Integrand size = 24, antiderivative size = 607 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {4 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}+\frac {2 \sqrt {2} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
-2*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2) /(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-4*e*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))*( c*x^2+b*x+a)^(1/2)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)+2*(c^2 *d^2+b^2*e^2-c*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)*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+ b^2))^(1/2)/(a*e^2-b*d*e+c*d^2)^2/(-4*a*c+b^2)^(1/2)/(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*(-b*e+2*c*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)*(-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)/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+ a)^(1/2)
Result contains complex when optimal does not.
Time = 17.58 (sec) , antiderivative size = 1172, normalized size of antiderivative = 1.93 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {4 e^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}}} \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (a+x (b+c x))-2 e \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}}} \left (\left (b^2-4 a c\right ) e^3 (a+x (b+c x))+(d+e x) \left (b^3 e^2+b^2 c e (-2 d+e x)+b c \left (-3 a e^2+c d (d-2 e x)\right )+2 c^2 \left (c d^2 x+a e (2 d-e x)\right )\right )\right )-i \sqrt {2} \left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} 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 )-i \sqrt {2} \left (b^3 e^3-b^2 e^2 \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+b c e \left (-4 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}\right )+c \left (-c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e^2 \left (8 c d+3 \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )\right ) (d+e x)^{3/2} \sqrt {\frac {-2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}+2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (d-e x)}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {\frac {2 a e^2+d \sqrt {\left (b^2-4 a c\right ) e^2}-2 c d e x+e \sqrt {\left (b^2-4 a c\right ) e^2} x+b e (-d+e x)}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \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 )}{\left (b^2-4 a c\right ) e \left (c d^2+e (-b d+a e)\right )^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} \sqrt {a+x (b+c x)}} \]
(4*e^2*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)* e^2])]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(a + x*(b + c*x)) - 2*e*Sqr t[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*((b ^2 - 4*a*c)*e^3*(a + x*(b + c*x)) + (d + e*x)*(b^3*e^2 + b^2*c*e*(-2*d + e *x) + b*c*(-3*a*e^2 + c*d*(d - 2*e*x)) + 2*c^2*(c*d^2*x + a*e*(2*d - e*x)) )) - I*Sqrt[2]*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e))*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c) *e^2] + 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2*c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/(( -2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*EllipticE[I*ArcSinh[(S qrt[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]))] - I*Sqrt[2]*(b^3*e^3 - b^2*e^2*(2*c*d + Sq rt[(b^2 - 4*a*c)*e^2]) + b*c*e*(-4*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]) + c* (-(c*d^2*Sqrt[(b^2 - 4*a*c)*e^2]) + a*e^2*(8*c*d + 3*Sqrt[(b^2 - 4*a*c)*e^ 2])))*(d + e*x)^(3/2)*Sqrt[(-2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] + 2*c*d*e *x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(d - e*x))/((2*c*d - b*e + Sqrt[(b^ 2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[(2*a*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2] - 2 *c*d*e*x + e*Sqrt[(b^2 - 4*a*c)*e^2]*x + b*e*(-d + e*x))/((-2*c*d + b*e...
Time = 0.87 (sec) , antiderivative size = 642, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1165, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {e \left (-2 e b^2+c d b+6 a c e+c (2 c d-b e) x\right )}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {-2 e b^2+c d b+6 a c e+c (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {c \left (d e b^2+c d^2 b+a e^2 b-8 a c d e+2 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {c \int \frac {d e b^2+c d^2 b+a e^2 b-8 a c d e+2 \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (\frac {2 \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}-\frac {c \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) - (e*((4*(c^2*d^2 + b ^2*e^2 - c*e*(b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2 )*Sqrt[d + e*x]) - (c*((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c^2*d^2 + b^2*e^2 - c *e*(b*d + 3*a*e))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c) )]*EllipticE[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[(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]*(2*c*d - b*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)))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
3.25.74.3.1 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_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*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*Simp[b*c*d*e*(2*p - m + 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]
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. \(1848\) vs. \(2(551)=1102\).
Time = 4.16 (sec) , antiderivative size = 1849, normalized size of antiderivative = 3.05
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1849\) |
| default | \(\text {Expression too large to display}\) | \(4415\) |
((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*c*e*(2 /(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+ 2*a*b^2*c*d^2*e^2-8*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^ 2*c^2*d^4)*(3*a*c*e^2-b^2*e^2+b*c*d*e-c^2*d^2)*x^2+(7*a*b*c*e^3-2*a*c^2*d* e^2-2*b^3*e^3+b^2*c*d*e^2+b*c^2*d^2*e-2*c^3*d^3)/(4*a^3*c*e^4-a^2*b^2*e^4- 8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^3+2*a*b^2*c*d^2*e^2-8*a*b*c^ 2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e-b^2*c^2*d^4)/c/e*x+(4*a^2*c* e^3-a*b^2*e^3+3*a*b*c*d*e^2-4*a*c^2*d^2*e-b^3*d*e^2+2*b^2*c*d^2*e-b*c^2*d^ 3)/(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a*b^3*d*e^ 3+2*a*b^2*c*d^2*e^2-8*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3*c*d^3*e- b^2*c^2*d^4)/c/e)/((x^3+(b*e+c*d)/c/e*x^2+(a*e+b*d)/c/e*x+a*d/c/e)*c*e)^(1 /2)+2*(-(15*a*b*c*e^3-12*a*c^2*d*e^2-4*b^3*e^3+3*b^2*c*d*e^2+3*b*c^2*d^2*e -4*c^3*d^3)/(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d^2*e^2+2*a *b^3*d*e^3+2*a*b^2*c*d^2*e^2-8*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2*e^2+2*b^3 *c*d^3*e-b^2*c^2*d^4)+2*(7*a*b*c*e^3-2*a*c^2*d*e^2-2*b^3*e^3+b^2*c*d*e^2+b *c^2*d^2*e-2*c^3*d^3)/(4*a^3*c*e^4-a^2*b^2*e^4-8*a^2*b*c*d*e^3+8*a^2*c^2*d ^2*e^2+2*a*b^3*d*e^3+2*a*b^2*c*d^2*e^2-8*a*b*c^2*d^3*e+4*a*c^3*d^4-b^4*d^2 *e^2+2*b^3*c*d^3*e-b^2*c^2*d^4))*(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^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.15 (sec) , antiderivative size = 1614, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
-2/3*((2*a*c^3*d^4 - 3*a*b*c^2*d^3*e - 3*(a*b^2*c - 6*a^2*c^2)*d^2*e^2 + ( 2*a*b^3 - 9*a^2*b*c)*d*e^3 + (2*c^4*d^3*e - 3*b*c^3*d^2*e^2 - 3*(b^2*c^2 - 6*a*c^3)*d*e^3 + (2*b^3*c - 9*a*b*c^2)*e^4)*x^3 + (2*c^4*d^4 - b*c^3*d^3* e - 6*(b^2*c^2 - 3*a*c^3)*d^2*e^2 - (b^3*c - 9*a*b*c^2)*d*e^3 + (2*b^4 - 9 *a*b^2*c)*e^4)*x^2 + (2*b*c^3*d^4 - (3*b^2*c^2 - 2*a*c^3)*d^3*e - 3*(b^3*c - 5*a*b*c^2)*d^2*e^2 + 2*(b^4 - 6*a*b^2*c + 9*a^2*c^2)*d*e^3 + (2*a*b^3 - 9*a^2*b*c)*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)) + 6*(a*c^3*d^3*e - a*b*c^2*d^2*e^2 + (a*b^2*c - 3*a^2*c^2)*d* e^3 + (c^4*d^2*e^2 - b*c^3*d*e^3 + (b^2*c^2 - 3*a*c^3)*e^4)*x^3 + (c^4*d^3 *e - 3*a*c^3*d*e^3 + (b^3*c - 3*a*b*c^2)*e^4)*x^2 + (b*c^3*d^3*e - (b^2*c^ 2 - a*c^3)*d^2*e^2 + (b^3*c - 4*a*b*c^2)*d*e^3 + (a*b^2*c - 3*a^2*c^2)*e^4 )*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*(b*c^3*d^3*e - 2*(b^2*c^2 - 2*a*c^3)*d^2*e^2 + (b^3 *c - 3*a*b*c^2)*d*e^3 + (a*b^2*c - 4*a^2*c^2)*e^4 + 2*(c^4*d^2*e^2 - b*...
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]