Integrand size = 28, antiderivative size = 339 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 (3 B c d-4 b B e+5 A c e) \sqrt {d+e x} \sqrt {b x+c x^2}}{15 c^2}+\frac {2 B (d+e x)^{3/2} \sqrt {b x+c x^2}}{5 c}+\frac {2 \sqrt {-b} \left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}-\frac {2 \sqrt {-b} d (c d-b e) (3 B c d-4 b B e+5 A c e) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right ),\frac {b e}{c d}\right )}{15 c^{5/2} e \sqrt {d+e x} \sqrt {b x+c x^2}} \]
2/15*(10*A*c*e*(-b*e+2*c*d)+B*(8*b^2*e^2-13*b*c*d*e+3*c^2*d^2))*EllipticE( c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^( 1/2)*(e*x+d)^(1/2)/c^(5/2)/e/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)-2/15*d*(-b* e+c*d)*(5*A*c*e-4*B*b*e+3*B*c*d)*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e /c/d)^(1/2))*(-b)^(1/2)*x^(1/2)*(1+c*x/b)^(1/2)*(1+e*x/d)^(1/2)/c^(5/2)/e/ (e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/5*B*(e*x+d)^(3/2)*(c*x^2+b*x)^(1/2)/c+2/ 15*(5*A*c*e-4*B*b*e+3*B*c*d)*(e*x+d)^(1/2)*(c*x^2+b*x)^(1/2)/c^2
Result contains complex when optimal does not.
Time = 18.18 (sec) , antiderivative size = 356, normalized size of antiderivative = 1.05 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\frac {2 \sqrt {x} \left (\frac {\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\right ) (b+c x) (d+e x)}{c e \sqrt {x}}+\sqrt {x} (b+c x) (d+e x) (5 A c e+B (6 c d-4 b e+3 c e x))+i \sqrt {\frac {b}{c}} \left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c d e+8 b^2 e^2\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x E\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-\frac {i \sqrt {\frac {b}{c}} (-c d+b e) \left (15 A c^2 d+8 b^2 B e-b c (9 B d+10 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right ),\frac {c d}{b e}\right )}{b}\right )}{15 c^2 \sqrt {x (b+c x)} \sqrt {d+e x}} \]
(2*Sqrt[x]*(((10*A*c*e*(2*c*d - b*e) + B*(3*c^2*d^2 - 13*b*c*d*e + 8*b^2*e ^2))*(b + c*x)*(d + e*x))/(c*e*Sqrt[x]) + Sqrt[x]*(b + c*x)*(d + e*x)*(5*A *c*e + B*(6*c*d - 4*b*e + 3*c*e*x)) + I*Sqrt[b/c]*(10*A*c*e*(2*c*d - b*e) + B*(3*c^2*d^2 - 13*b*c*d*e + 8*b^2*e^2))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(e* x)]*x*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - (I*Sqrt[b/c]* (-(c*d) + b*e)*(15*A*c^2*d + 8*b^2*B*e - b*c*(9*B*d + 10*A*e))*Sqrt[1 + b/ (c*x)]*Sqrt[1 + d/(e*x)]*x*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/( b*e)])/b))/(15*c^2*Sqrt[x*(b + c*x)]*Sqrt[d + e*x])
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.
| \(\displaystyle \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 \int -\frac {\sqrt {d+e x} ((b B-5 A c) d-(3 B c d-4 b B e+5 A c e) x)}{2 \sqrt {c x^2+b x}}dx}{5 c}+\frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\int \frac {\sqrt {d+e x} ((b B-5 A c) d-(3 B c d-4 b B e+5 A c e) x)}{\sqrt {c x^2+b x}}dx}{5 c}\) |
| \(\Big \downarrow \) 1236 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {2 \int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{2 \sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {\frac {\int -\frac {d \left (4 B e b^2-c (6 B d+5 A e) b+15 A c^2 d\right )+\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 B \sqrt {b x+c x^2} (d+e x)^{3/2}}{5 c}-\frac {-\frac {\int -\frac {d \left (-4 B e b^2+6 B c d b+5 A c e b-15 A c^2 d\right )-\left (10 A c e (2 c d-b e)+B \left (3 c^2 d^2-13 b c e d+8 b^2 e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x}}dx}{3 c}-\frac {2 \sqrt {b x+c x^2} \sqrt {d+e x} (5 A c e-4 b B e+3 B c d)}{3 c}}{5 c}\) |
3.13.65.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 2))), x] + Simp[1/(c*(m + 2*p + 2)) Int[(d + e*x)^(m - 1 )*(a + b*x + c*x^2)^p*Simp[m*(c*d*f - a*e*g) + d*(2*c*f - b*g)*(p + 1) + (m *(c*e*f + c*d*g - b*e*g) + e*(p + 1)*(2*c*f - b*g))*x, x], x], x] /; FreeQ[ {a, b, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) && !(IGtQ[m, 0] && EqQ[f, 0])
Time = 0.51 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.53
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (e x +d \right ) x \left (c x +b \right )}\, \left (\frac {2 B e x \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{5 c}+\frac {2 \left (A \,e^{2}+2 B d e -\frac {2 B e \left (2 b e +2 c d \right )}{5 c}\right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}{3 c e}+\frac {2 \left (A \,d^{2}-\frac {\left (A \,e^{2}+2 B d e -\frac {2 B e \left (2 b e +2 c d \right )}{5 c}\right ) b d}{3 c e}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}+\frac {2 \left (2 A d e +B \,d^{2}-\frac {3 B e b d}{5 c}-\frac {2 \left (A \,e^{2}+2 B d e -\frac {2 B e \left (2 b e +2 c d \right )}{5 c}\right ) \left (b e +c d \right )}{3 c e}\right ) b \sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) E\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d F\left (\sqrt {\frac {\left (x +\frac {b}{c}\right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+b d x}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\) | \(518\) |
| default | \(\text {Expression too large to display}\) | \(1144\) |
((e*x+d)*x*(c*x+b))^(1/2)/(x*(c*x+b))^(1/2)/(e*x+d)^(1/2)*(2/5*B*e/c*x*(c* e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2/3*(A*e^2+2*B*d*e-2/5*B*e/c*(2*b*e+2*c *d))/c/e*(c*e*x^3+b*e*x^2+c*d*x^2+b*d*x)^(1/2)+2*(A*d^2-1/3*(A*e^2+2*B*d*e -2/5*B*e/c*(2*b*e+2*c*d))/c/e*b*d)*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*(2*A*d*e+B*d^2-3/5*B*e/c*b *d-2/3*(A*e^2+2*B*d*e-2/5*B*e/c*(2*b*e+2*c*d))/c/e*(b*e+c*d))*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)*((-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 = 0.13 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=-\frac {2 \, {\left ({\left (3 \, B c^{3} d^{3} + {\left (8 \, B b c^{2} - 25 \, A c^{3}\right )} d^{2} e - {\left (17 \, B b^{2} c - 25 \, A b c^{2}\right )} d e^{2} + 2 \, {\left (4 \, B b^{3} - 5 \, A b^{2} c\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right ) + 3 \, {\left (3 \, B c^{3} d^{2} e - {\left (13 \, B b c^{2} - 20 \, A c^{3}\right )} d e^{2} + 2 \, {\left (4 \, B b^{2} c - 5 \, A b c^{2}\right )} e^{3}\right )} \sqrt {c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} - b c d e + b^{2} e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {4 \, {\left (2 \, c^{3} d^{3} - 3 \, b c^{2} d^{2} e - 3 \, b^{2} c d e^{2} + 2 \, b^{3} e^{3}\right )}}{27 \, c^{3} e^{3}}, \frac {3 \, c e x + c d + b e}{3 \, c e}\right )\right ) - 3 \, {\left (3 \, B c^{3} e^{3} x + 6 \, B c^{3} d e^{2} - {\left (4 \, B b c^{2} - 5 \, A c^{3}\right )} e^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {e x + d}\right )}}{45 \, c^{4} e^{2}} \]
-2/45*((3*B*c^3*d^3 + (8*B*b*c^2 - 25*A*c^3)*d^2*e - (17*B*b^2*c - 25*A*b* c^2)*d*e^2 + 2*(4*B*b^3 - 5*A*b^2*c)*e^3)*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^2*e - (13*B*b*c^2 - 20*A*c^3)*d*e^2 + 2*(4*B*b^2*c - 5*A*b *c^2)*e^3)*sqrt(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), 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*e^3*x + 6*B*c^3*d*e^2 - (4*B*b *c^2 - 5*A*c^3)*e^3)*sqrt(c*x^2 + b*x)*sqrt(e*x + d))/(c^4*e^2)
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (A + B x\right ) \left (d + e x\right )^{\frac {3}{2}}}{\sqrt {x \left (b + c x\right )}}\, dx \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]
\[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int { \frac {{\left (B x + A\right )} {\left (e x + d\right )}^{\frac {3}{2}}}{\sqrt {c x^{2} + b x}} \,d x } \]
Timed out. \[ \int \frac {(A+B x) (d+e x)^{3/2}}{\sqrt {b x+c x^2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}}{\sqrt {c\,x^2+b\,x}} \,d x \]