Integrand size = 29, antiderivative size = 591 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\frac {2 (B d-A e) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2}}-\frac {2 \left (2 A e (2 c d-b e)-B \left (c d^2+e (b d-3 a e)\right )\right ) \sqrt {a+b x+c x^2}}{3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (2 A e (2 c d-b e)-B \left (c d^2+e (b d-3 a e)\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (B d-A e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 e \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
2/3*(-A*e+B*d)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(3/2)-2/3*( 2*A*e*(-b*e+2*c*d)-B*(c*d^2+e*(-3*a*e+b*d)))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b* d*e+c*d^2)^2/(e*x+d)^(1/2)+1/3*(2*A*e*(-b*e+2*c*d)-B*(c*d^2+e*(-3*a*e+b*d) ))*EllipticE(1/2*((b+2*c*x+(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2 ^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2 ^(1/2)*(-4*a*c+b^2)^(1/2)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1 /2)/e/(a*e^2-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(1/2)/(c*(e*x+d)/(2*c*d-e*(b+(-4 *a*c+b^2)^(1/2))))^(1/2)+2/3*(-A*e+B*d)*EllipticF(1/2*((b+2*c*x+(-4*a*c+b^ 2)^(1/2))/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*e*(-4*a*c+b^2)^(1/2)/(2*c* d-e*(b+(-4*a*c+b^2)^(1/2))))^(1/2))*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-c*(c*x^2+ b*x+a)/(-4*a*c+b^2))^(1/2)*(c*(e*x+d)/(2*c*d-e*(b+(-4*a*c+b^2)^(1/2))))^(1 /2)/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.35 (sec) , antiderivative size = 992, normalized size of antiderivative = 1.68 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right ) \left (-\frac {2 (-B d+A e)}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^2}-\frac {2 \left (-B c d^2-b B d e+4 A c d e-2 A b e^2+3 a B e^2\right )}{3 \left (c d^2-b d e+a e^2\right )^2 (d+e x)}\right )}{\sqrt {a+x (b+c x)}}+\frac {2 (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (-\left (\left (B c d^2+B e (b d-3 a e)+2 A e (-2 c d+b e)\right ) \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )\right )+\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {1+\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (B c d^2+B e (b d-3 a e)+2 A e (-2 c d+b e)\right ) E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\left (-8 a B c d e^2+B c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}-3 a B e^2 \sqrt {\left (b^2-4 a c\right ) e^2}-b^2 e^2 (B d+2 A e)+2 A c e \left (-3 c d^2+a e^2-2 d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+b e \left (2 A e \left (3 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+B \left (3 c d^2+3 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt {2} \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} \sqrt {d+e x}}\right )}{3 e^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+x (b+c x)} \sqrt {\frac {(d+e x)^2 \left (c \left (-1+\frac {d}{d+e x}\right )^2+\frac {e \left (b-\frac {b d}{d+e x}+\frac {a e}{d+e x}\right )}{d+e x}\right )}{e^2}}} \]
(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*(-(B*d) + A*e))/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^2) - (2*(-(B*c*d^2) - b*B*d*e + 4*A*c*d*e - 2*A*b*e^2 + 3 *a*B*e^2))/(3*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]*(-((B*c*d^2 + B*e*(b*d - 3*a*e ) + 2*A*e*(-2*c*d + b*e))*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x ) + (a*e)/(d + e*x)))/(d + e*x))) + ((I/2)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2* (c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(B*c*d^2 + B*e*(b*d - 3*a *e) + 2*A*e*(-2*c*d + b*e))*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d *e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -(( -2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)* e^2]))] - (-8*a*B*c*d*e^2 + B*c*d^2*Sqrt[(b^2 - 4*a*c)*e^2] - 3*a*B*e^2*Sq rt[(b^2 - 4*a*c)*e^2] - b^2*e^2*(B*d + 2*A*e) + 2*A*c*e*(-3*c*d^2 + a*e^2 - 2*d*Sqrt[(b^2 - 4*a*c)*e^2]) + b*e*(2*A*e*(3*c*d + Sqrt[(b^2 - 4*a*c)*e^ 2]) + B*(3*c*d^2 + 3*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcS inh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a* c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c *d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[d + e*x])))/(3*e^...
Time = 0.91 (sec) , antiderivative size = 612, normalized size of antiderivative = 1.04, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1237, 27, 1237, 27, 1269, 1172, 321, 327}
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.
| \(\displaystyle \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {b B d-3 A c d+2 A b e-3 a B e-c (B d-A e) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {b B d-3 A c d+2 A b e-3 a B e-c (B d-A e) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {2 \int \frac {c \left (3 A c d^2+4 a B e d-b (2 B d+A e) d-a A e^2-\left (B c d^2+B e (b d-3 a e)-2 A e (2 c d-b e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \int \frac {3 A c d^2+4 a B e d-b (2 B d+A e) d-a A e^2-\left (B c d^2+B e (b d-3 a e)-2 A e (2 c d-b e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \left (\frac {(B d-A e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (B d-A e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (B d-A e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 \sqrt {a+b x+c x^2} (B d-A e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (B d-A e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (B e (b d-3 a e)-2 A e (2 c d-b e)+B c d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\) |
(2*(B*d - A*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x) ^(3/2)) - ((-2*(B*c*d^2 + B*e*(b*d - 3*a*e) - 2*A*e*(2*c*d - b*e))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) - (c*(-((Sqrt[2]*S qrt[b^2 - 4*a*c]*(B*c*d^2 + B*e*(b*d - 3*a*e) - 2*A*e*(2*c*d - b*e))*Sqrt[ d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqr t[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^ 2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(c*e*Sqrt[(c*(d + e*x) )/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2])) + (2*Sqrt[2 ]*Sqrt[b^2 - 4*a*c]*(B*d - A*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e*x)) /(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c] )*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])))/(c*d^2 - b*d*e + a*e^2) )/(3*(c*d^2 - b*d*e + a*e^2))
3.27.34.3.1 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[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 )/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e *Rt[b^2 - 4*a*c, 2])))^m)) Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e }, x] && EqQ[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]
Leaf count of result is larger than twice the leaf count of optimal. \(1108\) vs. \(2(527)=1054\).
Time = 3.76 (sec) , antiderivative size = 1109, normalized size of antiderivative = 1.88
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1109\) |
| default | \(\text {Expression too large to display}\) | \(11733\) |
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/3/e^2/ (a*e^2-b*d*e+c*d^2)*(A*e-B*d)*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1 /2)/(x+d/e)^2+2/3*(c*e*x^2+b*e*x+a*e)/e/(a*e^2-b*d*e+c*d^2)^2*(2*A*b*e^2-4 *A*c*d*e-3*B*a*e^2+B*b*d*e+B*c*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e))^(1/2)+2* (-1/3*c/e*(A*e-B*d)/(a*e^2-b*d*e+c*d^2)+1/3/e*(b*e-c*d)*(2*A*b*e^2-4*A*c*d *e-3*B*a*e^2+B*b*d*e+B*c*d^2)/(a*e^2-b*d*e+c*d^2)^2-1/3*b/(a*e^2-b*d*e+c*d ^2)^2*(2*A*b*e^2-4*A*c*d*e-3*B*a*e^2+B*b*d*e+B*c*d^2))*(d/e-1/2*(b+(-4*a*c +b^2)^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2 /c*(-b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(( x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2) /(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1 /2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/( -d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2))-2/3*c*(2*A*b*e^2-4*A*c*d*e-3*B *a*e^2+B*b*d*e+B*c*d^2)/(a*e^2-b*d*e+c*d^2)^2*(d/e-1/2*(b+(-4*a*c+b^2)^(1/ 2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(- 4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+ (-4*a*c+b^2)^(1/2))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3 +b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2 )))*EllipticE(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-d/e+1/ 2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.13 (sec) , antiderivative size = 1028, normalized size of antiderivative = 1.74 \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \]
2/9*((B*c^2*d^5 - (4*B*b*c - 5*A*c^2)*d^4*e + (B*b^2 + (9*B*a - 5*A*b)*c)* d^3*e^2 - (3*B*a*b - 2*A*b^2 + 3*A*a*c)*d^2*e^3 + (B*c^2*d^3*e^2 - (4*B*b* c - 5*A*c^2)*d^2*e^3 + (B*b^2 + (9*B*a - 5*A*b)*c)*d*e^4 - (3*B*a*b - 2*A* b^2 + 3*A*a*c)*e^5)*x^2 + 2*(B*c^2*d^4*e - (4*B*b*c - 5*A*c^2)*d^3*e^2 + ( B*b^2 + (9*B*a - 5*A*b)*c)*d^2*e^3 - (3*B*a*b - 2*A*b^2 + 3*A*a*c)*d*e^4)* x)*sqrt(c*e)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^ 2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3* (B*c^2*d^4*e - (3*B*a - 2*A*b)*c*d^2*e^3 + (B*b*c - 4*A*c^2)*d^3*e^2 + (B* c^2*d^2*e^3 - (3*B*a - 2*A*b)*c*e^5 + (B*b*c - 4*A*c^2)*d*e^4)*x^2 + 2*(B* c^2*d^3*e^2 - (3*B*a - 2*A*b)*c*d*e^4 + (B*b*c - 4*A*c^2)*d^2*e^3)*x)*sqrt (c*e)*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2 ), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^ 2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6 *a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e )/(c*e))) + 3*(2*B*c^2*d^3*e^2 - 5*A*c^2*d^2*e^3 - A*a*c*e^5 - (2*B*a - 3* A*b)*c*d*e^4 + (B*c^2*d^2*e^3 - (3*B*a - 2*A*b)*c*e^5 + (B*b*c - 4*A*c^2)* d*e^4)*x)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d))/(c^3*d^6*e^2 - 2*b*c^2*d^5* e^3 - 2*a*b*c*d^3*e^5 + a^2*c*d^2*e^6 + (b^2*c + 2*a*c^2)*d^4*e^4 + (c^...
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{\frac {5}{2}} \sqrt {a + b x + c x^{2}}}\, dx \]
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {B x + A}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{5/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^{5/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]