Integrand size = 29, antiderivative size = 583 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=-\frac {2 A}{b d (d+e x)^{3/2} \sqrt {b x-c x^2}}+\frac {2 c (b B d+2 A c d+A b e) x}{b^2 d (c d+b e) (d+e x)^{3/2} \sqrt {b x-c x^2}}-\frac {2 e \left (6 A c^2 d^2-b^2 e (B d-4 A e)+3 b c d (B d+2 A e)\right ) \sqrt {b x-c x^2}}{3 b^2 d^2 (c d+b e)^2 (d+e x)^{3/2}}-\frac {2 e \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)-2 b^3 e^2 (B d-4 A e)+3 b c^2 d^2 (B d+3 A e)\right ) \sqrt {b x-c x^2}}{3 b^2 d^3 (c d+b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {c} \left (6 A c^3 d^3-b^2 c d e (7 B d-19 A e)-2 b^3 e^2 (B d-4 A e)+3 b c^2 d^2 (B d+3 A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{3 b^{3/2} d^3 (c d+b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}+\frac {2 \sqrt {c} \left (6 A c^2 d^2-b^2 e (B d-4 A e)+3 b c d (B d+2 A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{3 b^{3/2} d^2 (c d+b e)^2 \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:
-2*A/b/d/(e*x+d)^(3/2)/(-c*x^2+b*x)^(1/2)+2*c*(A*b*e+2*A*c*d+B*b*d)*x/b^2/ d/(b*e+c*d)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(1/2)-2/3*e*(6*A*c^2*d^2-b^2*e*(-4* A*e+B*d)+3*b*c*d*(2*A*e+B*d))*(-c*x^2+b*x)^(1/2)/b^2/d^2/(b*e+c*d)^2/(e*x+ d)^(3/2)-2/3*e*(6*A*c^3*d^3-b^2*c*d*e*(-19*A*e+7*B*d)-2*b^3*e^2*(-4*A*e+B* d)+3*b*c^2*d^2*(3*A*e+B*d))*(-c*x^2+b*x)^(1/2)/b^2/d^3/(b*e+c*d)^3/(e*x+d) ^(1/2)-2/3*c^(1/2)*(6*A*c^3*d^3-b^2*c*d*e*(-19*A*e+7*B*d)-2*b^3*e^2*(-4*A* e+B*d)+3*b*c^2*d^2*(3*A*e+B*d))*x^(1/2)*(1-c*x/b)^(1/2)*(e*x+d)^(1/2)*Elli pticE(c^(1/2)*x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/b^(3/2)/d^3/(b*e+c*d)^3/(1 +e*x/d)^(1/2)/(-c*x^2+b*x)^(1/2)+2/3*c^(1/2)*(6*A*c^2*d^2-b^2*e*(-4*A*e+B* d)+3*b*c*d*(2*A*e+B*d))*x^(1/2)*(1-c*x/b)^(1/2)*(1+e*x/d)^(1/2)*EllipticF( c^(1/2)*x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/b^(3/2)/d^2/(b*e+c*d)^2/(e*x+d)^ (1/2)/(-c*x^2+b*x)^(1/2)
Result contains complex when optimal does not.
Time = 28.44 (sec) , antiderivative size = 525, normalized size of antiderivative = 0.90 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\frac {2 \left (\sqrt {-\frac {b}{c}} \left (b^2 d e^2 (B d-A e) (c d+b e) x (b-c x)+b^2 e^2 (-5 A e (2 c d+b e)+B d (7 c d+2 b e)) x (b-c x) (d+e x)+3 c^3 (b B+A c) d^3 x (d+e x)^2+3 A (c d+b e)^3 (-b+c x) (d+e x)^2\right )+(d+e x) \left (\sqrt {-\frac {b}{c}} \left (6 A c^3 d^3+3 b c^2 d^2 (B d+3 A e)+2 b^3 e^2 (-B d+4 A e)+b^2 c d e (-7 B d+19 A e)\right ) (b-c x) (d+e x)+i b e \left (6 A c^3 d^3+3 b c^2 d^2 (B d+3 A e)+2 b^3 e^2 (-B d+4 A e)+b^2 c d e (-7 B d+19 A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )-i b e (c d+b e) \left (3 A c^2 d^2+2 b^2 e (-B d+4 A e)+3 b c d (-2 B d+5 A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )\right )\right )}{3 b^2 \sqrt {-\frac {b}{c}} d^3 (c d+b e)^3 \sqrt {x (b-c x)} (d+e x)^{3/2}} \] Input:
Integrate[(A + B*x)/((d + e*x)^(5/2)*(b*x - c*x^2)^(3/2)),x]
Output:
(2*(Sqrt[-(b/c)]*(b^2*d*e^2*(B*d - A*e)*(c*d + b*e)*x*(b - c*x) + b^2*e^2* (-5*A*e*(2*c*d + b*e) + B*d*(7*c*d + 2*b*e))*x*(b - c*x)*(d + e*x) + 3*c^3 *(b*B + A*c)*d^3*x*(d + e*x)^2 + 3*A*(c*d + b*e)^3*(-b + c*x)*(d + e*x)^2) + (d + e*x)*(Sqrt[-(b/c)]*(6*A*c^3*d^3 + 3*b*c^2*d^2*(B*d + 3*A*e) + 2*b^ 3*e^2*(-(B*d) + 4*A*e) + b^2*c*d*e*(-7*B*d + 19*A*e))*(b - c*x)*(d + e*x) + I*b*e*(6*A*c^3*d^3 + 3*b*c^2*d^2*(B*d + 3*A*e) + 2*b^3*e^2*(-(B*d) + 4*A *e) + b^2*c*d*e*(-7*B*d + 19*A*e))*Sqrt[1 - b/(c*x)]*Sqrt[1 + d/(e*x)]*x^( 3/2)*EllipticE[I*ArcSinh[Sqrt[-(b/c)]/Sqrt[x]], -((c*d)/(b*e))] - I*b*e*(c *d + b*e)*(3*A*c^2*d^2 + 2*b^2*e*(-(B*d) + 4*A*e) + 3*b*c*d*(-2*B*d + 5*A* e))*Sqrt[1 - b/(c*x)]*Sqrt[1 + d/(e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[- (b/c)]/Sqrt[x]], -((c*d)/(b*e))])))/(3*b^2*Sqrt[-(b/c)]*d^3*(c*d + b*e)^3* Sqrt[x*(b - c*x)]*(d + e*x)^(3/2))
Time = 1.83 (sec) , antiderivative size = 591, normalized size of antiderivative = 1.01, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1235, 27, 1237, 27, 1237, 27, 1269, 1169, 122, 120, 127, 126}
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.
\(\displaystyle \int \frac {A+B x}{\left (b x-c x^2\right )^{3/2} (d+e x)^{5/2}} \, dx\) |
\(\Big \downarrow \) 1235 |
\(\displaystyle \frac {2 \int \frac {e (b (b B d-3 A c d-4 A b e)+3 c (b B d+2 A c d+A b e) x)}{2 (d+e x)^{5/2} \sqrt {b x-c x^2}}dx}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \int \frac {b (b B d-3 A c d-4 A b e)+3 c (b B d+2 A c d+A b e) x}{(d+e x)^{5/2} \sqrt {b x-c x^2}}dx}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 1237 |
\(\displaystyle \frac {e \left (\frac {2 \int -\frac {b \left (-2 e (B d-4 A e) b^2-3 c d (2 B d-5 A e) b+3 A c^2 d^2\right )-c \left (-e (B d-4 A e) b^2+3 c d (B d+2 A e) b+6 A c^2 d^2\right ) x}{2 (d+e x)^{3/2} \sqrt {b x-c x^2}}dx}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \left (-\frac {\int \frac {b \left (-2 e (B d-4 A e) b^2-3 c d (2 B d-5 A e) b+3 A c^2 d^2\right )-c \left (-e (B d-4 A e) b^2+3 c d (B d+2 A e) b+6 A c^2 d^2\right ) x}{(d+e x)^{3/2} \sqrt {b x-c x^2}}dx}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 1237 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \int -\frac {c \left (b d \left (e (B d-4 A e) b^2+9 c d (B d-A e) b+3 A c^2 d^2\right )-\left (-2 e^2 (B d-4 A e) b^3-c d e (7 B d-19 A e) b^2+3 c^2 d^2 (B d+3 A e) b+6 A c^3 d^3\right ) x\right )}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{d (b e+c d)}+\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \int \frac {b d \left (e (B d-4 A e) b^2+9 c d (B d-A e) b+3 A c^2 d^2\right )-\left (-2 e^2 (B d-4 A e) b^3-c d e (7 B d-19 A e) b^2+3 c^2 d^2 (B d+3 A e) b+6 A c^3 d^3\right ) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d (b e+c d) \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}-\frac {\left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}\right )}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 1169 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {b-c x} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}\right )}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 122 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 120 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 127 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
\(\Big \downarrow \) 126 |
\(\displaystyle \frac {e \left (-\frac {\frac {2 \sqrt {b x-c x^2} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {2 \sqrt {b} d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (-2 b^3 e^2 (B d-4 A e)-b^2 c d e (7 B d-19 A e)+3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}}{3 d (b e+c d)}-\frac {2 \sqrt {b x-c x^2} \left (b^2 (-e) (B d-4 A e)+3 b c d (2 A e+B d)+6 A c^2 d^2\right )}{3 d (d+e x)^{3/2} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{b^2 d \sqrt {b x-c x^2} (d+e x)^{3/2} (b e+c d)}\) |
Input:
Int[(A + B*x)/((d + e*x)^(5/2)*(b*x - c*x^2)^(3/2)),x]
Output:
(-2*(A*b*(c*d + b*e) - c*(b*B*d + 2*A*c*d + A*b*e)*x))/(b^2*d*(c*d + b*e)* (d + e*x)^(3/2)*Sqrt[b*x - c*x^2]) + (e*((-2*(6*A*c^2*d^2 - b^2*e*(B*d - 4 *A*e) + 3*b*c*d*(B*d + 2*A*e))*Sqrt[b*x - c*x^2])/(3*d*(c*d + b*e)*(d + e* x)^(3/2)) - ((2*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) - 2*b^3*e^2*(B*d - 4*A*e) + 3*b*c^2*d^2*(B*d + 3*A*e))*Sqrt[b*x - c*x^2])/(d*(c*d + b*e)*S qrt[d + e*x]) - (c*((-2*Sqrt[b]*(6*A*c^3*d^3 - b^2*c*d*e*(7*B*d - 19*A*e) - 2*b^3*e^2*(B*d - 4*A*e) + 3*b*c^2*d^2*(B*d + 3*A*e))*Sqrt[x]*Sqrt[1 - (c *x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[b]], -((b*e)/ (c*d))])/(Sqrt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x - c*x^2]) + (2*Sqrt[b]*d*(c *d + b*e)*(6*A*c^2*d^2 - b^2*e*(B*d - 4*A*e) + 3*b*c*d*(B*d + 2*A*e))*Sqrt [x]*Sqrt[1 - (c*x)/b]*Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x]) /Sqrt[b]], -((b*e)/(c*d))])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[b*x - c*x^2])))/ (d*(c*d + b*e)))/(3*d*(c*d + b*e))))/(b^2*d*(c*d + b*e))
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[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[2*(Sqrt[e]/b)*Rt[-b/d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[- b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] && Gt Q[e, 0] && !LtQ[-b/d, 0]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)]) ) Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /; FreeQ[{b , c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[Sqrt[x]*(Sqrt[b + c*x]/Sqrt[b*x + c*x^2]) Int[(d + e*x)^m/(Sqrt[x]* Sqrt[b + c*x]), x], x] /; FreeQ[{b, c, d, e}, x] && NeQ[c*d - b*e, 0] && Eq Q[m^2, 1/4]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + b* x + c*x^2)^(p + 1)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((m + 1) *(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 5.51 (sec) , antiderivative size = 823, normalized size of antiderivative = 1.41
method | result | size |
elliptic | \(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (-\frac {2 \left (-c e \,x^{2}-c d x \right ) c^{2} \left (A c +B b \right )}{\left (b e +c d \right )^{3} b^{2} \sqrt {\left (x -\frac {b}{c}\right ) \left (-c e \,x^{2}-c d x \right )}}-\frac {2 \left (A e -B d \right ) \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 d^{2} \left (b e +c d \right )^{2} \left (x +\frac {d}{e}\right )^{2}}-\frac {2 \left (-c e \,x^{2}+b e x \right ) e \left (5 A b \,e^{2}+10 A c d e -2 B b d e -7 B c \,d^{2}\right )}{3 d^{3} \left (b e +c d \right )^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+b e x \right )}}-\frac {2 \left (-c e \,x^{2}+b e x -c d x +b d \right ) A}{d^{3} b^{2} \sqrt {x \left (-c e \,x^{2}+b e x -c d x +b d \right )}}+\frac {2 \left (\frac {c^{2} \left (A c +B b \right )}{\left (b e +c d \right )^{2} b^{2}}-\frac {c^{3} d \left (A c +B b \right )}{\left (b e +c d \right )^{3} b^{2}}+\frac {\left (A e -B d \right ) e c}{3 d^{2} \left (b e +c d \right )^{2}}-\frac {e \left (5 A b \,e^{2}+10 A c d e -2 B b d e -7 B c \,d^{2}\right )}{3 \left (b e +c d \right )^{2} d^{3}}+\frac {b \,e^{2} \left (5 A b \,e^{2}+10 A c d e -2 B b d e -7 B c \,d^{2}\right )}{3 d^{3} \left (b e +c d \right )^{3}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}+\frac {2 \left (-\frac {c^{3} e \left (A c +B b \right )}{\left (b e +c d \right )^{3} b^{2}}-\frac {c \,e^{2} \left (5 A b \,e^{2}+10 A c d e -2 B b d e -7 B c \,d^{2}\right )}{3 d^{3} \left (b e +c d \right )^{3}}-\frac {A c e}{b^{2} d^{3}}\right ) d \sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}\, \sqrt {\frac {x -\frac {b}{c}}{-\frac {d}{e}-\frac {b}{c}}}\, \sqrt {-\frac {e x}{d}}\, \left (\left (-\frac {d}{e}-\frac {b}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )+\frac {b \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {d}{e}\right ) e}{d}}, \sqrt {-\frac {d}{e \left (-\frac {d}{e}-\frac {b}{c}\right )}}\right )}{c}\right )}{e \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}\right )}{\sqrt {e x +d}\, \sqrt {x \left (-c x +b \right )}}\) | \(823\) |
default | \(\text {Expression too large to display}\) | \(2948\) |
Input:
int((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+b*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/(e*x+d)^(1/2)*((-c*x+b)*x*(e*x+d))^(1/2)/(x*(-c*x+b))^(1/2)*(-2*(-c*e*x^ 2-c*d*x)/(b*e+c*d)^3*c^2/b^2*(A*c+B*b)/((x-b/c)*(-c*e*x^2-c*d*x))^(1/2)-2/ 3/d^2/(b*e+c*d)^2*(A*e-B*d)*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)/(x+d/e) ^2-2/3*(-c*e*x^2+b*e*x)/d^3/(b*e+c*d)^3*e*(5*A*b*e^2+10*A*c*d*e-2*B*b*d*e- 7*B*c*d^2)/((x+d/e)*(-c*e*x^2+b*e*x))^(1/2)-2*(-c*e*x^2+b*e*x-c*d*x+b*d)*A /d^3/b^2/(x*(-c*e*x^2+b*e*x-c*d*x+b*d))^(1/2)+2*(c^2/(b*e+c*d)^2*(A*c+B*b) /b^2-c^3*d/(b*e+c*d)^3/b^2*(A*c+B*b)+1/3*(A*e-B*d)*e*c/d^2/(b*e+c*d)^2-1/3 *e/(b*e+c*d)^2*(5*A*b*e^2+10*A*c*d*e-2*B*b*d*e-7*B*c*d^2)/d^3+1/3*b*e^2/d^ 3/(b*e+c*d)^3*(5*A*b*e^2+10*A*c*d*e-2*B*b*d*e-7*B*c*d^2))*d/e*((x+d/e)/d*e )^(1/2)*((x-b/c)/(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c*e*x^3+b*e*x^2-c*d*x^ 2+b*d*x)^(1/2)*EllipticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1/2))+2*(- c^3*e*(A*c+B*b)/(b*e+c*d)^3/b^2-1/3*c*e^2*(5*A*b*e^2+10*A*c*d*e-2*B*b*d*e- 7*B*c*d^2)/d^3/(b*e+c*d)^3-A*c*e/b^2/d^3)*d/e*((x+d/e)/d*e)^(1/2)*((x-b/c) /(-d/e-b/c))^(1/2)*(-e*x/d)^(1/2)/(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)*( (-d/e-b/c)*EllipticE(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1/2))+b/c*Elli pticF(((x+d/e)/d*e)^(1/2),(-d/e/(-d/e-b/c))^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 1784 vs. \(2 (521) = 1042\).
Time = 0.17 (sec) , antiderivative size = 1784, normalized size of antiderivative = 3.06 \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+b*x)^(3/2),x, algorithm="fricas")
Output:
2/9*(((8*A*b^4*c*e^6 - 3*(B*b*c^4 + 2*A*c^5)*d^4*e^2 - (17*B*b^2*c^3 + 12* A*b*c^4)*d^3*e^3 - (8*B*b^3*c^2 - 17*A*b^2*c^3)*d^2*e^4 - (2*B*b^4*c - 23* A*b^3*c^2)*d*e^5)*x^4 - (8*A*b^5*e^6 + 6*(B*b*c^4 + 2*A*c^5)*d^5*e + (31*B *b^2*c^3 + 18*A*b*c^4)*d^4*e^2 - (B*b^3*c^2 + 46*A*b^2*c^3)*d^3*e^3 - (4*B *b^4*c + 29*A*b^3*c^2)*d^2*e^4 - (2*B*b^5 - 7*A*b^4*c)*d*e^5)*x^3 - (11*B* b^2*c^3*d^5*e + 16*A*b^5*d*e^5 + 3*(B*b*c^4 + 2*A*c^5)*d^6 - (26*B*b^3*c^2 + 41*A*b^2*c^3)*d^4*e^2 - (14*B*b^4*c - 11*A*b^3*c^2)*d^3*e^3 - 2*(2*B*b^ 5 - 19*A*b^4*c)*d^2*e^4)*x^2 - (8*A*b^5*d^2*e^4 - 3*(B*b^2*c^3 + 2*A*b*c^4 )*d^6 - (17*B*b^3*c^2 + 12*A*b^2*c^3)*d^5*e - (8*B*b^4*c - 17*A*b^3*c^2)*d ^4*e^2 - (2*B*b^5 - 23*A*b^4*c)*d^3*e^3)*x)*sqrt(-c*e)*weierstrassPInverse (4/3*(c^2*d^2 + b*c*d*e + b^2*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 + 3*b*c^2*d ^2*e - 3*b^2*c*d*e^2 - 2*b^3*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d - b*e)/(c* e)) - 3*((8*A*b^3*c^2*e^6 + 3*(B*b*c^4 + 2*A*c^5)*d^3*e^3 - (7*B*b^2*c^3 - 9*A*b*c^4)*d^2*e^4 - (2*B*b^3*c^2 - 19*A*b^2*c^3)*d*e^5)*x^4 - (8*A*b^4*c *e^6 - 6*(B*b*c^4 + 2*A*c^5)*d^4*e^2 + (17*B*b^2*c^3 - 12*A*b*c^4)*d^3*e^3 - (3*B*b^3*c^2 + 29*A*b^2*c^3)*d^2*e^4 - (2*B*b^4*c - 3*A*b^3*c^2)*d*e^5) *x^3 - (16*A*b^4*c*d*e^5 - 3*(B*b*c^4 + 2*A*c^5)*d^5*e + (13*B*b^2*c^3 + 3 *A*b*c^4)*d^4*e^2 - (12*B*b^3*c^2 + A*b^2*c^3)*d^3*e^3 - 2*(2*B*b^4*c - 15 *A*b^3*c^2)*d^2*e^4)*x^2 - (8*A*b^4*c*d^2*e^4 + 3*(B*b^2*c^3 + 2*A*b*c^4)* d^5*e - (7*B*b^3*c^2 - 9*A*b^2*c^3)*d^4*e^2 - (2*B*b^4*c - 19*A*b^3*c^2...
\[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (- x \left (- b + c x\right )\right )^{\frac {3}{2}} \left (d + e x\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((B*x+A)/(e*x+d)**(5/2)/(-c*x**2+b*x)**(3/2),x)
Output:
Integral((A + B*x)/((-x*(-b + c*x))**(3/2)*(d + e*x)**(5/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+b*x)^(3/2),x, algorithm="maxima")
Output:
integrate((B*x + A)/((-c*x^2 + b*x)^(3/2)*(e*x + d)^(5/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + b x\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+b*x)^(3/2),x, algorithm="giac")
Output:
integrate((B*x + A)/((-c*x^2 + b*x)^(3/2)*(e*x + d)^(5/2)), x)
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (b\,x-c\,x^2\right )}^{3/2}\,{\left (d+e\,x\right )}^{5/2}} \,d x \] Input:
int((A + B*x)/((b*x - c*x^2)^(3/2)*(d + e*x)^(5/2)),x)
Output:
int((A + B*x)/((b*x - c*x^2)^(3/2)*(d + e*x)^(5/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{5/2} \left (b x-c x^2\right )^{3/2}} \, dx=\text {too large to display} \] Input:
int((B*x+A)/(e*x+d)^(5/2)/(-c*x^2+b*x)^(3/2),x)
Output:
( - 2*sqrt(d + e*x)*sqrt(b - c*x)*a + 5*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d**3 + 3*sqrt(x)*b**2*d**2*e*x + 3*sqrt(x)*b**2*d* e**2*x**2 + sqrt(x)*b**2*e**3*x**3 - 2*sqrt(x)*b*c*d**3*x - 6*sqrt(x)*b*c* d**2*e*x**2 - 6*sqrt(x)*b*c*d*e**2*x**3 - 2*sqrt(x)*b*c*e**3*x**4 + sqrt(x )*c**2*d**3*x**2 + 3*sqrt(x)*c**2*d**2*e*x**3 + 3*sqrt(x)*c**2*d*e**2*x**4 + sqrt(x)*c**2*e**3*x**5),x)*a*b*c*d**2*e + 10*sqrt(x)*int((sqrt(d + e*x) *sqrt(b - c*x)*x)/(sqrt(x)*b**2*d**3 + 3*sqrt(x)*b**2*d**2*e*x + 3*sqrt(x) *b**2*d*e**2*x**2 + sqrt(x)*b**2*e**3*x**3 - 2*sqrt(x)*b*c*d**3*x - 6*sqrt (x)*b*c*d**2*e*x**2 - 6*sqrt(x)*b*c*d*e**2*x**3 - 2*sqrt(x)*b*c*e**3*x**4 + sqrt(x)*c**2*d**3*x**2 + 3*sqrt(x)*c**2*d**2*e*x**3 + 3*sqrt(x)*c**2*d*e **2*x**4 + sqrt(x)*c**2*e**3*x**5),x)*a*b*c*d*e**2*x + 5*sqrt(x)*int((sqrt (d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d**3 + 3*sqrt(x)*b**2*d**2*e*x + 3*sqrt(x)*b**2*d*e**2*x**2 + sqrt(x)*b**2*e**3*x**3 - 2*sqrt(x)*b*c*d**3*x - 6*sqrt(x)*b*c*d**2*e*x**2 - 6*sqrt(x)*b*c*d*e**2*x**3 - 2*sqrt(x)*b*c*e **3*x**4 + sqrt(x)*c**2*d**3*x**2 + 3*sqrt(x)*c**2*d**2*e*x**3 + 3*sqrt(x) *c**2*d*e**2*x**4 + sqrt(x)*c**2*e**3*x**5),x)*a*b*c*e**3*x**2 - 5*sqrt(x) *int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**2*d**3 + 3*sqrt(x)*b**2*d **2*e*x + 3*sqrt(x)*b**2*d*e**2*x**2 + sqrt(x)*b**2*e**3*x**3 - 2*sqrt(x)* b*c*d**3*x - 6*sqrt(x)*b*c*d**2*e*x**2 - 6*sqrt(x)*b*c*d*e**2*x**3 - 2*sqr t(x)*b*c*e**3*x**4 + sqrt(x)*c**2*d**3*x**2 + 3*sqrt(x)*c**2*d**2*e*x**...