Integrand size = 29, antiderivative size = 741 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=-\frac {2 A}{3 b d \sqrt {d+e x} \left (b x-c x^2\right )^{3/2}}-\frac {2 (3 b B d+6 A c d-4 A b e) x}{3 b^2 d^2 \sqrt {d+e x} \left (b x-c x^2\right )^{3/2}}+\frac {2 c \left (8 A c^2 d^2+b^2 e (3 B d-4 A e)+b c d (4 B d+3 A e)\right ) x^2}{3 b^3 d^2 (c d+b e) \sqrt {d+e x} \left (b x-c x^2\right )^{3/2}}+\frac {2 c \left (16 A c^3 d^3+15 b^2 B c d^2 e+b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (B d+3 A e)\right ) x}{3 b^4 d^2 (c d+b e)^2 \sqrt {d+e x} \sqrt {b x-c x^2}}-\frac {2 e \left (16 A c^4 d^4+b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)+8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {b x-c x^2}}{3 b^4 d^3 (c d+b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {c} \left (16 A c^4 d^4+b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)+8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{3 b^{7/2} d^3 (c d+b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x-c x^2}}+\frac {2 \sqrt {c} \left (16 A c^3 d^3+15 b^2 B c d^2 e+b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{3 b^{7/2} d^2 (c d+b e)^2 \sqrt {d+e x} \sqrt {b x-c x^2}} \] Output:
-2/3*A/b/d/(e*x+d)^(1/2)/(-c*x^2+b*x)^(3/2)-2/3*(-4*A*b*e+6*A*c*d+3*B*b*d) *x/b^2/d^2/(e*x+d)^(1/2)/(-c*x^2+b*x)^(3/2)+2/3*c*(8*A*c^2*d^2+b^2*e*(-4*A *e+3*B*d)+b*c*d*(3*A*e+4*B*d))*x^2/b^3/d^2/(b*e+c*d)/(e*x+d)^(1/2)/(-c*x^2 +b*x)^(3/2)+2/3*c*(16*A*c^3*d^3+15*b^2*B*c*d^2*e+b^3*e^2*(-4*A*e+3*B*d)+8* b*c^2*d^2*(3*A*e+B*d))*x/b^4/d^2/(b*e+c*d)^2/(e*x+d)^(1/2)/(-c*x^2+b*x)^(1 /2)-2/3*e*(16*A*c^4*d^4+b^3*c*d*e^2*(-7*A*e+9*B*d)+2*b^4*e^3*(-4*A*e+3*B*d )+8*b*c^3*d^3*(4*A*e+B*d)+b^2*c^2*d^2*e*(9*A*e+19*B*d))*(-c*x^2+b*x)^(1/2) /b^4/d^3/(b*e+c*d)^3/(e*x+d)^(1/2)-2/3*c^(1/2)*(16*A*c^4*d^4+b^3*c*d*e^2*( -7*A*e+9*B*d)+2*b^4*e^3*(-4*A*e+3*B*d)+8*b*c^3*d^3*(4*A*e+B*d)+b^2*c^2*d^2 *e*(9*A*e+19*B*d))*x^(1/2)*(1-c*x/b)^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2) *x^(1/2)/b^(1/2),(-b*e/c/d)^(1/2))/b^(7/2)/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)*(16*A*c^3*d^3+15*b^2*B*c*d^2*e+b^3*e^2*(-4 *A*e+3*B*d)+8*b*c^2*d^2*(3*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^(7/2)/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 = 30.66 (sec) , antiderivative size = 644, normalized size of antiderivative = 0.87 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\frac {2 \left (-\sqrt {-\frac {b}{c}} \left (3 b^4 e^4 (B d-A e) x^2 (b-c x)^2-b c^3 (b B+A c) d^3 (c d+b e) x^2 (d+e x)+A b d (c d+b e)^3 (b-c x)^2 (d+e x)+(c d+b e)^3 (3 b B d+8 A c d-5 A b e) x (b-c x)^2 (d+e x)+c^3 d^3 \left (8 A c^2 d+10 b^2 B e+b c (5 B d+13 A e)\right ) x^2 (-b+c x) (d+e x)\right )+x (b-c x) \left (\sqrt {-\frac {b}{c}} \left (16 A c^4 d^4+b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)+8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) (b-c x) (d+e x)+i b e \left (16 A c^4 d^4+b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)+8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right )|-\frac {c d}{b e}\right )-i b e (c d+b e) \left (8 A c^3 d^3+2 b^3 e^2 (3 B d-4 A e)+3 b^2 c d e (2 B d-A e)+b c^2 d^2 (4 B d+9 A e)\right ) \sqrt {1-\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {b}{c}}}{\sqrt {x}}\right ),-\frac {c d}{b e}\right )\right )\right )}{3 b^4 \sqrt {-\frac {b}{c}} d^3 (c d+b e)^3 (x (b-c x))^{3/2} \sqrt {d+e x}} \] Input:
Integrate[(A + B*x)/((d + e*x)^(3/2)*(b*x - c*x^2)^(5/2)),x]
Output:
(2*(-(Sqrt[-(b/c)]*(3*b^4*e^4*(B*d - A*e)*x^2*(b - c*x)^2 - b*c^3*(b*B + A *c)*d^3*(c*d + b*e)*x^2*(d + e*x) + A*b*d*(c*d + b*e)^3*(b - c*x)^2*(d + e *x) + (c*d + b*e)^3*(3*b*B*d + 8*A*c*d - 5*A*b*e)*x*(b - c*x)^2*(d + e*x) + c^3*d^3*(8*A*c^2*d + 10*b^2*B*e + b*c*(5*B*d + 13*A*e))*x^2*(-b + c*x)*( d + e*x))) + x*(b - c*x)*(Sqrt[-(b/c)]*(16*A*c^4*d^4 + b^3*c*d*e^2*(9*B*d - 7*A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) + 8*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^2 *d^2*e*(19*B*d + 9*A*e))*(b - c*x)*(d + e*x) + I*b*e*(16*A*c^4*d^4 + b^3*c *d*e^2*(9*B*d - 7*A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) + 8*b*c^3*d^3*(B*d + 4* A*e) + b^2*c^2*d^2*e*(19*B*d + 9*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)*(8*A*c^3*d^3 + 2*b^3*e^2*(3*B*d - 4*A*e) + 3*b^2*c*d*e*(2*B* d - A*e) + b*c^2*d^2*(4*B*d + 9*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 ^4*Sqrt[-(b/c)]*d^3*(c*d + b*e)^3*(x*(b - c*x))^(3/2)*Sqrt[d + e*x])
Time = 2.35 (sec) , antiderivative size = 727, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {1235, 27, 1235, 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.
| \(\displaystyle \int \frac {A+B x}{\left (b x-c x^2\right )^{5/2} (d+e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \int \frac {e (3 B d-4 A e) b^2+c d (4 B d+3 A e) b+8 A c^2 d^2+5 c e (b B d+2 A c d+A b e) x}{2 (d+e x)^{3/2} \left (b x-c x^2\right )^{3/2}}dx}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {e (3 B d-4 A e) b^2+c d (4 B d+3 A e) b+8 A c^2 d^2+5 c e (b B d+2 A c d+A b e) x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{3/2}}dx}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {\frac {2 \int -\frac {e \left (b \left (2 e^2 (3 B d-4 A e) b^3+3 c d e (2 B d-A e) b^2+c^2 d^2 (4 B d+9 A e) b+8 A c^3 d^3\right )-c \left (e^2 (3 B d-4 A e) b^3+15 B c d^2 e b^2+8 c^2 d^2 (B d+3 A e) b+16 A c^3 d^3\right ) x\right )}{2 (d+e x)^{3/2} \sqrt {b x-c x^2}}dx}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \int \frac {b \left (2 e^2 (3 B d-4 A e) b^3+3 c d e (2 B d-A e) b^2+c^2 d^2 (4 B d+9 A e) b+8 A c^3 d^3\right )-c \left (e^2 (3 B d-4 A e) b^3+15 B c d^2 e b^2+8 c^2 d^2 (B d+3 A e) b+16 A c^3 d^3\right ) x}{(d+e x)^{3/2} \sqrt {b x-c x^2}}dx}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \int -\frac {c \left (b d \left (-e^2 (3 B d-4 A e) b^3+3 c d e (3 B d+A e) b^2+c^2 d^2 (4 B d+15 A e) b+8 A c^3 d^3\right )-\left (2 e^3 (3 B d-4 A e) b^4+c d e^2 (9 B d-7 A e) b^3+c^2 d^2 e (19 B d+9 A e) b^2+8 c^3 d^3 (B d+4 A e) b+16 A c^4 d^4\right ) x\right )}{2 \sqrt {d+e x} \sqrt {b x-c x^2}}dx}{d (b e+c d)}+\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \int \frac {b d \left (-e^2 (3 B d-4 A e) b^3+3 c d e (3 B d+A e) b^2+c^2 d^2 (4 B d+15 A e) b+8 A c^3 d^3\right )-\left (2 e^3 (3 B d-4 A e) b^4+c d e^2 (9 B d-7 A e) b^3+c^2 d^2 e (19 B d+9 A e) b^2+8 c^3 d^3 (B d+4 A e) b+16 A c^4 d^4\right ) x}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d (b e+c d) \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x-c x^2}}dx}{e}-\frac {\left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x-c x^2}}dx}{e}\right )}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {b-c x} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b-c x}}dx}{e \sqrt {b x-c x^2}}\right )}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {1-\frac {c x}{b}}}dx}{e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {b-c x} (b e+c d) \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) \int \frac {1}{\sqrt {x} \sqrt {b-c x} \sqrt {d+e x}}dx}{e \sqrt {b x-c x^2}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) \int \frac {1}{\sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {-\frac {e \left (\frac {2 \sqrt {b x-c x^2} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{d \sqrt {d+e x} (b e+c d)}-\frac {c \left (\frac {2 \sqrt {b} d \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {\frac {e x}{d}+1} (b e+c d) \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right ),-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {b} \sqrt {x} \sqrt {1-\frac {c x}{b}} \sqrt {d+e x} \left (2 b^4 e^3 (3 B d-4 A e)+b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)+8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )|-\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x-c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (b e+c d)}\right )}{b^2 d (b e+c d)}-\frac {2 \left (b (b e+c d) \left (b^2 e (3 B d-4 A e)+b c d (3 A e+4 B d)+8 A c^2 d^2\right )-c x \left (b^3 e^2 (3 B d-4 A e)+8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{b^2 d \sqrt {b x-c x^2} \sqrt {d+e x} (b e+c d)}}{3 b^2 d (b e+c d)}-\frac {2 (A b (b e+c d)-c x (A b e+2 A c d+b B d))}{3 b^2 d \left (b x-c x^2\right )^{3/2} \sqrt {d+e x} (b e+c d)}\) |
Input:
Int[(A + B*x)/((d + e*x)^(3/2)*(b*x - c*x^2)^(5/2)),x]
Output:
(-2*(A*b*(c*d + b*e) - c*(b*B*d + 2*A*c*d + A*b*e)*x))/(3*b^2*d*(c*d + b*e )*Sqrt[d + e*x]*(b*x - c*x^2)^(3/2)) + ((-2*(b*(c*d + b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 4*A*e) + b*c*d*(4*B*d + 3*A*e)) - c*(16*A*c^3*d^3 + 15*b^2* B*c*d^2*e + b^3*e^2*(3*B*d - 4*A*e) + 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(b^2* d*(c*d + b*e)*Sqrt[d + e*x]*Sqrt[b*x - c*x^2]) - (e*((2*(16*A*c^4*d^4 + b^ 3*c*d*e^2*(9*B*d - 7*A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) + 8*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e))*Sqrt[b*x - c*x^2])/(d*(c*d + b*e )*Sqrt[d + e*x]) - (c*((-2*Sqrt[b]*(16*A*c^4*d^4 + b^3*c*d*e^2*(9*B*d - 7* A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) + 8*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2 *e*(19*B*d + 9*A*e))*Sqrt[x]*Sqrt[1 - (c*x)/b]*Sqrt[d + e*x]*EllipticE[Arc Sin[(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)*(16*A*c^3*d^3 + 15*b^2*B *c*d^2*e + b^3*e^2*(3*B*d - 4*A*e) + 8*b*c^2*d^2*(B*d + 3*A*e))*Sqrt[x]*Sq rt[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))))/(b^2*d*(c*d + b*e)))/(3*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 = 6.10 (sec) , antiderivative size = 912, normalized size of antiderivative = 1.23
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-c x +b \right ) x \left (e x +d \right )}\, \left (\frac {2 c \left (A c +B b \right ) \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 b^{3} \left (b e +c d \right )^{2} \left (x -\frac {b}{c}\right )^{2}}-\frac {2 \left (-c e \,x^{2}-c d x \right ) c^{2} \left (13 A c e b +8 A \,c^{2} d +10 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e +c d \right )^{3} \sqrt {\left (x -\frac {b}{c}\right ) \left (-c e \,x^{2}-c d x \right )}}+\frac {2 \left (-c e \,x^{2}+b e x \right ) e^{3} \left (A e -B d \right )}{\left (b e +c d \right )^{3} d^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (-c e \,x^{2}+b e x \right )}}-\frac {2 A \sqrt {-c e \,x^{3}+b e \,x^{2}-c d \,x^{2}+b d x}}{3 d^{2} b^{3} x^{2}}+\frac {2 \left (-c e \,x^{2}+b e x -c d x +b d \right ) \left (5 A b e -8 A c d -3 B b d \right )}{3 d^{3} b^{4} \sqrt {x \left (-c e \,x^{2}+b e x -c d x +b d \right )}}+\frac {2 \left (-\frac {\left (A c +B b \right ) c^{2} e}{3 \left (b e +c d \right )^{2} b^{3}}+\frac {c^{2} \left (13 A c e b +8 A \,c^{2} d +10 b^{2} B e +5 B b c d \right )}{3 \left (b e +c d \right )^{2} b^{4}}-\frac {c^{3} d \left (13 A c e b +8 A \,c^{2} d +10 b^{2} B e +5 B b c d \right )}{3 b^{4} \left (b e +c d \right )^{3}}+\frac {e^{3} \left (A e -B d \right )}{\left (b e +c d \right )^{2} d^{3}}-\frac {b \,e^{4} \left (A e -B d \right )}{\left (b e +c d \right )^{3} d^{3}}+\frac {A c e}{3 b^{3} d^{2}}\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 (13 A c e b +8 A \,c^{2} d +10 b^{2} B e +5 B b c d \right )}{3 \left (b e +c d \right )^{3} b^{4}}+\frac {c \,e^{4} \left (A e -B d \right )}{d^{3} \left (b e +c d \right )^{3}}+\frac {c e \left (5 A b e -8 A c d -3 B b d \right )}{3 b^{4} 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 )}}\) | \(912\) |
| default | \(\text {Expression too large to display}\) | \(3935\) |
Input:
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(5/2),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)*(2/3/b^3/(b* e+c*d)^2*c*(A*c+B*b)*(-c*e*x^3+b*e*x^2-c*d*x^2+b*d*x)^(1/2)/(x-b/c)^2-2/3* (-c*e*x^2-c*d*x)/b^4/(b*e+c*d)^3*c^2*(13*A*b*c*e+8*A*c^2*d+10*B*b^2*e+5*B* b*c*d)/((x-b/c)*(-c*e*x^2-c*d*x))^(1/2)+2*(-c*e*x^2+b*e*x)/(b*e+c*d)^3*e^3 /d^3*(A*e-B*d)/((x+d/e)*(-c*e*x^2+b*e*x))^(1/2)-2/3*A/d^2/b^3*(-c*e*x^3+b* e*x^2-c*d*x^2+b*d*x)^(1/2)/x^2+2/3*(-c*e*x^2+b*e*x-c*d*x+b*d)/d^3/b^4*(5*A *b*e-8*A*c*d-3*B*b*d)/(x*(-c*e*x^2+b*e*x-c*d*x+b*d))^(1/2)+2*(-1/3*(A*c+B* b)*c^2*e/(b*e+c*d)^2/b^3+1/3*c^2/(b*e+c*d)^2*(13*A*b*c*e+8*A*c^2*d+10*B*b^ 2*e+5*B*b*c*d)/b^4-1/3*c^3*d/b^4/(b*e+c*d)^3*(13*A*b*c*e+8*A*c^2*d+10*B*b^ 2*e+5*B*b*c*d)+e^3/(b*e+c*d)^2*(A*e-B*d)/d^3-b*e^4/(b*e+c*d)^3/d^3*(A*e-B* d)+1/3/b^3/d^2*A*c*e)*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*(-1/3*c^3*e*(13*A*b*c*e+8*A*c^2*d+10*B *b^2*e+5*B*b*c*d)/(b*e+c*d)^3/b^4+c*e^4*(A*e-B*d)/d^3/(b*e+c*d)^3+1/3*c*e* (5*A*b*e-8*A*c*d-3*B*b*d)/b^4/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*EllipticF(( (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. 2207 vs. \(2 (669) = 1338\).
Time = 0.21 (sec) , antiderivative size = 2207, normalized size of antiderivative = 2.98 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(5/2),x, algorithm="fricas")
Output:
-2/9*(((8*A*b^5*c^2*e^6 + 8*(B*b*c^6 + 2*A*c^7)*d^5*e + (23*B*b^2*c^5 + 40 *A*b*c^6)*d^4*e^2 + (17*B*b^3*c^4 + 22*A*b^2*c^5)*d^3*e^3 - (12*B*b^4*c^3 + 7*A*b^3*c^4)*d^2*e^4 - (6*B*b^5*c^2 - 11*A*b^4*c^3)*d*e^5)*x^5 - (16*A*b ^6*c*e^6 - 8*(B*b*c^6 + 2*A*c^7)*d^6 - (7*B*b^2*c^5 + 8*A*b*c^6)*d^5*e + 2 9*(B*b^3*c^4 + 2*A*b^2*c^5)*d^4*e^2 + (46*B*b^4*c^3 + 51*A*b^3*c^4)*d^3*e^ 3 - (18*B*b^5*c^2 + 25*A*b^4*c^3)*d^2*e^4 - 2*(6*B*b^6*c - 7*A*b^5*c^2)*d* e^5)*x^4 - (29*A*b^5*c^2*d^2*e^4 - 8*A*b^7*e^6 + 16*(B*b^2*c^5 + 2*A*b*c^6 )*d^6 + 2*(19*B*b^3*c^4 + 32*A*b^2*c^5)*d^5*e + (11*B*b^4*c^3 + 4*A*b^3*c^ 4)*d^4*e^2 - (41*B*b^5*c^2 + 36*A*b^4*c^3)*d^3*e^3 + (6*B*b^7 + 5*A*b^6*c) *d*e^5)*x^3 + (8*A*b^7*d*e^5 + 8*(B*b^3*c^4 + 2*A*b^2*c^5)*d^6 + (23*B*b^4 *c^3 + 40*A*b^3*c^4)*d^5*e + (17*B*b^5*c^2 + 22*A*b^4*c^3)*d^4*e^2 - (12*B *b^6*c + 7*A*b^5*c^2)*d^3*e^3 - (6*B*b^7 - 11*A*b^6*c)*d^2*e^4)*x^2)*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^4*c^3*e^6 - 8*(B*b*c^6 + 2*A*c^7) *d^4*e^2 - (19*B*b^2*c^5 + 32*A*b*c^6)*d^3*e^3 - 9*(B*b^3*c^4 + A*b^2*c^5) *d^2*e^4 - (6*B*b^4*c^3 - 7*A*b^3*c^4)*d*e^5)*x^5 - (3*B*b^2*c^5*d^4*e^2 + 16*A*b^5*c^2*e^6 + 8*(B*b*c^6 + 2*A*c^7)*d^5*e - (29*B*b^3*c^4 + 55*A*b^2 *c^5)*d^3*e^3 - (12*B*b^4*c^3 + 25*A*b^3*c^4)*d^2*e^4 - 6*(2*B*b^5*c^2 - A *b^4*c^3)*d*e^5)*x^4 + (8*A*b^6*c*e^6 + 16*(B*b^2*c^5 + 2*A*b*c^6)*d^5*...
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((B*x+A)/(e*x+d)**(3/2)/(-c*x**2+b*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + b x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(5/2),x, algorithm="maxima")
Output:
integrate((B*x + A)/((-c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\int { \frac {B x + A}{{\left (-c x^{2} + b x\right )}^{\frac {5}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(5/2),x, algorithm="giac")
Output:
integrate((B*x + A)/((-c*x^2 + b*x)^(5/2)*(e*x + d)^(3/2)), x)
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\int \frac {A+B\,x}{{\left (b\,x-c\,x^2\right )}^{5/2}\,{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:
int((A + B*x)/((b*x - c*x^2)^(5/2)*(d + e*x)^(3/2)),x)
Output:
int((A + B*x)/((b*x - c*x^2)^(5/2)*(d + e*x)^(3/2)), x)
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (b x-c x^2\right )^{5/2}} \, dx=\text {too large to display} \] Input:
int((B*x+A)/(e*x+d)^(3/2)/(-c*x^2+b*x)^(5/2),x)
Output:
( - 2*sqrt(d + e*x)*sqrt(b - c*x)*a - 6*sqrt(d + e*x)*sqrt(b - c*x)*b*x + 15*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**3*d**2 + 2*sqrt (x)*b**3*d*e*x + sqrt(x)*b**3*e**2*x**2 - 3*sqrt(x)*b**2*c*d**2*x - 6*sqrt (x)*b**2*c*d*e*x**2 - 3*sqrt(x)*b**2*c*e**2*x**3 + 3*sqrt(x)*b*c**2*d**2*x **2 + 6*sqrt(x)*b*c**2*d*e*x**3 + 3*sqrt(x)*b*c**2*e**2*x**4 - sqrt(x)*c** 3*d**2*x**3 - 2*sqrt(x)*c**3*d*e*x**4 - sqrt(x)*c**3*e**2*x**5),x)*b**3*c* d*e*x + 15*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x)*b**3*d**2 + 2*sqrt(x)*b**3*d*e*x + sqrt(x)*b**3*e**2*x**2 - 3*sqrt(x)*b**2*c*d**2*x - 6*sqrt(x)*b**2*c*d*e*x**2 - 3*sqrt(x)*b**2*c*e**2*x**3 + 3*sqrt(x)*b*c** 2*d**2*x**2 + 6*sqrt(x)*b*c**2*d*e*x**3 + 3*sqrt(x)*b*c**2*e**2*x**4 - sqr t(x)*c**3*d**2*x**3 - 2*sqrt(x)*c**3*d*e*x**4 - sqrt(x)*c**3*e**2*x**5),x) *b**3*c*e**2*x**2 - 30*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c*x)*x)/(sqrt(x )*b**3*d**2 + 2*sqrt(x)*b**3*d*e*x + sqrt(x)*b**3*e**2*x**2 - 3*sqrt(x)*b* *2*c*d**2*x - 6*sqrt(x)*b**2*c*d*e*x**2 - 3*sqrt(x)*b**2*c*e**2*x**3 + 3*s qrt(x)*b*c**2*d**2*x**2 + 6*sqrt(x)*b*c**2*d*e*x**3 + 3*sqrt(x)*b*c**2*e** 2*x**4 - sqrt(x)*c**3*d**2*x**3 - 2*sqrt(x)*c**3*d*e*x**4 - sqrt(x)*c**3*e **2*x**5),x)*b**2*c**2*d*e*x**2 - 30*sqrt(x)*int((sqrt(d + e*x)*sqrt(b - c *x)*x)/(sqrt(x)*b**3*d**2 + 2*sqrt(x)*b**3*d*e*x + sqrt(x)*b**3*e**2*x**2 - 3*sqrt(x)*b**2*c*d**2*x - 6*sqrt(x)*b**2*c*d*e*x**2 - 3*sqrt(x)*b**2*c*e **2*x**3 + 3*sqrt(x)*b*c**2*d**2*x**2 + 6*sqrt(x)*b*c**2*d*e*x**3 + 3*s...