Integrand size = 24, antiderivative size = 609 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=-\frac {2 e \sqrt {a+b x+c x^2}}{5 \left (c d^2-b d e+a e^2\right ) (d+e x)^{5/2}}-\frac {8 e (2 c d-b e) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^2 (d+e x)^{3/2}}-\frac {2 e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {a+b x+c x^2}}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{15 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:
-2/5*e*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(5/2)-8/15*e*(-b*e+ 2*c*d)*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(3/2)-2/15*e*(23* c^2*d^2+8*b^2*e^2-c*e*(9*a*e+23*b*d))*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d ^2)^3/(e*x+d)^(1/2)+1/15*2^(1/2)*(-4*a*c+b^2)^(1/2)*(23*c^2*d^2+8*b^2*e^2- c*e*(9*a*e+23*b*d))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*El lipticE(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))/(a*e^2-b*d*e+c*d^2)^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)^(1/2)-8/15 *2^(1/2)*(-4*a*c+b^2)^(1/2)*(-b*e+2*c*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)*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))/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)/(c* x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 30.04 (sec) , antiderivative size = 983, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]
Output:
(Sqrt[d + e*x]*(a + b*x + c*x^2)*((-2*e)/(5*(c*d^2 - b*d*e + a*e^2)*(d + e *x)^3) + (8*e*(-2*c*d + b*e))/(15*(c*d^2 - b*d*e + a*e^2)^2*(d + e*x)^2) + (2*e*(-23*c^2*d^2 + 23*b*c*d*e - 8*b^2*e^2 + 9*a*c*e^2))/(15*(c*d^2 - b*d *e + a*e^2)^3*(d + e*x))))/Sqrt[a + x*(b + c*x)] + (2*(d + e*x)^(3/2)*Sqrt [a + b*x + c*x^2]*((23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*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])*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*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]))] + (-30*c^3*d^3 + 8*b^2*e^2*(b*e - Sqrt[(b^2 - 4*a*c)*e^2]) - c^2*d*(-45*b*d*e - 34*a*e^2 + 23*d*Sqrt[(b^2 - 4*a*c)*e^2]) + c*e*(-31*b^2*d*e - 17*a*b*e^2 + 23*b*d*Sqrt[(b^2 - 4*a*c )*e^2] + 9*a*e*Sqrt[(b^2 - 4*a*c)*e^2]))*EllipticF[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]))]))/(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])))/(15*e*(c*d^2 - b*d*e +...
Time = 1.06 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1167, 27, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx\) |
| \(\Big \downarrow \) 1167 |
| \(\displaystyle -\frac {2 \int -\frac {5 c d-4 b e-3 c e x}{2 (d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 c d-4 b e-3 c e x}{(d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {-\frac {2 \int -\frac {15 c^2 d^2+8 b^2 e^2-c e (19 b d+9 a e)-4 c e (2 c d-b e) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {15 c^2 d^2+8 b^2 e^2-c e (19 b d+9 a e)-4 c e (2 c d-b e) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {\frac {-\frac {2 \int -\frac {c \left (15 c^2 d^3-c e (11 b d+17 a e) d+4 b e^2 (b d+a e)+e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {c \int \frac {15 c^2 d^3-c e (11 b d+17 a e) d+4 b e^2 (b d+a e)+e \left (23 c^2 d^2+8 b^2 e^2-c e (23 b d+9 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {\frac {\frac {c \left (\left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx-4 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx\right )}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {\frac {\frac {c \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {8 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{a e^2-b d e+c d^2}-\frac {2 e \sqrt {a+b x+c x^2} \left (-c e (9 a e+23 b d)+8 b^2 e^2+23 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}-\frac {8 e \sqrt {a+b x+c x^2} (2 c d-b e)}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}}{5 \left (a e^2-b d e+c d^2\right )}-\frac {2 e \sqrt {a+b x+c x^2}}{5 (d+e x)^{5/2} \left (a e^2-b d e+c d^2\right )}\) |
Input:
Int[1/((d + e*x)^(7/2)*Sqrt[a + b*x + c*x^2]),x]
Output:
(-2*e*Sqrt[a + b*x + c*x^2])/(5*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(5/2)) + ((-8*e*(2*c*d - b*e)*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d*e + a*e^2)*(d + e*x)^(3/2)) + ((-2*e*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*a*e))*Sq rt[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]*(23*c^2*d^2 + 8*b^2*e^2 - c*e*(23*b*d + 9*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*Sqrt[(c*(d + e*x))/( 2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (8*Sqrt[2]*Sq rt[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*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)))/(5*(c*d^2 - b*d*e + a*e^2))
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[e*(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)*Simp[c*d*(m + 1) - b*e*(m + p + 2) - c*e*(m + 2*p + 3)*x, x]*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[m , -1] && ((LtQ[m, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]) || (SumSimp lerQ[m, 1] && IntegerQ[p]) || ILtQ[Simplify[m + 2*p + 3], 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. \(1157\) vs. \(2(543)=1086\).
Time = 11.92 (sec) , antiderivative size = 1158, normalized size of antiderivative = 1.90
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1158\) |
| default | \(\text {Expression too large to display}\) | \(14312\) |
Input:
int(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x,method=_RETURNVERBOSE)
Output:
((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(-2/5/e^2/ (a*e^2-b*d*e+c*d^2)*(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 )^3+8/15*(b*e-2*c*d)/e/(a*e^2-b*d*e+c*d^2)^2*(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+2/15*(c*e*x^2+b*e*x+a*e)/(a*e^2-b*d*e+c*d^2)^ 3*(9*a*c*e^2-8*b^2*e^2+23*b*c*d*e-23*c^2*d^2)/((x+d/e)*(c*e*x^2+b*e*x+a*e) )^(1/2)+2*(4/15*c*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)^2+1/15*(b*e-c*d)*(9*a*c* e^2-8*b^2*e^2+23*b*c*d*e-23*c^2*d^2)/(a*e^2-b*d*e+c*d^2)^3-1/15*b*e/(a*e^2 -b*d*e+c*d^2)^3*(9*a*c*e^2-8*b^2*e^2+23*b*c*d*e-23*c^2*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))-2/15*c*e*(9*a*c*e^2-8*b^ 2*e^2+23*b*c*d*e-23*c^2*d^2)/(a*e^2-b*d*e+c*d^2)^3*(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),(...
Leaf count of result is larger than twice the leaf count of optimal. 1403 vs. \(2 (551) = 1102\).
Time = 0.15 (sec) , antiderivative size = 1403, normalized size of antiderivative = 2.30 \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")
Output:
2/45*((22*c^3*d^6 - 33*b*c^2*d^5*e + 3*(9*b^2*c - 14*a*c^2)*d^4*e^2 - (8*b ^3 - 21*a*b*c)*d^3*e^3 + (22*c^3*d^3*e^3 - 33*b*c^2*d^2*e^4 + 3*(9*b^2*c - 14*a*c^2)*d*e^5 - (8*b^3 - 21*a*b*c)*e^6)*x^3 + 3*(22*c^3*d^4*e^2 - 33*b* c^2*d^3*e^3 + 3*(9*b^2*c - 14*a*c^2)*d^2*e^4 - (8*b^3 - 21*a*b*c)*d*e^5)*x ^2 + 3*(22*c^3*d^5*e - 33*b*c^2*d^4*e^2 + 3*(9*b^2*c - 14*a*c^2)*d^3*e^3 - (8*b^3 - 21*a*b*c)*d^2*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*(23*c^3*d^5*e - 23*b*c^2*d^4*e^2 + (8*b^2*c - 9*a*c^2)*d^3*e^3 + (23*c^3*d^2*e^4 - 23*b*c^2*d*e^5 + (8*b^2*c - 9*a*c^2) *e^6)*x^3 + 3*(23*c^3*d^3*e^3 - 23*b*c^2*d^2*e^4 + (8*b^2*c - 9*a*c^2)*d*e ^5)*x^2 + 3*(23*c^3*d^4*e^2 - 23*b*c^2*d^3*e^3 + (8*b^2*c - 9*a*c^2)*d^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*(34*c^3*d^4*e^2 - 41*b*c^2*d^3*e^3 - 10*a*b*c*d*e ^5 + 3*a^2*c*e^6 + 5*(3*b^2*c + a*c^2)*d^2*e^4 + (23*c^3*d^2*e^4 - 23*b*c^ 2*d*e^5 + (8*b^2*c - 9*a*c^2)*e^6)*x^2 + 2*(27*c^3*d^3*e^3 - 29*b*c^2*d...
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {7}{2}} \sqrt {a + b x + c x^{2}}}\, dx \] Input:
integrate(1/(e*x+d)**(7/2)/(c*x**2+b*x+a)**(1/2),x)
Output:
Integral(1/((d + e*x)**(7/2)*sqrt(a + b*x + c*x**2)), x)
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)), x)
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^(7/2)), x)
Timed out. \[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{7/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:
int(1/((d + e*x)^(7/2)*(a + b*x + c*x^2)^(1/2)),x)
Output:
int(1/((d + e*x)^(7/2)*(a + b*x + c*x^2)^(1/2)), x)
\[ \int \frac {1}{(d+e x)^{7/2} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{c \,e^{4} x^{6}+b \,e^{4} x^{5}+4 c d \,e^{3} x^{5}+a \,e^{4} x^{4}+4 b d \,e^{3} x^{4}+6 c \,d^{2} e^{2} x^{4}+4 a d \,e^{3} x^{3}+6 b \,d^{2} e^{2} x^{3}+4 c \,d^{3} e \,x^{3}+6 a \,d^{2} e^{2} x^{2}+4 b \,d^{3} e \,x^{2}+c \,d^{4} x^{2}+4 a \,d^{3} e x +b \,d^{4} x +a \,d^{4}}d x \] Input:
int(1/(e*x+d)^(7/2)/(c*x^2+b*x+a)^(1/2),x)
Output:
int((sqrt(d + e*x)*sqrt(a + b*x + c*x**2))/(a*d**4 + 4*a*d**3*e*x + 6*a*d* *2*e**2*x**2 + 4*a*d*e**3*x**3 + a*e**4*x**4 + b*d**4*x + 4*b*d**3*e*x**2 + 6*b*d**2*e**2*x**3 + 4*b*d*e**3*x**4 + b*e**4*x**5 + c*d**4*x**2 + 4*c*d **3*e*x**3 + 6*c*d**2*e**2*x**4 + 4*c*d*e**3*x**5 + c*e**4*x**6),x)