Integrand size = 45, antiderivative size = 995 \[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\frac {2 \left (8 b^3 C e^3-3 b c e^2 (b C d+4 b B e-a C e)+c^3 d \left (8 C d^2-3 e (4 B d-7 A e)\right )+3 c^2 e \left (a e (C d-5 B e)-b \left (C d^2-2 B d e-7 A e^2\right )\right )+3 c e \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right ) x\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3}}{315 c^3 e^3}+\frac {2 \left (3 B c-\frac {2 C (c d+b e)}{e}\right ) \left (a+b x+c x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3}}{21 c^2}+\frac {2 C (d+e x) \left (a+b x+c x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3}}{9 c e}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\frac {\left (8 c^2 d^2-2 b^2 e^2-3 c e (b d-2 a e)\right ) \left (8 b^2 C e^2-c e (b C d+12 b B e+7 a C e)-c^2 \left (2 C d^2-3 e (B d+7 A e)\right )\right )}{e}-5 c (2 c d-b e) \left (6 b^2 C d e+c e (21 A c d-5 a C d-3 a B e)+b \left (2 a C e^2-c d (C d+9 B e)\right )\right )\right ) \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+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 )}{315 c^4 e^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \left (a+b x+c x^2\right )}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right ) \left (8 b^3 C e^3-3 c^2 e^2 (b B d+2 a C d-7 A b e-10 a B e)+3 b c e^2 (b C d-4 b B e-9 a C e)-2 c^3 d \left (8 C d^2-3 e (4 B d-7 A e)\right )\right ) \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}} \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+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 )}{315 c^4 e^4 (d+e x) \left (a+b x+c x^2\right )} \] Output:
2/315*(8*b^3*C*e^3-3*b*c*e^2*(4*B*b*e-C*a*e+C*b*d)+c^3*d*(8*C*d^2-3*e*(-7* A*e+4*B*d))+3*c^2*e*(a*e*(-5*B*e+C*d)-b*(-7*A*e^2-2*B*d*e+C*d^2))+3*c*e*(8 *b^2*C*e^2-c*e*(12*B*b*e+7*C*a*e+C*b*d)-c^2*(2*C*d^2-3*e*(7*A*e+B*d)))*x)* (a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)/c^3/e^3+2/21*(3*B*c-2*C*(b*e +c*d)/e)*(c*x^2+b*x+a)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)/c^2+2 /9*C*(e*x+d)*(c*x^2+b*x+a)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)/c /e+1/315*2^(1/2)*(-4*a*c+b^2)^(1/2)*((8*c^2*d^2-2*b^2*e^2-3*c*e*(-2*a*e+b* d))*(8*b^2*C*e^2-c*e*(12*B*b*e+7*C*a*e+C*b*d)-c^2*(2*C*d^2-3*e*(7*A*e+B*d) ))/e-5*c*(-b*e+2*c*d)*(6*b^2*C*d*e+c*e*(21*A*c*d-3*B*a*e-5*C*a*d)+b*(2*a*C *e^2-c*d*(9*B*e+C*d))))*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(a*d+(a*e+b* d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)*EllipticE(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^ (1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2) )*e))^(1/2))/c^4/e^3/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2)/(c *x^2+b*x+a)-2/315*2^(1/2)*(-4*a*c+b^2)^(1/2)*(a*e^2-b*d*e+c*d^2)*(8*b^3*C* e^3-3*c^2*e^2*(-7*A*b*e-10*B*a*e+B*b*d+2*C*a*d)+3*b*c*e^2*(-4*B*b*e-9*C*a* e+C*b*d)-2*c^3*d*(8*C*d^2-3*e*(-7*A*e+4*B*d)))*(c*(e*x+d)/(2*c*d-(b+(-4*a* c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*(a*d+(a*e+b* d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(-4*a*c+b^2)^ (1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2) )*e))^(1/2))/c^4/e^4/(e*x+d)/(c*x^2+b*x+a)
Result contains complex when optimal does not.
Time = 18.10 (sec) , antiderivative size = 15681, normalized size of antiderivative = 15.76 \[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\text {Result too large to show} \] Input:
Integrate[(A + B*x + C*x^2)*Sqrt[a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c *e*x^3],x]
Output:
Result too large to show
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.
| \(\displaystyle \int \left (A+B x+C x^2\right ) \sqrt {x (a e+b d)+a d+x^2 (b e+c d)+c e x^3} \, dx\) |
| \(\Big \downarrow \) 2526 |
| \(\displaystyle \frac {\int -\left ((b C d-3 A c e+a C e+(2 c C d-3 B c e+2 b C e) x) \sqrt {c e x^3+(c d+b e) x^2+(b d+a e) x+a d}\right )dx}{3 c e}+\frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int (b C d-3 A c e+a C e+(2 c C d-3 B c e+2 b C e) x) \sqrt {c e x^3+(c d+b e) x^2+(b d+a e) x+a d}dx}{3 c e}\) |
| \(\Big \downarrow \) 2490 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {3 c e (b C d-3 A c e+a C e)-(c d+b e) (2 c C d-3 B c e+2 b C e)}{3 c e}+(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right )\right ) \sqrt {c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}}d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{3 c e}+(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {c e \left (\frac {c d+b e}{3 c e}+x\right )^3+\frac {\left (3 c e (b d+a e)-(c d+b e)^2\right ) \left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}+\frac {(2 c d-b e) \left (c^2 d^2-2 b^2 e^2-c e (b d-9 a e)\right )}{27 c^2 e^2}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 B e d+9 A e^2\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {2 C \left (x (a e+b d)+a d+x^2 (b e+c d)+c e x^3\right )^{3/2}}{9 c e}-\frac {\int \left (\frac {\sqrt {\frac {\left (-2 c d+b e-3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (-c^2 d^2+2 b^2 e^2-9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2+c e (b d-9 a e)+3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}} \left (-\left (\left (2 C d^2-3 e (B d-3 A e)\right ) c^2\right )-e (b C d-3 b B e-3 a C e) c-2 b^2 C e^2\right )}{9 \sqrt {3} c e}+\frac {(2 c C d-3 B c e+2 b C e) \left (\frac {c d+b e}{3 c e}+x\right ) \sqrt {\frac {\left (2 c d-b e+3 c e \left (\frac {c d+b e}{3 c e}+x\right )\right ) \left (c^2 d^2-2 b^2 e^2+9 c^2 e^2 \left (\frac {c d+b e}{3 c e}+x\right )^2-c e (b d-9 a e)-3 c e (2 c d-b e) \left (\frac {c d+b e}{3 c e}+x\right )\right )}{c^2 e^2}}}{3 \sqrt {3}}\right )d\left (\frac {c d+b e}{3 c e}+x\right )}{3 c e}\) |
Input:
Int[(A + B*x + C*x^2)*Sqrt[a*d + (b*d + a*e)*x + (c*d + b*e)*x^2 + c*e*x^3 ],x]
Output:
$Aborted
Int[(P3_)^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> With[{a = Coeff[P3 , x, 0], b = Coeff[P3, x, 1], c = Coeff[P3, x, 2], d = Coeff[P3, x, 3]}, Su bst[Int[((3*d*e - c*f)/(3*d) + f*x)^m*Simp[(2*c^3 - 9*b*c*d + 27*a*d^2)/(27 *d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, x + c/(3*d)] /; NeQ[c , 0]] /; FreeQ[{e, f, m, p}, x] && PolyQ[P3, x, 3]
Int[(Pm_)*(Qn_)^(p_), x_Symbol] :> With[{m = Expon[Pm, x], n = Expon[Qn, x] }, Simp[Coeff[Pm, x, m]*(Qn^(p + 1)/(n*(p + 1)*Coeff[Qn, x, n])), x] + Simp [1/(n*Coeff[Qn, x, n]) Int[ExpandToSum[n*Coeff[Qn, x, n]*Pm - Coeff[Pm, x , m]*D[Qn, x], x]*Qn^p, x], x] /; EqQ[m, n - 1]] /; FreeQ[p, x] && PolyQ[Pm , x] && PolyQ[Qn, x] && NeQ[p, -1]
Time = 8.19 (sec) , antiderivative size = 1698, normalized size of antiderivative = 1.71
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1698\) |
| default | \(\text {Expression too large to display}\) | \(3190\) |
| risch | \(\text {Expression too large to display}\) | \(7090\) |
Input:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x,method=_ RETURNVERBOSE)
Output:
2/9*C*x^3*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/7*(B*c*e+C*b*e +C*c*d-2/9*(4*b*e+4*c*d)*C)/c/e*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a *d)^(1/2)+2/5*(A*c*e+B*b*e+B*c*d+C*a*e+C*b*d-2/9*C*(7/2*a*e+7/2*b*d)-2/7*( B*c*e+C*b*e+C*c*d-2/9*(4*b*e+4*c*d)*C)/c/e*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b *e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/3*(A*b*e+A*c*d+B*a*e+B*b*d+1/3*C*a *d-2/7*(B*c*e+C*b*e+C*c*d-2/9*(4*b*e+4*c*d)*C)/c/e*(5/2*a*e+5/2*b*d)-2/5*( A*c*e+B*b*e+B*c*d+C*a*e+C*b*d-2/9*C*(7/2*a*e+7/2*b*d)-2/7*(B*c*e+C*b*e+C*c *d-2/9*(4*b*e+4*c*d)*C)/c/e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3 +b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(a*d*A-2/5*(A*c*e+B*b*e+B*c*d+C* a*e+C*b*d-2/9*C*(7/2*a*e+7/2*b*d)-2/7*(B*c*e+C*b*e+C*c*d-2/9*(4*b*e+4*c*d) *C)/c/e*(3*b*e+3*c*d))/c/e*a*d-2/3*(A*b*e+A*c*d+B*a*e+B*b*d+1/3*C*a*d-2/7* (B*c*e+C*b*e+C*c*d-2/9*(4*b*e+4*c*d)*C)/c/e*(5/2*a*e+5/2*b*d)-2/5*(A*c*e+B *b*e+B*c*d+C*a*e+C*b*d-2/9*C*(7/2*a*e+7/2*b*d)-2/7*(B*c*e+C*b*e+C*c*d-2/9* (4*b*e+4*c*d)*C)/c/e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(1/2*a*e+1/2*b* d))*(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*(-b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2*(-b+(-4*a*c+b ^2)^(1/2))/c))^(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)*El lipticF(((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*(-b+(-4*a*c+b^2)^(1/2))/c))^(1/2))+2*(A...
Time = 0.11 (sec) , antiderivative size = 1036, normalized size of antiderivative = 1.04 \[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\text {Too large to display} \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x, a lgorithm="fricas")
Output:
2/945*((16*C*c^5*d^5 - 8*(2*C*b*c^4 + 3*B*c^5)*d^4*e - (5*C*b^2*c^3 - 42*A *c^5 - 3*(10*C*a + 9*B*b)*c^4)*d^3*e^2 - (5*C*b^3*c^2 + 3*(22*B*a + 21*A*b )*c^4 - 3*(7*C*a*b + 4*B*b^2)*c^3)*d^2*e^3 - (16*C*b^4*c - 378*A*a*c^4 + 3 *(22*C*a^2 + 41*B*a*b + 21*A*b^2)*c^3 - 3*(28*C*a*b^2 + 9*B*b^3)*c^2)*d*e^ 4 + (16*C*b^5 - 9*(10*B*a^2 + 21*A*a*b)*c^3 + 3*(41*C*a^2*b + 41*B*a*b^2 + 14*A*b^3)*c^2 - 24*(4*C*a*b^3 + B*b^4)*c)*e^5)*sqrt(c*e)*weierstrassPInve rse(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*(16*C*c^5*d^4*e - 8*(C*b*c^4 + 3*B*c^5)*d^3*e^2 - 3*(2*C*b^2*c^3 - 14*A*c^5 - (6*C*a + 5*B*b)*c^4)*d^2* e^3 - (8*C*b^3*c^2 + 6*(8*B*a + 7*A*b)*c^4 - 15*(2*C*a*b + B*b^2)*c^3)*d*e ^4 + (16*C*b^4*c - 126*A*a*c^4 + 3*(14*C*a^2 + 29*B*a*b + 14*A*b^2)*c^3 - 24*(3*C*a*b^2 + B*b^3)*c^2)*e^5)*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), 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*(35*C*c^5*e^5*x^3 + 8*C *c^5*d^3*e^2 - 3*(C*b*c^4 + 4*B*c^5)*d^2*e^3 - (3*C*b^2*c^3 - 21*A*c^5 - 2 *(4*C*a + 3*B*b)*c^4)*d*e^4 + (8*C*b^3*c^2 + 3*(10*B*a + 7*A*b)*c^4 - 3...
\[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int \sqrt {\left (d + e x\right ) \left (a + b x + c x^{2}\right )} \left (A + B x + C x^{2}\right )\, dx \] Input:
integrate((C*x**2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x**2+c*e*x**3)**(1/2), x)
Output:
Integral(sqrt((d + e*x)*(a + b*x + c*x**2))*(A + B*x + C*x**2), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int { \sqrt {c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x} {\left (C x^{2} + B x + A\right )} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x, a lgorithm="maxima")
Output:
integrate(sqrt(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)*(C*x^2 + B *x + A), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int { \sqrt {c e x^{3} + {\left (c d + b e\right )} x^{2} + a d + {\left (b d + a e\right )} x} {\left (C x^{2} + B x + A\right )} \,d x } \] Input:
integrate((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x, a lgorithm="giac")
Output:
integrate(sqrt(c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)*(C*x^2 + B *x + A), x)
Timed out. \[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int \left (C\,x^2+B\,x+A\right )\,\sqrt {c\,e\,x^3+\left (b\,e+c\,d\right )\,x^2+\left (a\,e+b\,d\right )\,x+a\,d} \,d x \] Input:
int((A + B*x + C*x^2)*(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^(1 /2),x)
Output:
int((A + B*x + C*x^2)*(a*d + x*(a*e + b*d) + x^2*(b*e + c*d) + c*e*x^3)^(1 /2), x)
\[ \int \left (A+B x+C x^2\right ) \sqrt {a d+(b d+a e) x+(c d+b e) x^2+c e x^3} \, dx=\int \left (C \,x^{2}+B x +A \right ) \sqrt {a d +\left (a e +b d \right ) x +\left (b e +c d \right ) x^{2}+c e \,x^{3}}d x \] Input:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x)
Output:
int((C*x^2+B*x+A)*(a*d+(a*e+b*d)*x+(b*e+c*d)*x^2+c*e*x^3)^(1/2),x)