Integrand size = 24, antiderivative size = 585 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (9 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (2 c d^2+a e^2\right )-c \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {16 \sqrt {2} (2 c d-b e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:
-2/3*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^(3/2)-2/3*(e*x+d)^ (1/2)*(9*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-4*b*c*(a*e^2+2*c*d^2)-c*(16*c^2*d^2 +b^2*e^2-4*c*e*(-3*a*e+4*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/(c*x^ 2+b*x+a)^(1/2)-1/3*2^(1/2)*(16*c^2*d^2+b^2*e^2-4*c*e*(-3*a*e+4*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)^(3/2)/(a*e^2-b*d*e+c*d^2)/(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)+16/3*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))/(-4*a*c+b^2)^(3/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
Result contains complex when optimal does not.
Time = 21.43 (sec) , antiderivative size = 1323, normalized size of antiderivative = 2.26 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:
Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^(5/2),x]
Output:
(Sqrt[d + e*x]*(a + b*x + c*x^2)^3*((-2*(b + 2*c*x))/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^2) + (2*(8*b*c^2*d^2 - 9*b^2*c*d*e + 4*a*c^2*d*e + b^3*e^2 + 4*a*b*c*e^2 + 16*c^3*d^2*x - 16*b*c^2*d*e*x + b^2*c*e^2*x + 12*a*c^2*e^2* x))/(3*(-b^2 + 4*a*c)^2*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(5/2) + ((d + e*x)^(3/2)*(a + b*x + c*x^2)^(5/2)*(-4*Sqrt[(c* d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(16*c^2* d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*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 + Sqrt[(b^2 - 4*a*c)*e^2])*(16*c^2*d^2 + b^2*e^2 + 4*c*e*(-4*b*d + 3*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]*(-(b^3*e^3) + b^2*e^2*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2]) + 4*b* (a*c*e^3 - 4*c*d*e*Sqrt[(b^2 - 4*a*c)*e^2]) + 4*c*(4*c*d^2*Sqrt[(b^2 - 4*a *c)*e^2] + a*e^2*(-2*c*d + 3*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*(-1 ...
Time = 0.88 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1163, 27, 1235, 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.
\(\displaystyle \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 1163 |
\(\displaystyle \frac {2 \int -\frac {8 c d-b e+6 c e x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {8 c d-b e+6 c e x}{\sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1235 |
\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int \frac {c e \left (7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {c e \int \frac {7 d e b^2-8 \left (c d^2+a e^2\right ) b+4 a c d e-\left (16 c^2 d^2+b^2 e^2-4 c e (4 b d-3 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1269 |
\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {c e \left (\frac {8 (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 1172 |
\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {c e \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 321 |
\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {c e \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-c x \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right )-4 b c \left (a e^2+2 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+9 b^2 c d e\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {c e \left (\frac {16 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right )}-\frac {2 (b+2 c x) \sqrt {d+e x}}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}\) |
Input:
Int[Sqrt[d + e*x]/(a + b*x + c*x^2)^(5/2),x]
Output:
(-2*(b + 2*c*x)*Sqrt[d + e*x])/(3*(b^2 - 4*a*c)*(a + b*x + c*x^2)^(3/2)) - ((2*Sqrt[d + e*x]*(9*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - 4*b*c*(2*c*d^2 + a*e^2) - c*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*x))/((b^2 - 4*a *c)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x + c*x^2]) - (c*e*(-((Sqrt[2]*Sqrt [b^2 - 4*a*c]*(16*c^2*d^2 + b^2*e^2 - 4*c*e*(4*b*d - 3*a*e))*Sqrt[d + e*x] *Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticE[ArcSin[Sqrt[(b + S qrt[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])) + (16*Sqrt[2]*Sqrt[ b^2 - 4*a*c]*(2*c*d - b*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e*x))/(2*c *d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c ))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c ]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)] )/(c*e*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])))/((b^2 - 4*a*c)*(c*d^2 - b*d* e + a*e^2)))/(3*(b^2 - 4*a*c))
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*(b + 2*c*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)* (b^2 - 4*a*c))), x] - Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1 )*(b*e*m + 2*c*d*(2*p + 3) + 2*c*e*(m + 2*p + 3)*x)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && LtQ[p, -1] && GtQ[m, 0] && (LtQ[ m, 1] || (ILtQ[m + 2*p + 3, 0] && NeQ[m, 2])) && 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[(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[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. \(1485\) vs. \(2(525)=1050\).
Time = 1.93 (sec) , antiderivative size = 1486, normalized size of antiderivative = 2.54
method | result | size |
elliptic | \(\text {Expression too large to display}\) | \(1486\) |
default | \(\text {Expression too large to display}\) | \(13071\) |
Input:
int((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),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)*((4/3/(4*a *c-b^2)/c*x+2/3/c^2/(4*a*c-b^2)*b)*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a* d)^(1/2)/(a/c+b/c*x+x^2)^2-2*(c*e*x+c*d)*(-1/3*(12*a*c*e^2+b^2*e^2-16*b*c* d*e+16*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)/(4*a*c-b^2)*x-1/3*(4*a*b*c*e^2+4*a*c^2*d*e+b^3*e^2-9*b^2*c*d*e+8*b *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 )/(4*a*c-b^2)/c)/((a/c+b/c*x+x^2)*(c*e*x+c*d))^(1/2)+2*(-1/3*(4*a*b*c*e^3- 32*a*c^2*d*e^2-b^3*e^3+32*b*c^2*d^2*e-32*c^3*d^3)/(4*a*c-b^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)-1/3*e*(4*a*b*c*e^2+4 *a*c^2*d*e+b^3*e^2-9*b^2*c*d*e+8*b*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)/(4*a*c-b^2)-2/3*c*d*(12*a*c*e^2+b^2*e^ 2-16*b*c*d*e+16*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)/(4*a*c-b^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))-2/3*c*e*(12*a*c*e^2+b^2*e^2-16*b*c*d*e+16*c^2*d^2)/(4*a *c-b^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^...
Leaf count of result is larger than twice the leaf count of optimal. 1722 vs. \(2 (533) = 1066\).
Time = 0.15 (sec) , antiderivative size = 1722, normalized size of antiderivative = 2.94 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
Output:
2/9*((16*a^2*c^3*d^3 - 24*a^2*b*c^2*d^2*e + (16*c^5*d^3 - 24*b*c^4*d^2*e + 6*(b^2*c^3 + 4*a*c^4)*d*e^2 + (b^3*c^2 - 12*a*b*c^3)*e^3)*x^4 + 6*(a^2*b^ 2*c + 4*a^3*c^2)*d*e^2 + (a^2*b^3 - 12*a^3*b*c)*e^3 + 2*(16*b*c^4*d^3 - 24 *b^2*c^3*d^2*e + 6*(b^3*c^2 + 4*a*b*c^3)*d*e^2 + (b^4*c - 12*a*b^2*c^2)*e^ 3)*x^3 + (16*(b^2*c^3 + 2*a*c^4)*d^3 - 24*(b^3*c^2 + 2*a*b*c^3)*d^2*e + 6* (b^4*c + 6*a*b^2*c^2 + 8*a^2*c^3)*d*e^2 + (b^5 - 10*a*b^3*c - 24*a^2*b*c^2 )*e^3)*x^2 + 2*(16*a*b*c^3*d^3 - 24*a*b^2*c^2*d^2*e + 6*(a*b^3*c + 4*a^2*b *c^2)*d*e^2 + (a*b^4 - 12*a^2*b^2*c)*e^3)*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*(16*a^2*c^3*d^2*e - 16*a^2*b*c^2 *d*e^2 + (16*c^5*d^2*e - 16*b*c^4*d*e^2 + (b^2*c^3 + 12*a*c^4)*e^3)*x^4 + (a^2*b^2*c + 12*a^3*c^2)*e^3 + 2*(16*b*c^4*d^2*e - 16*b^2*c^3*d*e^2 + (b^3 *c^2 + 12*a*b*c^3)*e^3)*x^3 + (16*(b^2*c^3 + 2*a*c^4)*d^2*e - 16*(b^3*c^2 + 2*a*b*c^3)*d*e^2 + (b^4*c + 14*a*b^2*c^2 + 24*a^2*c^3)*e^3)*x^2 + 2*(16* a*b*c^3*d^2*e - 16*a*b^2*c^2*d*e^2 + (a*b^3*c + 12*a^2*b*c^2)*e^3)*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 ...
\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d + e x}}{\left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**(5/2),x)
Output:
Integral(sqrt(d + e*x)/(a + b*x + c*x**2)**(5/2), x)
\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
Output:
integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^(5/2), x)
\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
Output:
integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^(5/2), x)
Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {d+e\,x}}{{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:
int((d + e*x)^(1/2)/(a + b*x + c*x^2)^(5/2),x)
Output:
int((d + e*x)^(1/2)/(a + b*x + c*x^2)^(5/2), x)
\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {\sqrt {e x +d}}{\left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}d x \] Input:
int((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x)
Output:
int((e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x)