Integrand size = 29, antiderivative size = 705 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 e \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {a+b x+c x^2}}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x}}-\frac {\sqrt {2} \left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} (b B d-2 A c d+A b e-2 a B e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \]
2*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d)+c*(A*b*e-2*A*c*d-2*B*a*e+B*b*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)+2* e*(b^2*e*(-2*A*e+B*d)-2*c*(-3*A*a*e^2+A*c*d^2+4*B*a*d*e)+b*(2*A*c*d*e+B*a* e^2+B*c*d^2))*(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)-(b^2*e*(-2*A*e+B*d)-2*c*(-3*A*a*e^2+A*c*d^2+4*B*a*d*e)+b*(2*A*c*d *e+B*a*e^2+B*c*d^2))*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*(A*b*e-2*A*c*d-2*B*a*e+B*b*d)*E llipticF(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 = 31.38 (sec) , antiderivative size = 1267, normalized size of antiderivative = 1.80 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d+e x} \left (a+b x+c x^2\right )^2 \left (-\frac {2 e^2 (-B d+A e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac {2 \left (A b c^2 d^2-2 a B c^2 d^2-2 A b^2 c d e+2 a b B c d e+4 a A c^2 d e+A b^3 e^2-a b^2 B e^2-3 a A b c e^2+2 a^2 B c e^2-b B c^2 d^2 x+2 A c^3 d^2 x-2 A b c^2 d e x+4 a B c^2 d e x+A b^2 c e^2 x-a b B c e^2 x-2 a A c^2 e^2 x\right )}{\left (-b^2+4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right )}{(a+x (b+c x))^{3/2}}-\frac {2 (d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2} \left (-\left (\left (b^2 e (B d-2 A e)-2 c \left (A c d^2+4 a B d e-3 a A e^2\right )+b \left (B c d^2+2 A c d e+a B e^2\right )\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^2 e (-B d+2 A e)+2 c \left (A c d^2+4 a B d e-3 a A e^2\right )-b \left (B c d^2+2 A c d e+a B e^2\right )\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 (b^3 e^2 (-B d+2 A e)+b \left (c d \sqrt {\left (b^2-4 a c\right ) e^2} (B d+2 A e)+a e^2 \left (4 B c d-8 A c e+B \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )-b^2 e \left (2 A e \left (2 c d+\sqrt {\left (b^2-4 a c\right ) e^2}\right )+B \left (-3 c d^2+a e^2-d \sqrt {\left (b^2-4 a c\right ) e^2}\right )\right )-2 c \left (-2 a^2 B e^3+A c d^2 \sqrt {\left (b^2-4 a c\right ) e^2}+a e \left (6 B c d^2-8 A c d e+4 B d \sqrt {\left (b^2-4 a c\right ) e^2}-3 A e \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 )}{\left (-b^2+4 a c\right ) e \left (c d^2-b d e+a e^2\right )^2 (a+x (b+c x))^{3/2} \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*((-2*e^2*(-(B*d) + A*e))/((c*d^2 - b*d* e + a*e^2)^2*(d + e*x)) + (2*(A*b*c^2*d^2 - 2*a*B*c^2*d^2 - 2*A*b^2*c*d*e + 2*a*b*B*c*d*e + 4*a*A*c^2*d*e + A*b^3*e^2 - a*b^2*B*e^2 - 3*a*A*b*c*e^2 + 2*a^2*B*c*e^2 - b*B*c^2*d^2*x + 2*A*c^3*d^2*x - 2*A*b*c^2*d*e*x + 4*a*B* c^2*d*e*x + A*b^2*c*e^2*x - a*b*B*c*e^2*x - 2*a*A*c^2*e^2*x))/((-b^2 + 4*a *c)*(c*d^2 - b*d*e + a*e^2)^2*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(3/2) - (2*(d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*(-((b^2*e*(B*d - 2*A*e) - 2* c*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2) + b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*( 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 + Sq rt[(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 + Sqr t[(b^2 - 4*a*c)*e^2])*(b^2*e*(-(B*d) + 2*A*e) + 2*c*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2) - b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*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]))] + (b^3*e^2*(-(B*d) + 2*A*e) + b*(c*d*Sqrt[(b^ 2 - 4*a*c)*e^2]*(B*d + 2*A*e) + a*e^2*(4*B*c*d - 8*A*c*e + B*Sqrt[(b^2 - 4 *a*c)*e^2])) - b^2*e*(2*A*e*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + B*(-3*c*d^ 2 + a*e^2 - d*Sqrt[(b^2 - 4*a*c)*e^2])) - 2*c*(-2*a^2*B*e^3 + A*c*d^2*S...
Time = 1.15 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1235, 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.
| \(\displaystyle \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {e \left ((B d-2 A e) b^2+(A c d+a B e) b-6 a c (B d-A e)-c (b B d-2 A c d+A b e-2 a B e) x\right )}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \int \frac {(B d-2 A e) b^2+(A c d+a B e) b-6 a c (B d-A e)-c (b B d-2 A c d+A b e-2 a B e) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {2 \int -\frac {c \left (d (2 B d-A e) b^2-\left (A c d^2+2 a B e d+a A e^2\right ) b-2 a \left (3 B c d^2-4 A c e d-a B e^2\right )+\left (e (B d-2 A e) b^2+\left (B c d^2+2 A c e d+a B e^2\right ) b-2 c \left (A c d^2+4 a B e d-3 a A e^2\right )\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {c \int \frac {d (2 B d-A e) b^2-\left (A c d^2+2 a B e d+a A e^2\right ) b-2 a \left (3 B c d^2-4 A c e d-a B e^2\right )+\left (e (B d-2 A e) b^2+\left (B c d^2+2 A c e d+a B e^2\right ) b-2 c \left (A c d^2+4 a B e d-3 a A e^2\right )\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {c \left (\frac {\left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (a e^2-b d e+c d^2\right ) (-2 a B e+A b e-2 A c d+b B d) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {2 \left (a B (2 c d-b e)-A \left (-e b^2+c d b+2 a c e\right )+c (b B d-2 A c d+A b e-2 a B e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {e \left (\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (e (B d-2 A e) b^2+\left (B c d^2+2 A c e d+a B e^2\right ) b-2 c \left (A c d^2+4 a B e d-3 a A e^2\right )\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (b B d-2 A c d+A b e-2 a B e) \left (c d^2-b e d+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}-\frac {2 \left (e (B d-2 A e) b^2+\left (B c d^2+2 A c e d+a B e^2\right ) b-2 c \left (A c d^2+4 a B e d-3 a A e^2\right )\right ) \sqrt {c x^2+b x+a}}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {2 \left (-A \left (2 a c e+b^2 (-e)+b c d\right )+c x (-2 a B e+A b e-2 A c d+b B d)+a B (2 c d-b e)\right )}{\left (b^2-4 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} (-2 a B e+A b e-2 A c d+b B d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} \left (b \left (a B e^2+2 A c d e+B c d^2\right )-2 c \left (-3 a A e^2+4 a B d e+A c d^2\right )+b^2 e (B d-2 A e)\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}\) |
(2*(a*B*(2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e) + c*(b*B*d - 2*A*c*d + A*b*e - 2*a*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*((-2*(b^2*e*(B*d - 2*A*e) - 2*c*(A*c*d^2 + 4* a*B*d*e - 3*a*A*e^2) + b*(B*c*d^2 + 2*A*c*d*e + a*B*e^2))*Sqrt[a + b*x + c *x^2])/((c*d^2 - b*d*e + a*e^2)*Sqrt[d + e*x]) + (c*((Sqrt[2]*Sqrt[b^2 - 4 *a*c]*(b^2*e*(B*d - 2*A*e) - 2*c*(A*c*d^2 + 4*a*B*d*e - 3*a*A*e^2) + b*(B* c*d^2 + 2*A*c*d*e + a*B*e^2))*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]*(b*B*d - 2*A*c*d + A *b*e - 2*a*B*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d - (b + S qrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Ellipti cF[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))
3.27.39.3.1 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[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(2168\) vs. \(2(649)=1298\).
Time = 5.54 (sec) , antiderivative size = 2169, normalized size of antiderivative = 3.08
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(2169\) |
| default | \(\text {Expression too large to display}\) | \(8357\) |
((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*(( 6*A*a*c*e^2-2*A*b^2*e^2+2*A*b*c*d*e-2*A*c^2*d^2+B*a*b*e^2-8*B*a*c*d*e+B*b^ 2*d*e+B*b*c*d^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)*x^2+(7*A*a*b*c*e^3-2*A*a*c^2*d*e^2-2*A*b^3*e^3+ A*b^2*c*d*e^2+A*b*c^2*d^2*e-2*A*c^3*d^3-2*B*a^2*c*e^3+B*a*b^2*e^3-5*B*a*b* c*d*e^2-2*B*a*c^2*d^2*e+B*b^3*d*e^2+B*b*c^2*d^3)/c/e/(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)*x+(4*A*a^2* c*e^3-A*a*b^2*e^3+3*A*a*b*c*d*e^2-4*A*a*c^2*d^2*e-A*b^3*d*e^2+2*A*b^2*c*d^ 2*e-A*b*c^2*d^3-6*B*a^2*c*d*e^2+2*B*a*b^2*d*e^2-2*B*a*b*c*d^2*e+2*B*a*c^2* d^3)/c/e/(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))/((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*a*b*c*e^3-12*A*a*c^2*d*e^2-4*A*b^3*e^3+3*A*b^2*c*d*e^2+3*A *b*c^2*d^2*e-4*A*c^3*d^3-6*B*a^2*c*e^3+2*B*a*b^2*e^3-8*B*a*b*c*d*e^2+2*B*a *c^2*d^2*e+2*B*b^3*d*e^2-2*B*b^2*c*d^2*e+2*B*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)+2*(7*A*a *b*c*e^3-2*A*a*c^2*d*e^2-2*A*b^3*e^3+A*b^2*c*d*e^2+A*b*c^2*d^2*e-2*A*c^...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 2251, normalized size of antiderivative = 3.19 \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \]
2/3*(((B*a*b*c^2 - 2*A*a*c^3)*d^4 - (4*B*a*b^2*c - (10*B*a^2 + 3*A*a*b)*c^ 2)*d^3*e + (B*a*b^3 - 18*A*a^2*c^2 - (B*a^2*b - 3*A*a*b^2)*c)*d^2*e^2 + (B *a^2*b^2 - 2*A*a*b^3 - 3*(2*B*a^3 - 3*A*a^2*b)*c)*d*e^3 + ((B*b*c^3 - 2*A* c^4)*d^3*e - (4*B*b^2*c^2 - (10*B*a + 3*A*b)*c^3)*d^2*e^2 + (B*b^3*c - 18* A*a*c^3 - (B*a*b - 3*A*b^2)*c^2)*d*e^3 - (3*(2*B*a^2 - 3*A*a*b)*c^2 - (B*a *b^2 - 2*A*b^3)*c)*e^4)*x^3 + ((B*b*c^3 - 2*A*c^4)*d^4 - (3*B*b^2*c^2 - (1 0*B*a + A*b)*c^3)*d^3*e - 3*(B*b^3*c + 6*A*a*c^3 - (3*B*a*b + 2*A*b^2)*c^2 )*d^2*e^2 + (B*b^4 + A*b^3*c - 3*(2*B*a^2 + 3*A*a*b)*c^2)*d*e^3 + (B*a*b^3 - 2*A*b^4 - 3*(2*B*a^2*b - 3*A*a*b^2)*c)*e^4)*x^2 + ((B*b^2*c^2 - 2*A*b*c ^3)*d^4 - (4*B*b^3*c + 2*A*a*c^3 - (11*B*a*b + 3*A*b^2)*c^2)*d^3*e + (B*b^ 4 + 5*(2*B*a^2 - 3*A*a*b)*c^2 - (5*B*a*b^2 - 3*A*b^3)*c)*d^2*e^2 + (2*B*a* b^3 - 2*A*b^4 - 18*A*a^2*c^2 - (7*B*a^2*b - 12*A*a*b^2)*c)*d*e^3 + (B*a^2* b^2 - 2*A*a*b^3 - 3*(2*B*a^3 - 3*A*a^2*b)*c)*e^4)*x)*sqrt(c*e)*weierstrass PInverse(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*a*b*c^2 - 2*A*a*c^3) *d^3*e + (B*a*b^2*c - 2*(4*B*a^2 - A*a*b)*c^2)*d^2*e^2 + (6*A*a^2*c^2 + (B *a^2*b - 2*A*a*b^2)*c)*d*e^3 + ((B*b*c^3 - 2*A*c^4)*d^2*e^2 + (B*b^2*c^2 - 2*(4*B*a - A*b)*c^3)*d*e^3 + (6*A*a*c^3 + (B*a*b - 2*A*b^2)*c^2)*e^4)*x^3 + ((B*b*c^3 - 2*A*c^4)*d^3*e + 2*(B*b^2*c^2 - 4*B*a*c^3)*d^2*e^2 + (B*...
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A + B x}{\left (d + e x\right )^{\frac {3}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {B x + A}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {A+B x}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {A+B\,x}{{\left (d+e\,x\right )}^{3/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \]