Integrand size = 24, antiderivative size = 723 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=-\frac {2 e}{3 \left (c d^2-b d e+a e^2\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2}}-\frac {8 e (2 c d-b e)}{3 \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {d+e x} \left (12 a c e (2 c d-b e)^2+\left (b c d-b^2 e+2 a c e\right ) \left (3 c^2 d^2+8 b^2 e^2-5 c e (3 b d+a e)\right )+c (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3 \sqrt {a+b x+c x^2}}+\frac {\sqrt {2} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^3 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \sqrt {b^2-4 a c} \left (c d^2-b d e+a e^2\right )^2 \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:
-2/3*e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(1/2)-8/3*e*(-b*e+2 *c*d)/(a*e^2-b*d*e+c*d^2)^2/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)-2/3*(e*x+d)^ (1/2)*(12*a*c*e*(-b*e+2*c*d)^2+(2*a*c*e-b^2*e+b*c*d)*(3*c^2*d^2+8*b^2*e^2- 5*c*e*(a*e+3*b*d))+c*(-b*e+2*c*d)*(3*c^2*d^2+8*b^2*e^2-c*e*(29*a*e+3*b*d)) *x)/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)^3/(c*x^2+b*x+a)^(1/2)+1/3*2^(1/2)*(-b *e+2*c*d)*(3*c^2*d^2+8*b^2*e^2-c*e*(29*a*e+3*b*d))*(e*x+d)^(1/2)*(-c*(c*x^ 2+b*x+a)/(-4*a*c+b^2))^(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))/(-4*a*c+b^2)^(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)-4/3*2^(1/2)*(3*c^2*d^2+2*b^2 *e^2-c*e*(5*a*e+3*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)*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))/(-4*a*c+b^2)^(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 = 23.59 (sec) , antiderivative size = 1525, normalized size of antiderivative = 2.11 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:
Integrate[1/((d + e*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]
Output:
(Sqrt[d + e*x]*(a + b*x + c*x^2)^2*((-2*e^3)/(3*(c*d^2 - b*d*e + a*e^2)^2* (d + e*x)^2) + (10*e^3*(-2*c*d + b*e))/(3*(c*d^2 - b*d*e + a*e^2)^3*(d + e *x)) - (2*(-(b*c^3*d^3) + 3*b^2*c^2*d^2*e - 6*a*c^3*d^2*e - 3*b^3*c*d*e^2 + 9*a*b*c^2*d*e^2 + b^4*e^3 - 4*a*b^2*c*e^3 + 2*a^2*c^2*e^3 - 2*c^4*d^3*x + 3*b*c^3*d^2*e*x - 3*b^2*c^2*d*e^2*x + 6*a*c^3*d*e^2*x + b^3*c*e^3*x - 3* a*b*c^2*e^3*x))/((-b^2 + 4*a*c)*(c*d^2 - b*d*e + a*e^2)^3*(a + b*x + c*x^2 ))))/(a + x*(b + c*x))^(3/2) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)*(4 *(-2*c*d + b*e)*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*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 *Sqrt[2]*(-2*c*d + b*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] - (2*a* e^2)/(d + e*x) - 2*c*d*(-1 + d/(d + e*x)) + b*e*(-1 + (2*d)/(d + e*x)))/(2 *c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*Sqrt[(Sqrt[(b^2 - 4*a*c)*e^2] + (2* a*e^2)/(d + e*x) + 2*c*d*(-1 + d/(d + e*x)) + b*(e - (2*d*e)/(d + e*x)))/( -2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*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]))])/Sqrt[d + e*x] + (I*Sqrt[2]*(-8*b^4*e^4 + b^3*e^3*(27 *c*d + 8*Sqrt[(b^2 - 4*a*c)*e^2]) - b^2*c*e^2*(27*c*d^2 - 37*a*e^2 + 19...
Time = 1.30 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {1165, 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.
| \(\displaystyle \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1165 |
| \(\displaystyle -\frac {2 \int \frac {e \left (-4 e b^2+3 c d b+10 a c e+3 c (2 c d-b e) x\right )}{2 (d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \int \frac {-4 e b^2+3 c d b+10 a c e+3 c (2 c d-b e) x}{(d+e x)^{5/2} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {-8 e^2 b^3+15 c d e b^2-c \left (3 c d^2-29 a e^2\right ) b-48 a c^2 d e-2 c \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x}{2 (d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {-8 e^2 b^3+15 c d e b^2-c \left (3 c d^2-29 a e^2\right ) b-48 a c^2 d e-2 c \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) x}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{3 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {2 \int \frac {c \left (4 d e^2 b^3-\left (9 c d^2 e-4 a e^3\right ) b^2-c d \left (3 c d^2+25 a e^2\right ) b+2 a c e \left (27 c d^2-5 a e^2\right )-(2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \int \frac {4 d e^2 b^3-\left (9 c d^2 e-4 a e^3\right ) b^2-c d \left (3 c d^2+25 a e^2\right ) b+2 a c e \left (27 c d^2-5 a e^2\right )-(2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \left (\frac {2 \left (a e^2-b d e+c d^2\right ) \left (-5 a c e^2+2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {(2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
| \(\Big \downarrow \) 1172 |
| \(\displaystyle -\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a}}-\frac {e \left (\frac {4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {c x^2+b x+a}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {-\frac {2 (2 c d-b e) \sqrt {c x^2+b x+a} \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \left (\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (3 c^2 d^2-3 b c e d+2 b^2 e^2-5 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}}{3 \left (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {2 \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2} \sqrt {c x^2+b x+a}}-\frac {e \left (\frac {4 \left (3 c^2 d^2+2 b^2 e^2-c e (3 b d+5 a e)\right ) \sqrt {c x^2+b x+a}}{3 \left (c d^2-b e d+a e^2\right ) (d+e x)^{3/2}}-\frac {-\frac {2 (2 c d-b e) \sqrt {c x^2+b x+a} \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right )}{\left (c d^2-b e d+a e^2\right ) \sqrt {d+e x}}-\frac {c \left (\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (3 c^2 d^2-3 b c e d+2 b^2 e^2-5 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (3 c^2 d^2+8 b^2 e^2-c e (3 b d+29 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}\right )}{c d^2-b e d+a e^2}}{3 \left (c d^2-b e d+a e^2\right )}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle -\frac {e \left (\frac {4 \sqrt {a+b x+c x^2} \left (-c e (5 a e+3 b d)+2 b^2 e^2+3 c^2 d^2\right )}{3 (d+e x)^{3/2} \left (a e^2-b d e+c d^2\right )}-\frac {-\frac {c \left (\frac {4 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (-5 a c e^2+2 b^2 e^2-3 b c d e+3 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{a e^2-b d e+c d^2}-\frac {2 \sqrt {a+b x+c x^2} (2 c d-b e) \left (-c e (29 a e+3 b d)+8 b^2 e^2+3 c^2 d^2\right )}{\sqrt {d+e x} \left (a e^2-b d e+c d^2\right )}}{3 \left (a e^2-b d e+c d^2\right )}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) (d+e x)^{3/2} \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}\) |
Input:
Int[1/((d + e*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x]
Output:
(-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)*(d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]) - (e*((4*(3*c^2*d^2 + 2*b^2*e^2 - c*e*(3*b*d + 5*a*e))*Sqrt[a + b*x + c*x^2])/(3*(c*d^2 - b*d *e + a*e^2)*(d + e*x)^(3/2)) - ((-2*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*a*e))*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*Sqr t[d + e*x]) - (c*(-((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(3*c^2*d^2 + 8*b^2*e^2 - c*e*(3*b*d + 29*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)/S qrt[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])) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*(3*c^2*d^2 - 3*b*c*d*e + 2*b^2*e^2 - 5*a*c*e^2)*Sqrt[(c*(d + e*x) )/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c ])*e)])/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])))/(c*d^2 - b*d*e + a*e^2 ))/(3*(c*d^2 - b*d*e + a*e^2))))/((b^2 - 4*a*c)*(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[(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. \(1963\) vs. \(2(657)=1314\).
Time = 12.08 (sec) , antiderivative size = 1964, normalized size of antiderivative = 2.72
| method | result | size |
| elliptic | \(\text {Expression too large to display}\) | \(1964\) |
| default | \(\text {Expression too large to display}\) | \(12895\) |
Input:
int(1/(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),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)*(-2/3*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+10/3*(c*e*x^2+b*e*x+a*e)*e^2/(a*e^2-b*d*e+c*d^2)^3*(b*e-2*c*d)/((x+d/e )*(c*e*x^2+b*e*x+a*e))^(1/2)-2*(c*e*x+c*d)*(-(b*e-2*c*d)*(3*a*c*e^2-b^2*e^ 2+b*c*d*e-c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e- b^2*c*d^2)/(a*e^2-b*d*e+c*d^2)^2*x+(2*a^2*c^2*e^3-4*a*b^2*c*e^3+9*a*b*c^2* d*e^2-6*a*c^3*d^2*e+b^4*e^3-3*b^3*c*d*e^2+3*b^2*c^2*d^2*e-b*c^3*d^3)/(4*a^ 2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)/c/(a*e^2-b*d* e+c*d^2)^2)/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(-1/3*c*e^2/(a*e^2-b*d*e +c*d^2)^2+5/3*e^2*(b*e-c*d)*(b*e-2*c*d)/(a*e^2-b*d*e+c*d^2)^3-5/3*b*e^3/(a *e^2-b*d*e+c*d^2)^3*(b*e-2*c*d)-(4*a^2*c^2*e^4-5*a*b^2*c*e^4+6*a*b*c^2*d*e ^3+b^4*e^4-b^3*c*d*e^3-3*b^2*c^2*d^2*e^2+6*b*c^3*d^3*e-4*c^4*d^4)/(4*a^2*c *e^2-a*b^2*e^2-4*a*b*c*d*e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)/(a*e^2-b*d*e+c*d ^2)^2+e*(2*a^2*c^2*e^3-4*a*b^2*c*e^3+9*a*b*c^2*d*e^2-6*a*c^3*d^2*e+b^4*e^3 -3*b^3*c*d*e^2+3*b^2*c^2*d^2*e-b*c^3*d^3)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d *e+4*a*c^2*d^2+b^3*d*e-b^2*c*d^2)/(a*e^2-b*d*e+c*d^2)^2-2*c*d*(b*e-2*c*d)* (3*a*c*e^2-b^2*e^2+b*c*d*e-c^2*d^2)/(4*a^2*c*e^2-a*b^2*e^2-4*a*b*c*d*e+4*a *c^2*d^2+b^3*d*e-b^2*c*d^2)/(a*e^2-b*d*e+c*d^2)^2)*(d/e-1/2*(b+(-4*a*c+b^2 )^(1/2))/c)*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*( -b+(-4*a*c+b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x...
Leaf count of result is larger than twice the leaf count of optimal. 3059 vs. \(2 (665) = 1330\).
Time = 0.23 (sec) , antiderivative size = 3059, normalized size of antiderivative = 4.23 \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {5}{2}} \left (a + b x + c x^{2}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(e*x+d)**(5/2)/(c*x**2+b*x+a)**(3/2),x)
Output:
Integral(1/((d + e*x)**(5/2)*(a + b*x + c*x**2)**(3/2)), x)
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="maxima")
Output:
integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(5/2)), x)
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {3}{2}} {\left (e x + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x, algorithm="giac")
Output:
integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(5/2)), x)
Timed out. \[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {1}{{\left (d+e\,x\right )}^{5/2}\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \,d x \] Input:
int(1/((d + e*x)^(5/2)*(a + b*x + c*x^2)^(3/2)),x)
Output:
int(1/((d + e*x)^(5/2)*(a + b*x + c*x^2)^(3/2)), x)
\[ \int \frac {1}{(d+e x)^{5/2} \left (a+b x+c x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}{c^{2} e^{3} x^{7}+2 b c \,e^{3} x^{6}+3 c^{2} d \,e^{2} x^{6}+2 a c \,e^{3} x^{5}+b^{2} e^{3} x^{5}+6 b c d \,e^{2} x^{5}+3 c^{2} d^{2} e \,x^{5}+2 a b \,e^{3} x^{4}+6 a c d \,e^{2} x^{4}+3 b^{2} d \,e^{2} x^{4}+6 b c \,d^{2} e \,x^{4}+c^{2} d^{3} x^{4}+a^{2} e^{3} x^{3}+6 a b d \,e^{2} x^{3}+6 a c \,d^{2} e \,x^{3}+3 b^{2} d^{2} e \,x^{3}+2 b c \,d^{3} x^{3}+3 a^{2} d \,e^{2} x^{2}+6 a b \,d^{2} e \,x^{2}+2 a c \,d^{3} x^{2}+b^{2} d^{3} x^{2}+3 a^{2} d^{2} e x +2 a b \,d^{3} x +a^{2} d^{3}}d x \] Input:
int(1/(e*x+d)^(5/2)/(c*x^2+b*x+a)^(3/2),x)
Output:
int((sqrt(d + e*x)*sqrt(a + b*x + c*x**2))/(a**2*d**3 + 3*a**2*d**2*e*x + 3*a**2*d*e**2*x**2 + a**2*e**3*x**3 + 2*a*b*d**3*x + 6*a*b*d**2*e*x**2 + 6 *a*b*d*e**2*x**3 + 2*a*b*e**3*x**4 + 2*a*c*d**3*x**2 + 6*a*c*d**2*e*x**3 + 6*a*c*d*e**2*x**4 + 2*a*c*e**3*x**5 + b**2*d**3*x**2 + 3*b**2*d**2*e*x**3 + 3*b**2*d*e**2*x**4 + b**2*e**3*x**5 + 2*b*c*d**3*x**3 + 6*b*c*d**2*e*x* *4 + 6*b*c*d*e**2*x**5 + 2*b*c*e**3*x**6 + c**2*d**3*x**4 + 3*c**2*d**2*e* x**5 + 3*c**2*d*e**2*x**6 + c**2*e**3*x**7),x)