1/2)-2/3*d^(1/2)*(2*a^3*C*d*f +a*b^2*(-4*A*d*f+B*c*f+B*d*e+6*C*c*e)-b^3*(3*B*c*e-2*A*(c*f+d*e))+a^2*b*(B *d*f-4*C*(c*f+d*e)))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(f*x+e)^(1/2)*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 ^2/(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) -2/3*(a^2*C*d*(-c*f+d*e)-b^2*(A*c*d*f+2*A*d^2*e-3*B*c*d*e+3*C*c^2*e)+a*b*( 3*(A*d^2+C*c^2)*f-B*d*(2*c*f+d*e)))*(b*(d*x+c)/(-a*d+b*c))^(1/2)*(b*(f*x+e )/(-a*f+b*e))^(1/2)*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^2/d^(1/2)/(a*d-b*c)^(3/2)/(-a*f+b*e)/(d*x+c )^(1/2)/(f*x+e)^(1/2)
Result contains complex when optimal does not.
Time = 27.50 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.09 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=-\frac {2 \left (b^2 \sqrt {-a+\frac {b c}{d}} (c+d x) (e+f x) \left (\left (A b^2+a (-b B+a C)\right ) (b c-a d) (b e-a f)+\left (-2 a^3 C d f-a b^2 (6 c C e+B d e+B c f-4 A d f)+b^3 (3 B c e-2 A (d e+c f))+a^2 b (-B d f+4 C (d e+c f))\right ) (a+b x)\right )+(a+b x) \left (b^2 \sqrt {-a+\frac {b c}{d}} \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)+b^3 (-3 B c e+2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) (c+d x) (e+f x)+i (b c-a d) f \left (2 a^3 C d f+a b^2 (6 c C e+B d e+B c f-4 A d f)+b^3 (-3 B c e+2 A (d e+c f))+a^2 b (B d f-4 C (d e+c f))\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )-i b (b c-a d) \left (a^2 C f (d e-c f)+b^2 \left (3 c C e^2+A d e f+c f (-3 B e+2 A f)\right )+a b \left (-3 C d e^2+f (2 B d e+B c f-3 A d f)\right )\right ) (a+b x)^{3/2} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-a+\frac {b c}{d}}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )\right )\right )}{3 b^3 \sqrt {-a+\frac {b c}{d}} (b c-a d)^2 (b e-a f)^2 (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}} \] Input:
Integrate[(A + B*x + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]), x]
Output:
(-2*(b^2*Sqrt[-a + (b*c)/d]*(c + d*x)*(e + f*x)*((A*b^2 + a*(-(b*B) + a*C) )*(b*c - a*d)*(b*e - a*f) + (-2*a^3*C*d*f - a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(-(B*d*f) + 4*C*(d*e + c*f)))*(a + b*x)) + (a + b*x)*(b^2*Sqrt[-a + (b*c)/d]*(2*a^3*C*d*f + a* b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(c + d*x)*(e + f*x) + I*(b*c - a*d)*f* (2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) + b^3*(-3*B*c*e + 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(d*e + c*f)))*(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)*(a^2*C*f*(d*e - c*f) + b^2*(3*c*C*e^2 + A*d*e*f + c*f *(-3*B*e + 2*A*f)) + a*b*(-3*C*d*e^2 + f*(2*B*d*e + B*c*f - 3*A*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^3*Sqrt[-a + (b*c)/d]*(b*c - a*d)^2*(b*e - a*f )^2*(a + b*x)^(3/2)*Sqrt[c + d*x]*Sqrt[e + f*x])
Time = 1.37 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {2117, 27, 169, 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.
\(\displaystyle \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx\) |
\(\Big \downarrow \) 2117 |
\(\displaystyle -\frac {2 \int -\frac {C (d e+c f) a^2-b (3 c C e+B d e+B c f-3 A d f) a+b^2 (3 B c e-2 A (d e+c f))+b \left (\frac {2 C d f a^2}{b}-3 C d e a-3 c C f a+B d f a+3 b c C e-A b d f\right ) x}{2 b (a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {C (d e+c f) a^2-b (3 c C e+B d e+B c f-3 A d f) a+b^2 (3 B c e-2 A (d e+c f))+b \left (\frac {2 C d f a^2}{b}+B d f a-3 C (d e+c f) a+b (3 c C e-A d f)\right ) x}{(a+b x)^{3/2} \sqrt {c+d x} \sqrt {e+f x}}dx}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {2 \int \frac {C d f (d e+c f) a^3-b \left (C \left (3 d^2 e^2+5 c d f e+3 c^2 f^2\right )+d f (3 A d f-2 B (d e+c f))\right ) a^2+b^2 \left (6 C e f c^2+d \left (6 C e^2-4 B f e+A f^2\right ) c+A d^2 e f\right ) a-b^3 c e (3 c C e-A d f)+d f \left (2 C d f a^3+b (B d f-4 C (d e+c f)) a^2+b^2 (6 c C e+B d e+B c f-4 A d f) a-b^3 (3 B c e-2 A (d e+c f))\right ) x}{2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {\int \frac {C d f (d e+c f) a^3-b \left (C \left (3 d^2 e^2+5 c d f e+3 c^2 f^2\right )+d f (3 A d f-2 B (d e+c f))\right ) a^2+b^2 \left (6 C e f c^2+d \left (6 C e^2-4 B f e+A f^2\right ) c+A d^2 e f\right ) a-b^3 c e (3 c C e-A d f)+d f \left (2 C d f a^3+b (B d f-4 C (d e+c f)) a^2+b^2 (6 c C e+B d e+B c f-4 A d f) a-b^3 (3 B c e-2 A (d e+c f))\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {(b e-a f) \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+d \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}}dx}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 124 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {(b e-a f) \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+\frac {d \sqrt {e+f x} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}}dx}{\sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {(b e-a f) \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}dx+\frac {2 \sqrt {d} \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {\frac {(b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}}dx}{\sqrt {c+d x}}+\frac {2 \sqrt {d} \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 131 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {\frac {(b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}dx}{\sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {d} \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
\(\Big \downarrow \) 130 |
\(\displaystyle \frac {\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right )}{\sqrt {a+b x} (b c-a d) (b e-a f)}-\frac {\frac {2 \sqrt {a d-b c} (b e-a f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (a^2 C d (d e-c f)+a b \left (3 f \left (A d^2+c^2 C\right )-B d (2 c f+d e)\right )-b^2 \left (A c d f+2 A d^2 e-3 B c d e+3 c^2 C e\right )\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {d} \sqrt {c+d x} \sqrt {e+f x}}+\frac {2 \sqrt {d} \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} \left (2 a^3 C d f+a^2 b (B d f-4 C (c f+d e))+a b^2 (-4 A d f+B c f+B d e+6 c C e)-b^3 (3 B c e-2 A (c f+d e))\right ) E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{b \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}}{(b c-a d) (b e-a f)}}{3 b (b c-a d) (b e-a f)}-\frac {2 \sqrt {c+d x} \sqrt {e+f x} \left (A b^2-a (b B-a C)\right )}{3 b (a+b x)^{3/2} (b c-a d) (b e-a f)}\) |
Input:
Int[(A + B*x + C*x^2)/((a + b*x)^(5/2)*Sqrt[c + d*x]*Sqrt[e + f*x]),x]
Output:
(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[c + d*x]*Sqrt[e + f*x])/(3*b*(b*c - a*d)* (b*e - a*f)*(a + b*x)^(3/2)) + ((2*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B*d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*( d*e + c*f)))*Sqrt[c + d*x]*Sqrt[e + f*x])/((b*c - a*d)*(b*e - a*f)*Sqrt[a + b*x]) - ((2*Sqrt[d]*Sqrt[-(b*c) + a*d]*(2*a^3*C*d*f + a*b^2*(6*c*C*e + B *d*e + B*c*f - 4*A*d*f) - b^3*(3*B*c*e - 2*A*(d*e + c*f)) + a^2*b*(B*d*f - 4*C*(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)*(a^2*C*d*(d*e - c*f) - b^2*(3*c^2*C*e - 3*B*c*d*e + 2*A*d^2*e + A*c*d*f) + a*b*(3*(c^2*C + A*d^2)*f - B*d*(d*e + 2*c*f)))*S qrt[(b*(c + d*x))/(b*c - a*d)]*Sqrt[(b*(e + f*x))/(b*e - a*f)]*EllipticF[A rcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], ((b*c - a*d)*f)/(d*(b*e - a*f))])/(b*Sqrt[d]*Sqrt[c + d*x]*Sqrt[e + f*x]))/((b*c - a*d)*(b*e - a* f)))/(3*b*(b*c - a*d)*(b*e - a*f))
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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[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*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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. \(1248\) vs. \(2(588)=1176\).
Time = 21.47 (sec) , antiderivative size = 1249, normalized size of antiderivative = 1.95
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1249\) |
default | \(\text {Expression too large to display}\) | \(13626\) |
Input:
int((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x,method=_RETU RNVERBOSE)
Output:
((f*x+e)*(b*x+a)*(d*x+c))^(1/2)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2)* (-2/3/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^3*(A*b^2-B*a*b+C*a^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)/(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 )^2/b^2*(4*A*a*b^2*d*f-2*A*b^3*c*f-2*A*b^3*d*e-B*a^2*b*d*f-B*a*b^2*c*f-B*a *b^2*d*e+3*B*b^3*c*e-2*C*a^3*d*f+4*C*a^2*b*c*f+4*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*(C/b^2-1/3*d*f/b^2*( A*b^2-B*a*b+C*a^2)/(a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)+1/3/b^2*(a*d*f-b*c*f- b*d*e)*(4*A*a*b^2*d*f-2*A*b^3*c*f-2*A*b^3*d*e-B*a^2*b*d*f-B*a*b^2*c*f-B*a* b^2*d*e+3*B*b^3*c*e-2*C*a^3*d*f+4*C*a^2*b*c*f+4*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)^2+1/3*(b*c*f+b*d*e)/(a^2*d*f-a*b*c*f-a* b*d*e+b^2*c*e)^2/b^2*(4*A*a*b^2*d*f-2*A*b^3*c*f-2*A*b^3*d*e-B*a^2*b*d*f-B* a*b^2*c*f-B*a*b^2*d*e+3*B*b^3*c*e-2*C*a^3*d*f+4*C*a^2*b*c*f+4*C*a^2*b*d*e- 6*C*a*b^2*c*e))*(c/d-a/b)*((x+c/d)/(c/d-a/b))^(1/2)*((x+e/f)/(-c/d+e/f))^( 1/2)*((x+a/b)/(-c/d+a/b))^(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+c/d)/(c/d-a/b))^(1/2),(( -c/d+a/b)/(-c/d+e/f))^(1/2))+2/3/b*d*f*(4*A*a*b^2*d*f-2*A*b^3*c*f-2*A*b^3* d*e-B*a^2*b*d*f-B*a*b^2*c*f-B*a*b^2*d*e+3*B*b^3*c*e-2*C*a^3*d*f+4*C*a^2*b* c*f+4*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)^2*(c/d- a/b)*((x+c/d)/(c/d-a/b))^(1/2)*((x+e/f)/(-c/d+e/f))^(1/2)*((x+a/b)/(-c/...
Leaf count of result is larger than twice the leaf count of optimal. 2344 vs. \(2 (587) = 1174\).
Time = 0.28 (sec) , antiderivative size = 2344, normalized size of antiderivative = 3.65 \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor ithm="fricas")
Output:
2/9*(3*(((5*C*a^2*b^4 - 2*B*a*b^5 - A*b^6)*c*d - 3*(C*a^3*b^3 - A*a*b^5)*d ^2)*e*f - (3*(C*a^3*b^3 - A*a*b^5)*c*d - (C*a^4*b^2 + 2*B*a^3*b^3 - 5*A*a^ 2*b^4)*d^2)*f^2 + ((3*(2*C*a*b^5 - B*b^6)*c*d - (4*C*a^2*b^4 - B*a*b^5 - 2 *A*b^6)*d^2)*e*f - ((4*C*a^2*b^4 - B*a*b^5 - 2*A*b^6)*c*d - (2*C*a^3*b^3 + B*a^2*b^4 - 4*A*a*b^5)*d^2)*f^2)*x)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(f*x + e) + ((9*C*a^2*b^4*c^2 - 3*(4*C*a^3*b^3 + B*a^2*b^4)*c*d + (5*C*a^4*b^2 + B*a^3*b^3 + 2*A*a^2*b^4)*d^2)*e^2 - (3*(4*C*a^3*b^3 + B*a^2*b^4)*c^2 - ( 13*C*a^4*b^2 + 11*B*a^3*b^3 + A*a^2*b^4)*c*d + (5*C*a^5*b + 4*B*a^4*b^2 + 5*A*a^3*b^3)*d^2)*e*f + ((5*C*a^4*b^2 + B*a^3*b^3 + 2*A*a^2*b^4)*c^2 - (5* C*a^5*b + 4*B*a^4*b^2 + 5*A*a^3*b^3)*c*d + (2*C*a^6 + B*a^5*b + 5*A*a^4*b^ 2)*d^2)*f^2 + ((9*C*b^6*c^2 - 3*(4*C*a*b^5 + B*b^6)*c*d + (5*C*a^2*b^4 + B *a*b^5 + 2*A*b^6)*d^2)*e^2 - (3*(4*C*a*b^5 + B*b^6)*c^2 - (13*C*a^2*b^4 + 11*B*a*b^5 + A*b^6)*c*d + (5*C*a^3*b^3 + 4*B*a^2*b^4 + 5*A*a*b^5)*d^2)*e*f + ((5*C*a^2*b^4 + B*a*b^5 + 2*A*b^6)*c^2 - (5*C*a^3*b^3 + 4*B*a^2*b^4 + 5 *A*a*b^5)*c*d + (2*C*a^4*b^2 + B*a^3*b^3 + 5*A*a^2*b^4)*d^2)*f^2)*x^2 + 2* ((9*C*a*b^5*c^2 - 3*(4*C*a^2*b^4 + B*a*b^5)*c*d + (5*C*a^3*b^3 + B*a^2*b^4 + 2*A*a*b^5)*d^2)*e^2 - (3*(4*C*a^2*b^4 + B*a*b^5)*c^2 - (13*C*a^3*b^3 + 11*B*a^2*b^4 + A*a*b^5)*c*d + (5*C*a^4*b^2 + 4*B*a^3*b^3 + 5*A*a^2*b^4)*d^ 2)*e*f + ((5*C*a^3*b^3 + B*a^2*b^4 + 2*A*a*b^5)*c^2 - (5*C*a^4*b^2 + 4*B*a ^3*b^3 + 5*A*a^2*b^4)*c*d + (2*C*a^5*b + B*a^4*b^2 + 5*A*a^3*b^3)*d^2)*...
\[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {A + B x + C x^{2}}{\left (a + b x\right )^{\frac {5}{2}} \sqrt {c + d x} \sqrt {e + f x}}\, dx \] Input:
integrate((C*x**2+B*x+A)/(b*x+a)**(5/2)/(d*x+c)**(1/2)/(f*x+e)**(1/2),x)
Output:
Integral((A + B*x + C*x**2)/((a + b*x)**(5/2)*sqrt(c + d*x)*sqrt(e + f*x)) , x)
\[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor ithm="maxima")
Output:
integrate((C*x^2 + B*x + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)
\[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int { \frac {C x^{2} + B x + A}{{\left (b x + a\right )}^{\frac {5}{2}} \sqrt {d x + c} \sqrt {f x + e}} \,d x } \] Input:
integrate((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x, algor ithm="giac")
Output:
integrate((C*x^2 + B*x + A)/((b*x + a)^(5/2)*sqrt(d*x + c)*sqrt(f*x + e)), x)
Timed out. \[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\int \frac {C\,x^2+B\,x+A}{\sqrt {e+f\,x}\,{\left (a+b\,x\right )}^{5/2}\,\sqrt {c+d\,x}} \,d x \] Input:
int((A + B*x + C*x^2)/((e + f*x)^(1/2)*(a + b*x)^(5/2)*(c + d*x)^(1/2)),x)
Output:
int((A + B*x + C*x^2)/((e + f*x)^(1/2)*(a + b*x)^(5/2)*(c + d*x)^(1/2)), x )
\[ \int \frac {A+B x+C x^2}{(a+b x)^{5/2} \sqrt {c+d x} \sqrt {e+f x}} \, dx=\text {too large to display} \] Input:
int((C*x^2+B*x+A)/(b*x+a)^(5/2)/(d*x+c)^(1/2)/(f*x+e)^(1/2),x)
Output:
(2*sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*b + 2*int((sqrt(e + f*x)*sqrt (c + d*x)*sqrt(a + b*x)*x**2)/(a**4*c*d*e*f + a**4*c*d*f**2*x + a**4*d**2* e*f*x + a**4*d**2*f**2*x**2 - a**3*b*c**2*e*f - a**3*b*c**2*f**2*x - a**3* b*c*d*e**2 + a**3*b*c*d*e*f*x + 2*a**3*b*c*d*f**2*x**2 - a**3*b*d**2*e**2* x + 2*a**3*b*d**2*e*f*x**2 + 3*a**3*b*d**2*f**2*x**3 - 3*a**2*b**2*c**2*e* f*x - 3*a**2*b**2*c**2*f**2*x**2 - 3*a**2*b**2*c*d*e**2*x - 3*a**2*b**2*c* d*e*f*x**2 - 3*a**2*b**2*d**2*e**2*x**2 + 3*a**2*b**2*d**2*f**2*x**4 - 3*a *b**3*c**2*e*f*x**2 - 3*a*b**3*c**2*f**2*x**3 - 3*a*b**3*c*d*e**2*x**2 - 5 *a*b**3*c*d*e*f*x**3 - 2*a*b**3*c*d*f**2*x**4 - 3*a*b**3*d**2*e**2*x**3 - 2*a*b**3*d**2*e*f*x**4 + a*b**3*d**2*f**2*x**5 - b**4*c**2*e*f*x**3 - b**4 *c**2*f**2*x**4 - b**4*c*d*e**2*x**3 - 2*b**4*c*d*e*f*x**4 - b**4*c*d*f**2 *x**5 - b**4*d**2*e**2*x**4 - b**4*d**2*e*f*x**5),x)*a**4*c*d**2*f**2 + in t((sqrt(e + f*x)*sqrt(c + d*x)*sqrt(a + b*x)*x**2)/(a**4*c*d*e*f + a**4*c* d*f**2*x + a**4*d**2*e*f*x + a**4*d**2*f**2*x**2 - a**3*b*c**2*e*f - a**3* b*c**2*f**2*x - a**3*b*c*d*e**2 + a**3*b*c*d*e*f*x + 2*a**3*b*c*d*f**2*x** 2 - a**3*b*d**2*e**2*x + 2*a**3*b*d**2*e*f*x**2 + 3*a**3*b*d**2*f**2*x**3 - 3*a**2*b**2*c**2*e*f*x - 3*a**2*b**2*c**2*f**2*x**2 - 3*a**2*b**2*c*d*e* *2*x - 3*a**2*b**2*c*d*e*f*x**2 - 3*a**2*b**2*d**2*e**2*x**2 + 3*a**2*b**2 *d**2*f**2*x**4 - 3*a*b**3*c**2*e*f*x**2 - 3*a*b**3*c**2*f**2*x**3 - 3*a*b **3*c*d*e**2*x**2 - 5*a*b**3*c*d*e*f*x**3 - 2*a*b**3*c*d*f**2*x**4 - 3*...