\(\int \frac {1}{\sqrt {d+e x} (a+b x+c x^2)^{5/2}} \, dx\) [674]

Optimal result

Integrand size = 24, antiderivative size = 705 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=-\frac {2 \sqrt {d+e x} \left (b c d-b^2 e+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {2 \sqrt {d+e x} \left (3 a c e (2 c d-b e)^2-\left (b c d-b^2 e+2 a c e\right ) \left (8 c^2 d^2-2 b^2 e^2-5 c e (b d-2 a e)\right )-2 c (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{3 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \sqrt {2} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^2 \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \left (16 c^2 d^2-b^2 e^2-4 c e (4 b d-5 a e)\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{3 \left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2/3*(e*x+d)^(1/2)*(b*c*d-b^2*e+2*a*c*e+c*(-b*e+2*c*d)*x)/(-4*a*c+b^2)/(a* 
e^2-b*d*e+c*d^2)/(c*x^2+b*x+a)^(3/2)-2/3*(e*x+d)^(1/2)*(3*a*c*e*(-b*e+2*c* 
d)^2-(2*a*c*e-b^2*e+b*c*d)*(8*c^2*d^2-2*b^2*e^2-5*c*e*(-2*a*e+b*d))-2*c*(- 
b*e+2*c*d)*(4*c^2*d^2-b^2*e^2-4*c*e*(-2*a*e+b*d))*x)/(-4*a*c+b^2)^2/(a*e^2 
-b*d*e+c*d^2)^2/(c*x^2+b*x+a)^(1/2)-2/3*2^(1/2)*(-b*e+2*c*d)*(4*c^2*d^2-b^ 
2*e^2-4*c*e*(-2*a*e+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)^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)+2/3*2^(1/2)*(16*c^2*d^2-b^2*e^2-4*c*e*(-5*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)*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)/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 23.58 (sec) , antiderivative size = 5566, normalized size of antiderivative = 7.90 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 776, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1165, 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.

\(\displaystyle \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1165

\(\displaystyle -\frac {2 \int \frac {8 c^2 d^2-2 b^2 e^2-5 c e (b d-2 a e)+3 c e (2 c d-b e) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {8 c^2 d^2-2 b^2 e^2-5 c e (b d-2 a e)+3 c e (2 c d-b e) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )^{3/2}}dx}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {\frac {2 \sqrt {d+e x} \left (-2 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-5 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right )+3 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac {2 \int -\frac {c e \left (d e^2 b^3-\left (11 c d^2 e-a e^3\right ) b^2+4 c d \left (2 c d^2+5 a e^2\right ) b-4 a c e \left (c d^2+5 a e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {c e \int \frac {d e^2 b^3-\left (11 c d^2 e-a e^3\right ) b^2+4 c d \left (2 c d^2+5 a e^2\right ) b-4 a c e \left (c d^2+5 a e^2\right )+2 (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {d+e x} \left (-2 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-5 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right )+3 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1269

\(\displaystyle -\frac {\frac {c e \left (\frac {2 (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (a e^2-b d e+c d^2\right ) \left (20 a c e^2-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {d+e x} \left (-2 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-5 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right )+3 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1172

\(\displaystyle -\frac {2 \sqrt {d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \sqrt {d+e x} \left (3 a c e (2 c d-b e)^2-2 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-2 b^2 e^2-5 c e (b d-2 a e)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}+\frac {c e \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^2 d^2-16 b c e d-b^2 e^2+20 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 \sqrt {d+e x} \left (-e b^2+c d b+2 a c e+c (2 c d-b e) x\right )}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \left (c x^2+b x+a\right )^{3/2}}-\frac {\frac {2 \sqrt {d+e x} \left (3 a c e (2 c d-b e)^2-2 c \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) x (2 c d-b e)-\left (-e b^2+c d b+2 a c e\right ) \left (8 c^2 d^2-2 b^2 e^2-5 c e (b d-2 a e)\right )\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right ) \sqrt {c x^2+b x+a}}+\frac {c e \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (2 c d-b e) \left (4 c^2 d^2-b^2 e^2-4 c e (b d-2 a e)\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (c d^2-b e d+a e^2\right ) \left (16 c^2 d^2-16 b c e d-b^2 e^2+20 a c e^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{\left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}}{3 \left (b^2-4 a c\right ) \left (c d^2-b e d+a e^2\right )}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {\frac {c e \left (\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right ) E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {a+b x+c x^2} \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \left (20 a c e^2-b^2 e^2-16 b c d e+16 c^2 d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {a+b x+c x^2}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}+\frac {2 \sqrt {d+e x} \left (-2 c x (2 c d-b e) \left (-4 c e (b d-2 a e)-b^2 e^2+4 c^2 d^2\right )-\left (2 a c e+b^2 (-e)+b c d\right ) \left (-5 c e (b d-2 a e)-2 b^2 e^2+8 c^2 d^2\right )+3 a c e (2 c d-b e)^2\right )}{\left (b^2-4 a c\right ) \sqrt {a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}}{3 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}-\frac {2 \sqrt {d+e x} \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2} \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[1/(Sqrt[d + e*x]*(a + b*x + c*x^2)^(5/2)),x]
 

Output:

(-2*Sqrt[d + e*x]*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/(3*(b^2 - 
 4*a*c)*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)^(3/2)) - ((2*Sqrt[d + e* 
x]*(3*a*c*e*(2*c*d - b*e)^2 - (b*c*d - b^2*e + 2*a*c*e)*(8*c^2*d^2 - 2*b^2 
*e^2 - 5*c*e*(b*d - 2*a*e)) - 2*c*(2*c*d - b*e)*(4*c^2*d^2 - b^2*e^2 - 4*c 
*e*(b*d - 2*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*((2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(4*c^2*d^2 - 
b^2*e^2 - 4*c*e*(b*d - 2*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*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)] 
*Sqrt[a + b*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a* 
e^2)*(16*c^2*d^2 - 16*b*c*d*e - b^2*e^2 + 20*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])))/((b^2 - 4*a*c)*(c*d^2 - b 
*d*e + a*e^2)))/(3*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

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])
 

rule 327

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]
 

rule 1165

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]
 

rule 1172

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]
 

rule 1235

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] 
)
 

rule 1269

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1788\) vs. \(2(645)=1290\).

Time = 10.43 (sec) , antiderivative size = 1789, normalized size of antiderivative = 2.54

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1789\)
default	\(\text {Expression too large to display}\)	\(19400\)

Input:

int(1/(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)*((-2/3/(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)*(b*e-2*c*d) 
/c*x+2/3*(2*a*c*e-b^2*e+b*c*d)/(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^2)*(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)*(2/3*(b*e-2*c*d)*(8*a*c*e^2-b^2*e^2-4*b* 
c*d*e+4*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)^2*x-1/3*(20*a^2*c^2*e^3-17*a*b^2*c*e^3+12*a*b*c^2*d*e^2+4*a*c^3*d 
^2*e+2*b^4*e^3+3*b^3*c*d*e^2-13*b^2*c^2*d^2*e+8*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)^2/c)/((a/c+b/c*x+x^2)*( 
c*e*x+c*d))^(1/2)+2*(2/3*(20*a^2*c^2*e^4-9*a*b^2*c*e^4-20*a*b*c^2*d*e^3+36 
*a*c^3*d^2*e^2+b^4*e^4+3*b^3*c*d*e^3+3*b^2*c^2*d^2*e^2-24*b*c^3*d^3*e+16*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)^ 
2-1/3*e*(20*a^2*c^2*e^3-17*a*b^2*c*e^3+12*a*b*c^2*d*e^2+4*a*c^3*d^2*e+2*b^ 
4*e^3+3*b^3*c*d*e^2-13*b^2*c^2*d^2*e+8*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)^2+4/3*c*d*(b*e-2*c*d)*(8*a*c*e^2 
-b^2*e^2-4*b*c*d*e+4*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)^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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2574 vs. \(2 (653) = 1306\).

Time = 0.18 (sec) , antiderivative size = 2574, normalized size of antiderivative = 3.65 \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="fricas")
 

Output:

2/9*((16*a^2*c^4*d^4 - 32*a^2*b*c^3*d^3*e + (13*a^2*b^2*c^2 + 44*a^3*c^3)* 
d^2*e^2 + (3*a^2*b^3*c - 44*a^3*b*c^2)*d*e^3 + (2*a^2*b^4 - 19*a^3*b^2*c + 
 60*a^4*c^2)*e^4 + (16*c^6*d^4 - 32*b*c^5*d^3*e + (13*b^2*c^4 + 44*a*c^5)* 
d^2*e^2 + (3*b^3*c^3 - 44*a*b*c^4)*d*e^3 + (2*b^4*c^2 - 19*a*b^2*c^3 + 60* 
a^2*c^4)*e^4)*x^4 + 2*(16*b*c^5*d^4 - 32*b^2*c^4*d^3*e + (13*b^3*c^3 + 44* 
a*b*c^4)*d^2*e^2 + (3*b^4*c^2 - 44*a*b^2*c^3)*d*e^3 + (2*b^5*c - 19*a*b^3* 
c^2 + 60*a^2*b*c^3)*e^4)*x^3 + (16*(b^2*c^4 + 2*a*c^5)*d^4 - 32*(b^3*c^3 + 
 2*a*b*c^4)*d^3*e + (13*b^4*c^2 + 70*a*b^2*c^3 + 88*a^2*c^4)*d^2*e^2 + (3* 
b^5*c - 38*a*b^3*c^2 - 88*a^2*b*c^3)*d*e^3 + (2*b^6 - 15*a*b^4*c + 22*a^2* 
b^2*c^2 + 120*a^3*c^3)*e^4)*x^2 + 2*(16*a*b*c^4*d^4 - 32*a*b^2*c^3*d^3*e + 
 (13*a*b^3*c^2 + 44*a^2*b*c^3)*d^2*e^2 + (3*a*b^4*c - 44*a^2*b^2*c^2)*d*e^ 
3 + (2*a*b^5 - 19*a^2*b^3*c + 60*a^3*b*c^2)*e^4)*x)*sqrt(c*e)*weierstrassP 
Inverse(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)) + 6*(8*a^2*c^4*d^3*e - 12*a^2 
*b*c^3*d^2*e^2 + 2*(a^2*b^2*c^2 + 8*a^3*c^3)*d*e^3 + (a^2*b^3*c - 8*a^3*b* 
c^2)*e^4 + (8*c^6*d^3*e - 12*b*c^5*d^2*e^2 + 2*(b^2*c^4 + 8*a*c^5)*d*e^3 + 
 (b^3*c^3 - 8*a*b*c^4)*e^4)*x^4 + 2*(8*b*c^5*d^3*e - 12*b^2*c^4*d^2*e^2 + 
2*(b^3*c^3 + 8*a*b*c^4)*d*e^3 + (b^4*c^2 - 8*a*b^2*c^3)*e^4)*x^3 + (8*(b^2 
*c^4 + 2*a*c^5)*d^3*e - 12*(b^3*c^3 + 2*a*b*c^4)*d^2*e^2 + 2*(b^4*c^2 +...
 

Sympy [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {d + e x} \left (a + b x + c x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(e*x+d)**(1/2)/(c*x**2+b*x+a)**(5/2),x)
 

Output:

Integral(1/(sqrt(d + e*x)*(a + b*x + c*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \] Input:

integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/((c*x^2 + b*x + a)^(5/2)*sqrt(e*x + d)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int { \frac {1}{{\left (c x^{2} + b x + a\right )}^{\frac {5}{2}} \sqrt {e x + d}} \,d x } \] Input:

integrate(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x, algorithm="giac")
 

Output:

integrate(1/((c*x^2 + b*x + a)^(5/2)*sqrt(e*x + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {d+e\,x}\,{\left (c\,x^2+b\,x+a\right )}^{5/2}} \,d x \] Input:

int(1/((d + e*x)^(1/2)*(a + b*x + c*x^2)^(5/2)),x)
                                                                                    
                                                                                    
 

Output:

int(1/((d + e*x)^(1/2)*(a + b*x + c*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt {d+e x} \left (a+b x+c x^2\right )^{5/2}} \, dx=\int \frac {1}{\sqrt {e x +d}\, \left (c \,x^{2}+b x +a \right )^{\frac {5}{2}}}d x \] Input:

int(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x)
 

Output:

int(1/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(5/2),x)