Integrand size = 28, antiderivative size = 510 \[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {2 (B d-A e) \sqrt {b x+c x^2}}{5 d (c d-b e) (d+e x)^{5/2}}-\frac {2 (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt {b x+c x^2}}{15 d^2 (c d-b e)^2 (d+e x)^{3/2}}+\frac {2 \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt {b x+c x^2}}{15 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {c} \left (B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 d^3 e (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {c} (4 A e (2 c d-b e)-B d (3 c d+b e)) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 d^2 e (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \]
-2/15*(B*d*(-2*b^2*e^2+7*b*c*d*e+3*c^2*d^2)-A*e*(8*b^2*e^2-23*b*c*d*e+23*c ^2*d^2))*EllipticE(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)* c^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(e*x+d)^(1/2)/d^3/e/(-b*e+c*d)^3/(1+e*x/d) ^(1/2)/(c*x^2+b*x)^(1/2)-2/15*(4*A*e*(-b*e+2*c*d)-B*d*(b*e+3*c*d))*Ellipti cF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*c^(1/2)*x^(1/2)* (1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/d^2/e/(-b*e+c*d)^2/(e*x+d)^(1/2)/(c*x^2+b* x)^(1/2)+2/5*(-A*e+B*d)*(c*x^2+b*x)^(1/2)/d/(-b*e+c*d)/(e*x+d)^(5/2)-2/15* (4*A*e*(-b*e+2*c*d)-B*d*(b*e+3*c*d))*(c*x^2+b*x)^(1/2)/d^2/(-b*e+c*d)^2/(e *x+d)^(3/2)+2/15*(B*d*(-2*b^2*e^2+7*b*c*d*e+3*c^2*d^2)-A*e*(8*b^2*e^2-23*b *c*d*e+23*c^2*d^2))*(c*x^2+b*x)^(1/2)/d^3/(-b*e+c*d)^3/(e*x+d)^(1/2)
Result contains complex when optimal does not.
Time = 24.04 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\frac {2 \left (b e x (b+c x) \left (3 d^2 (B d-A e) (c d-b e)^2+d (c d-b e) (4 A e (-2 c d+b e)+B d (3 c d+b e)) (d+e x)+\left (A e \left (-23 c^2 d^2+23 b c d e-8 b^2 e^2\right )+B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )\right ) (d+e x)^2\right )-\sqrt {\frac {b}{c}} c (d+e x)^2 \left (\sqrt {\frac {b}{c}} \left (A e \left (-23 c^2 d^2+23 b c d e-8 b^2 e^2\right )+B d \left (3 c^2 d^2+7 b c d e-2 b^2 e^2\right )\right ) (b+c x) (d+e x)-i b e \left (B d \left (-3 c^2 d^2-7 b c d e+2 b^2 e^2\right )+A e \left (23 c^2 d^2-23 b c d e+8 b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i e (c d-b e) \left (15 A c^2 d^2+2 b^2 e (B d+4 A e)-b c d (6 B d+19 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )\right )\right )}{15 b d^3 e (c d-b e)^3 \sqrt {x (b+c x)} (d+e x)^{5/2}} \]
(2*(b*e*x*(b + c*x)*(3*d^2*(B*d - A*e)*(c*d - b*e)^2 + d*(c*d - b*e)*(4*A* e*(-2*c*d + b*e) + B*d*(3*c*d + b*e))*(d + e*x) + (A*e*(-23*c^2*d^2 + 23*b *c*d*e - 8*b^2*e^2) + B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2))*(d + e*x)^2 ) - Sqrt[b/c]*c*(d + e*x)^2*(Sqrt[b/c]*(A*e*(-23*c^2*d^2 + 23*b*c*d*e - 8* b^2*e^2) + B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2))*(b + c*x)*(d + e*x) - I*b*e*(B*d*(-3*c^2*d^2 - 7*b*c*d*e + 2*b^2*e^2) + A*e*(23*c^2*d^2 - 23*b*c *d*e + 8*b^2*e^2))*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*e*(c*d - b*e)*(15*A*c^2*d^2 + 2*b^2*e*(B*d + 4*A*e) - b*c*d*(6*B*d + 19*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)])) )/(15*b*d^3*e*(c*d - b*e)^3*Sqrt[x*(b + c*x)]*(d + e*x)^(5/2))
Time = 0.96 (sec) , antiderivative size = 539, normalized size of antiderivative = 1.06, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1237, 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.
| \(\displaystyle \int \frac {A+B x}{\sqrt {b x+c x^2} (d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {2 \int \frac {b B d-5 A c d+4 A b e-3 c (B d-A e) x}{2 (d+e x)^{5/2} \sqrt {c x^2+b x}}dx}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\int \frac {b B d-5 A c d+4 A b e-3 c (B d-A e) x}{(d+e x)^{5/2} \sqrt {c x^2+b x}}dx}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {2 \int \frac {2 e (B d+4 A e) b^2-c d (6 B d+19 A e) b+15 A c^2 d^2-c (4 A e (2 c d-b e)-B d (3 c d+b e)) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\int \frac {2 e (B d+4 A e) b^2-c d (6 B d+19 A e) b+15 A c^2 d^2-c (4 A e (2 c d-b e)-B d (3 c d+b e)) x}{(d+e x)^{3/2} \sqrt {c x^2+b x}}dx}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}-\frac {2 \int -\frac {c \left (d \left (e (B d+4 A e) b^2-c d (9 B d+11 A e) b+15 A c^2 d^2\right )-\left (B d \left (3 c^2 d^2+7 b c e d-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c e d+8 b^2 e^2\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \int \frac {d \left (e (B d+4 A e) b^2-c d (9 B d+11 A e) b+15 A c^2 d^2\right )-\left (B d \left (3 c^2 d^2+7 b c e d-2 b^2 e^2\right )-A e \left (23 c^2 d^2-23 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {\left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x}}dx}{e}-\frac {d (c d-b e) (4 A e (2 c d-b e)-B d (b e+3 c d)) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{e}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 1169 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {\sqrt {x} \sqrt {b+c x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}}dx}{e \sqrt {b x+c x^2}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (4 A e (2 c d-b e)-B d (b e+3 c d)) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 122 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) \int \frac {\sqrt {\frac {e x}{d}+1}}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (4 A e (2 c d-b e)-B d (b e+3 c d)) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 120 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d \sqrt {x} \sqrt {b+c x} (c d-b e) (4 A e (2 c d-b e)-B d (b e+3 c d)) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}}dx}{e \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (4 A e (2 c d-b e)-B d (b e+3 c d)) \int \frac {1}{\sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1}}dx}{e \sqrt {b x+c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {2 \sqrt {b x+c x^2} (B d-A e)}{5 d (d+e x)^{5/2} (c d-b e)}-\frac {\frac {2 \sqrt {b x+c x^2} (4 A e (2 c d-b e)-B d (b e+3 c d))}{3 d (d+e x)^{3/2} (c d-b e)}-\frac {\frac {c \left (-\frac {2 \sqrt {-b} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right ) E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1}}-\frac {2 \sqrt {-b} d \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} (c d-b e) (4 A e (2 c d-b e)-B d (b e+3 c d)) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{\sqrt {c} e \sqrt {b x+c x^2} \sqrt {d+e x}}\right )}{d (c d-b e)}+\frac {2 \sqrt {b x+c x^2} \left (B d \left (-2 b^2 e^2+7 b c d e+3 c^2 d^2\right )-A e \left (8 b^2 e^2-23 b c d e+23 c^2 d^2\right )\right )}{d \sqrt {d+e x} (c d-b e)}}{3 d (c d-b e)}}{5 d (c d-b e)}\) |
(2*(B*d - A*e)*Sqrt[b*x + c*x^2])/(5*d*(c*d - b*e)*(d + e*x)^(5/2)) - ((2* (4*A*e*(2*c*d - b*e) - B*d*(3*c*d + b*e))*Sqrt[b*x + c*x^2])/(3*d*(c*d - b *e)*(d + e*x)^(3/2)) - ((2*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e* (23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2))*Sqrt[b*x + c*x^2])/(d*(c*d - b*e)*S qrt[d + e*x]) + (c*((-2*Sqrt[-b]*(B*d*(3*c^2*d^2 + 7*b*c*d*e - 2*b^2*e^2) - A*e*(23*c^2*d^2 - 23*b*c*d*e + 8*b^2*e^2))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqr t[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(Sq rt[c]*e*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) - (2*Sqrt[-b]*d*(c*d - b*e)*( 4*A*e*(2*c*d - b*e) - B*d*(3*c*d + b*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)])/(Sq rt[c]*e*Sqrt[d + e*x]*Sqrt[b*x + c*x^2])))/(d*(c*d - b*e)))/(3*d*(c*d - b* e)))/(5*d*(c*d - b*e))
3.13.70.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(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[(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 = 1.23 (sec) , antiderivative size = 826, normalized size of antiderivative = 1.62
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 \left (A e -B d \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 e^{3} d \left (b e -c d \right ) \left (x +\frac {d}{e}\right )^{3}}+\frac {2 \left (4 A b \,e^{2}-8 A c d e +B b d e +3 B c \,d^{2}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{15 e^{2} 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 ) \left (8 A \,b^{2} e^{3}-23 A b c d \,e^{2}+23 A \,c^{2} d^{2} e +2 B \,b^{2} d \,e^{2}-7 B b c \,d^{2} e -3 B \,c^{2} d^{3}\right )}{15 e \,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 (\frac {c \left (4 A b \,e^{2}-8 A c d e +B b d e +3 B c \,d^{2}\right )}{15 d^{2} \left (b e -c d \right )^{2} e}+\frac {8 A \,b^{2} e^{3}-23 A b c d \,e^{2}+23 A \,c^{2} d^{2} e +2 B \,b^{2} d \,e^{2}-7 B b c \,d^{2} e -3 B \,c^{2} d^{3}}{15 \left (b e -c d \right )^{2} e \,d^{3}}-\frac {b \left (8 A \,b^{2} e^{3}-23 A b c d \,e^{2}+23 A \,c^{2} d^{2} e +2 B \,b^{2} d \,e^{2}-7 B b c \,d^{2} e -3 B \,c^{2} d^{3}\right )}{15 d^{3} \left (b e -c d \right )^{3}}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}-\frac {2 \left (8 A \,b^{2} e^{3}-23 A b c d \,e^{2}+23 A \,c^{2} d^{2} e +2 B \,b^{2} d \,e^{2}-7 B b c \,d^{2} e -3 B \,c^{2} d^{3}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{15 d^{3} \left (b e -c d \right )^{3} \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(826\) |
| default | \(\text {Expression too large to display}\) | \(3863\) |
((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/5/e^3/d/(b*e- c*d)*(A*e-B*d)*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)/(x+d/e)^3+2/15*(4*A*b *e^2-8*A*c*d*e+B*b*d*e+3*B*c*d^2)/e^2/d^2/(b*e-c*d)^2*(c*e*x^3+b*e*x^2+c*d *x^2+b*d*x)^(1/2)/(x+d/e)^2+2/15*(c*e*x^2+b*e*x)/e/d^3/(b*e-c*d)^3*(8*A*b^ 2*e^3-23*A*b*c*d*e^2+23*A*c^2*d^2*e+2*B*b^2*d*e^2-7*B*b*c*d^2*e-3*B*c^2*d^ 3)/((x+d/e)*(c*e*x^2+b*e*x))^(1/2)+2*(1/15*c*(4*A*b*e^2-8*A*c*d*e+B*b*d*e+ 3*B*c*d^2)/d^2/(b*e-c*d)^2/e+1/15/(b*e-c*d)^2/e*(8*A*b^2*e^3-23*A*b*c*d*e^ 2+23*A*c^2*d^2*e+2*B*b^2*d*e^2-7*B*b*c*d^2*e-3*B*c^2*d^3)/d^3-1/15*b/d^3/( b*e-c*d)^3*(8*A*b^2*e^3-23*A*b*c*d*e^2+23*A*c^2*d^2*e+2*B*b^2*d*e^2-7*B*b* c*d^2*e-3*B*c^2*d^3))*b/c*((x+b/c)/b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*( -c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)*EllipticF(((x+b/c)/b*c )^(1/2),(-b/c/(-b/c+d/e))^(1/2))-2/15*(8*A*b^2*e^3-23*A*b*c*d*e^2+23*A*c^2 *d^2*e+2*B*b^2*d*e^2-7*B*b*c*d^2*e-3*B*c^2*d^3)/d^3/(b*e-c*d)^3*b*((x+b/c) /b*c)^(1/2)*((x+d/e)/(-b/c+d/e))^(1/2)*(-c*x/b)^(1/2)/(c*e*x^3+b*e*x^2+c*d *x^2+b*d*x)^(1/2)*((-b/c+d/e)*EllipticE(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/ e))^(1/2))-d/e*EllipticF(((x+b/c)/b*c)^(1/2),(-b/c/(-b/c+d/e))^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1356, normalized size of antiderivative = 2.66 \[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\text {Too large to display} \]
2/45*((3*B*c^3*d^7 - 8*A*b^3*d^3*e^4 - (17*B*b*c^2 - 22*A*c^3)*d^6*e + (8* B*b^2*c - 33*A*b*c^2)*d^5*e^2 - (2*B*b^3 - 27*A*b^2*c)*d^4*e^3 + (3*B*c^3* d^4*e^3 - 8*A*b^3*e^7 - (17*B*b*c^2 - 22*A*c^3)*d^3*e^4 + (8*B*b^2*c - 33* A*b*c^2)*d^2*e^5 - (2*B*b^3 - 27*A*b^2*c)*d*e^6)*x^3 + 3*(3*B*c^3*d^5*e^2 - 8*A*b^3*d*e^6 - (17*B*b*c^2 - 22*A*c^3)*d^4*e^3 + (8*B*b^2*c - 33*A*b*c^ 2)*d^3*e^4 - (2*B*b^3 - 27*A*b^2*c)*d^2*e^5)*x^2 + 3*(3*B*c^3*d^6*e - 8*A* b^3*d^2*e^5 - (17*B*b*c^2 - 22*A*c^3)*d^5*e^2 + (8*B*b^2*c - 33*A*b*c^2)*d ^4*e^3 - (2*B*b^3 - 27*A*b^2*c)*d^3*e^4)*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*(3*B*c^3*d^6*e - 8*A*b^2*c*d^3*e^4 + (7*B*b*c^2 - 23*A*c^3)*d^5*e^2 - (2*B*b^2*c - 23*A*b*c^2)*d^4*e^3 + (3*B*c^3*d^3*e^4 - 8*A*b^2*c*e^7 + ( 7*B*b*c^2 - 23*A*c^3)*d^2*e^5 - (2*B*b^2*c - 23*A*b*c^2)*d*e^6)*x^3 + 3*(3 *B*c^3*d^4*e^3 - 8*A*b^2*c*d*e^6 + (7*B*b*c^2 - 23*A*c^3)*d^3*e^4 - (2*B*b ^2*c - 23*A*b*c^2)*d^2*e^5)*x^2 + 3*(3*B*c^3*d^5*e^2 - 8*A*b^2*c*d^2*e^5 + (7*B*b*c^2 - 23*A*c^3)*d^4*e^3 - (2*B*b^2*c - 23*A*b*c^2)*d^3*e^4)*x)*sqr t(c*e)*weierstrassZeta(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), weierst rassPInverse(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 +...
\[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A + B x}{\sqrt {x \left (b + c x\right )} \left (d + e x\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{7/2} \sqrt {b x+c x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {c\,x^2+b\,x}\,{\left (d+e\,x\right )}^{7/2}} \,d x \]