\(\int \frac {\sqrt {d+e x}}{(a+b x+c x^2)^3} \, dx\) [548]

Optimal result

Integrand size = 22, antiderivative size = 634 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}-\frac {\sqrt {d+e x} \left (13 b^2 c d e-4 a c^2 d e-b^3 e^2-4 b c \left (3 c d^2+2 a e^2\right )-c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x\right )}{4 \left (b^2-4 a c\right )^2 \left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}-\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d-3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {c} \left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \left (c d^2-b d e+a e^2\right )} \] Output:

-1/2*(2*c*x+b)*(e*x+d)^(1/2)/(-4*a*c+b^2)/(c*x^2+b*x+a)^2-1/4*(e*x+d)^(1/2 
)*(13*b^2*c*d*e-4*a*c^2*d*e-b^3*e^2-4*b*c*(2*a*e^2+3*c*d^2)-c*(24*c^2*d^2+ 
b^2*e^2-4*c*e*(-5*a*e+6*b*d))*x)/(-4*a*c+b^2)^2/(a*e^2-b*d*e+c*d^2)/(c*x^2 
+b*x+a)-1/8*c^(1/2)*(96*c^3*d^3+b^2*(b+(-4*a*c+b^2)^(1/2))*e^3-8*c^2*d*e*( 
18*b*d-3*(-4*a*c+b^2)^(1/2)*d-13*a*e)+2*c*e^2*(23*b^2*d+10*a*(-4*a*c+b^2)^ 
(1/2)*e-2*b*(6*(-4*a*c+b^2)^(1/2)*d+13*a*e)))*arctanh(2^(1/2)*c^(1/2)*(e*x 
+d)^(1/2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2))*2^(1/2)/(-4*a*c+b^2)^(5/ 
2)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)^(1/2)/(a*e^2-b*d*e+c*d^2)+1/8*c^(1/2)* 
(96*c^3*d^3+b^2*(b-(-4*a*c+b^2)^(1/2))*e^3-8*c^2*d*e*(18*b*d+3*(-4*a*c+b^2 
)^(1/2)*d-13*a*e)+2*c*e^2*(23*b^2*d+12*b*(-4*a*c+b^2)^(1/2)*d-26*a*b*e-10* 
a*(-4*a*c+b^2)^(1/2)*e))*arctanh(2^(1/2)*c^(1/2)*(e*x+d)^(1/2)/(2*c*d-(b+( 
-4*a*c+b^2)^(1/2))*e)^(1/2))*2^(1/2)/(-4*a*c+b^2)^(5/2)/(2*c*d-(b+(-4*a*c+ 
b^2)^(1/2))*e)^(1/2)/(a*e^2-b*d*e+c*d^2)
 

Mathematica [A] (verified)

Time = 12.53 (sec) , antiderivative size = 580, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) (a+x (b+c x))^2}-\frac {\sqrt {d+e x} \left (b^3 e^2+b^2 c e (-13 d+e x)+4 b c \left (2 a e^2+3 c d (d-2 e x)\right )+4 c^2 \left (6 c d^2 x+a e (d+5 e x)\right )\right )}{4 \left (b^2-4 a c\right )^2 \left (-c d^2+e (b d-a e)\right ) (a+x (b+c x))}-\frac {\sqrt {c} \left (\frac {\left (96 c^3 d^3+b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^3+8 c^2 d e \left (-18 b d+3 \sqrt {b^2-4 a c} d+13 a e\right )+2 c e^2 \left (23 b^2 d+10 a \sqrt {b^2-4 a c} e-2 b \left (6 \sqrt {b^2-4 a c} d+13 a e\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e}}\right )}{\sqrt {2 c d+\left (-b+\sqrt {b^2-4 a c}\right ) e}}-\frac {\left (96 c^3 d^3+b^2 \left (b-\sqrt {b^2-4 a c}\right ) e^3-8 c^2 d e \left (18 b d+3 \sqrt {b^2-4 a c} d-13 a e\right )+2 c e^2 \left (23 b^2 d+12 b \sqrt {b^2-4 a c} d-26 a b e-10 a \sqrt {b^2-4 a c} e\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}}\right )}{4 \sqrt {2} \left (b^2-4 a c\right )^{5/2} \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + x*(b + c*x))^2) - (Sq 
rt[d + e*x]*(b^3*e^2 + b^2*c*e*(-13*d + e*x) + 4*b*c*(2*a*e^2 + 3*c*d*(d - 
 2*e*x)) + 4*c^2*(6*c*d^2*x + a*e*(d + 5*e*x))))/(4*(b^2 - 4*a*c)^2*(-(c*d 
^2) + e*(b*d - a*e))*(a + x*(b + c*x))) - (Sqrt[c]*(((96*c^3*d^3 + b^2*(b 
+ Sqrt[b^2 - 4*a*c])*e^3 + 8*c^2*d*e*(-18*b*d + 3*Sqrt[b^2 - 4*a*c]*d + 13 
*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[b^2 - 4 
*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - b 
*e + Sqrt[b^2 - 4*a*c]*e]])/Sqrt[2*c*d + (-b + Sqrt[b^2 - 4*a*c])*e] - ((9 
6*c^3*d^3 + b^2*(b - Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d + 3*Sqrt[b 
^2 - 4*a*c]*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 12*b*Sqrt[b^2 - 4*a*c]*d - 2 
6*a*b*e - 10*a*Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x 
])/Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/Sqrt[2*c*d - (b + Sqrt[b^2 - 
4*a*c])*e]))/(4*Sqrt[2]*(b^2 - 4*a*c)^(5/2)*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 1.32 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.95, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {1163, 27, 1235, 27, 1197, 27, 1480, 221}

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.

\(\displaystyle \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx\)

\(\Big \downarrow \) 1163

\(\displaystyle \frac {\int -\frac {12 c d-b e+10 c e x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{2 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {12 c d-b e+10 c e x}{\sqrt {d+e x} \left (c x^2+b x+a\right )^2}dx}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1235

\(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {48 c^3 d^3-4 c^2 e (15 b d-13 a e) d+b^3 e^3+b c e^2 (11 b d-16 a e)+c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x}{2 \sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {48 c^3 d^3-4 c^2 e (15 b d-13 a e) d+b^3 e^3+b c e^2 (11 b d-16 a e)+c e \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) x}{\sqrt {d+e x} \left (c x^2+b x+a\right )}dx}{2 \left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1197

\(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int \frac {e \left ((2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )+c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) (d+e x)\right )}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e \int \frac {(2 c d-b e) \left (12 c^2 d^2-b^2 e^2-4 c e (3 b d-4 a e)\right )+c \left (24 c^2 d^2+b^2 e^2-4 c e (6 b d-5 a e)\right ) (d+e x)}{c d^2-b e d+a e^2+c (d+e x)^2-(2 c d-b e) (d+e x)}d\sqrt {d+e x}}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 1480

\(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e \left (\frac {1}{2} c \left (-\frac {(2 c d-b e) \left (-4 c e (12 b d-13 a e)-b^2 e^2+48 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right ) \int \frac {1}{\frac {1}{2} \left (\left (b+\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}+\frac {c \left (-8 c^2 d e \left (-3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt {b^2-4 a c}+13 a e\right )+10 a e \sqrt {b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+96 c^3 d^3\right ) \int \frac {1}{\frac {1}{2} \left (\left (b-\sqrt {b^2-4 a c}\right ) e-2 c d\right )+c (d+e x)}d\sqrt {d+e x}}{2 e \sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {\sqrt {d+e x} \left (-c x \left (-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right )-4 b c \left (2 a e^2+3 c d^2\right )-4 a c^2 d e+b^3 \left (-e^2\right )+13 b^2 c d e\right )}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {e \left (-\frac {\sqrt {c} \left (-\frac {(2 c d-b e) \left (-4 c e (12 b d-13 a e)-b^2 e^2+48 c^2 d^2\right )}{e \sqrt {b^2-4 a c}}-4 c e (6 b d-5 a e)+b^2 e^2+24 c^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}\right )}{\sqrt {2} \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}-\frac {\sqrt {c} \left (-8 c^2 d e \left (-3 d \sqrt {b^2-4 a c}-13 a e+18 b d\right )+2 c e^2 \left (-2 b \left (6 d \sqrt {b^2-4 a c}+13 a e\right )+10 a e \sqrt {b^2-4 a c}+23 b^2 d\right )+b^2 e^3 \left (\sqrt {b^2-4 a c}+b\right )+96 c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}}\right )}{\left (b^2-4 a c\right ) \left (a e^2-b d e+c d^2\right )}}{4 \left (b^2-4 a c\right )}-\frac {(b+2 c x) \sqrt {d+e x}}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^2}\)

Input:

Int[Sqrt[d + e*x]/(a + b*x + c*x^2)^3,x]
 

Output:

-1/2*((b + 2*c*x)*Sqrt[d + e*x])/((b^2 - 4*a*c)*(a + b*x + c*x^2)^2) - ((S 
qrt[d + e*x]*(13*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 - 4*b*c*(3*c*d^2 + 2*a* 
e^2) - c*(24*c^2*d^2 + b^2*e^2 - 4*c*e*(6*b*d - 5*a*e))*x))/((b^2 - 4*a*c) 
*(c*d^2 - b*d*e + a*e^2)*(a + b*x + c*x^2)) - (e*(-((Sqrt[c]*(96*c^3*d^3 + 
 b^2*(b + Sqrt[b^2 - 4*a*c])*e^3 - 8*c^2*d*e*(18*b*d - 3*Sqrt[b^2 - 4*a*c] 
*d - 13*a*e) + 2*c*e^2*(23*b^2*d + 10*a*Sqrt[b^2 - 4*a*c]*e - 2*b*(6*Sqrt[ 
b^2 - 4*a*c]*d + 13*a*e)))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2* 
c*d - (b - Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*e*Sqrt[2*c*d 
 - (b - Sqrt[b^2 - 4*a*c])*e])) - (Sqrt[c]*(24*c^2*d^2 + b^2*e^2 - 4*c*e*( 
6*b*d - 5*a*e) - ((2*c*d - b*e)*(48*c^2*d^2 - b^2*e^2 - 4*c*e*(12*b*d - 13 
*a*e)))/(Sqrt[b^2 - 4*a*c]*e))*ArcTanh[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqr 
t[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]])/(Sqrt[2]*Sqrt[2*c*d - (b + Sqrt[b^2 
 - 4*a*c])*e])))/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)))/(4*(b^2 - 4*a*c) 
)
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 1163

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]
 

rule 1197

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)), x_Symbol] :> Simp[2   Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - 
b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; Fr 
eeQ[{a, b, c, d, e, f, g}, x]
 

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 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q))   Int[1/(b/2 
+ q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] 
 && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
 

Maple [A] (verified)

Time = 3.42 (sec) , antiderivative size = 778, normalized size of antiderivative = 1.23

method	result	size
pseudoelliptic	\(-\frac {10 e^{4} \left (-\left (\left (\frac {6 c^{2} d^{2}}{5}+e \left (a e -\frac {6 b d}{5}\right ) c +\frac {b^{2} e^{2}}{20}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-\frac {13 \left (b e -2 c d \right ) \left (\frac {12 c^{2} d^{2}}{13}+e \left (a e -\frac {12 b d}{13}\right ) c -\frac {b^{2} e^{2}}{52}\right )}{5}\right ) e \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} c \,\operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \left (\left (\left (\frac {6 c^{2} d^{2}}{5}+e \left (a e -\frac {6 b d}{5}\right ) c +\frac {b^{2} e^{2}}{20}\right ) \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+\frac {13 \left (b e -2 c d \right ) \left (\frac {12 c^{2} d^{2}}{13}+e \left (a e -\frac {12 b d}{13}\right ) c -\frac {b^{2} e^{2}}{52}\right )}{5}\right ) e \sqrt {2}\, \left (c \,x^{2}+b x +a \right )^{2} c \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}}\right )+\frac {8 \sqrt {e x +d}\, \sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (\frac {3 c^{4} d^{2} x^{3}}{2}+\frac {5 \left (\frac {a \,e^{2} x^{2}}{2}+\frac {d x \left (-6 b x +a \right ) e}{10}+d^{2} \left (\frac {9 b x}{10}+a \right )\right ) x \,c^{3}}{2}+\frac {\left (\left (\frac {1}{4} b^{2} x^{3}+7 a b \,x^{2}+9 a^{2} x \right ) e^{2}+d \left (-\frac {37}{4} b^{2} x^{2}-9 a b x +a^{2}\right ) e +5 \left (\frac {2 b x}{5}+a \right ) d^{2} b \right ) c^{2}}{4}+\left (\left (\frac {1}{8} b^{2} x^{2}+\frac {5}{16} a b x +a^{2}\right ) e^{2}-\frac {21 d \left (\frac {3 b x}{7}+a \right ) b e}{16}-\frac {b^{2} d^{2}}{8}\right ) b c -\frac {e \,b^{3} \left (\left (-b x +a \right ) e -2 b d \right )}{16}\right )}{5}\right )\right ) c^{3}}{\sqrt {\left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {\left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) c}\, \sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\, \left (-b e +2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \left (\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}+e \left (2 c x +b \right )\right )^{2} \left (b e -2 c d +\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}\right ) \left (\sqrt {-4 e^{2} \left (a c -\frac {b^{2}}{4}\right )}-e \left (2 c x +b \right )\right )^{2} \left (a c -\frac {b^{2}}{4}\right )^{2}}\)	\(778\)
derivativedivides	\(128 e^{5} c^{3} \left (-\frac {\frac {-\frac {\left (-6 b e +12 c d -5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}+\frac {\left (-6 b e +12 c d -7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{{\left (-e x -\frac {b e}{2 c}-\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}^{2}}-\frac {\left (-20 a c \,e^{2}+17 b^{2} e^{2}-48 b c d e +48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (4 b e -8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{4} \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {\frac {\left (-6 b e +12 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}-\frac {\left (-6 b e +12 c d +7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{{\left (-e x -\frac {b e}{2 c}+\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}^{2}}+\frac {\left (20 a c \,e^{2}-17 b^{2} e^{2}+48 b c d e -48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-4 b e +8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{4} \left (4 a c -b^{2}\right )^{2}}\right )\)	\(800\)
default	\(128 e^{5} c^{3} \left (-\frac {\frac {-\frac {\left (-6 b e +12 c d -5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d -\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}+\frac {\left (-6 b e +12 c d -7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{{\left (-e x -\frac {b e}{2 c}-\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}^{2}}-\frac {\left (-20 a c \,e^{2}+17 b^{2} e^{2}-48 b c d e +48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (4 b e -8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (b e -2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{4} \left (4 a c -b^{2}\right )^{2}}+\frac {\frac {\frac {\left (-6 b e +12 c d +5 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \left (e x +d \right )^{\frac {3}{2}}}{16 c \left (-b e +2 c d +\sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right )}-\frac {\left (-6 b e +12 c d +7 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, \sqrt {e x +d}}{32 c^{2}}}{{\left (-e x -\frac {b e}{2 c}+\frac {\sqrt {\left (-4 a c +b^{2}\right ) e^{2}}}{2 c}\right )}^{2}}+\frac {\left (20 a c \,e^{2}-17 b^{2} e^{2}+48 b c d e -48 c^{2} d^{2}+18 b e \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}-36 d \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\, c \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}\, c \sqrt {2}}{\sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}\right )}{4 \left (-4 b e +8 c d +4 \sqrt {-4 a c \,e^{2}+b^{2} e^{2}}\right ) \sqrt {\left (-b e +2 c d +\sqrt {-e^{2} \left (4 a c -b^{2}\right )}\right ) c}}}{16 c \sqrt {-e^{2} \left (4 a c -b^{2}\right )}\, e^{4} \left (4 a c -b^{2}\right )^{2}}\right )\)	\(800\)

Input:

int((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x,method=_RETURNVERBOSE)
 

Output:

-10*e^4/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)/((-b*e+2*c*d+(- 
4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-((6/5*c^2*d^2+e*(a*e-6/5*b*d)*c+1/2 
0*b^2*e^2)*(-4*e^2*(a*c-1/4*b^2))^(1/2)-13/5*(b*e-2*c*d)*(12/13*c^2*d^2+e* 
(a*e-12/13*b*d)*c-1/52*b^2*e^2))*e*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2 
))*c)^(1/2)*2^(1/2)*(c*x^2+b*x+a)^2*c*arctanh((e*x+d)^(1/2)*c*2^(1/2)/((-b 
*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+((-b*e+2*c*d+(-4*e^2*(a*c 
-1/4*b^2))^(1/2))*c)^(1/2)*(((6/5*c^2*d^2+e*(a*e-6/5*b*d)*c+1/20*b^2*e^2)* 
(-4*e^2*(a*c-1/4*b^2))^(1/2)+13/5*(b*e-2*c*d)*(12/13*c^2*d^2+e*(a*e-12/13* 
b*d)*c-1/52*b^2*e^2))*e*2^(1/2)*(c*x^2+b*x+a)^2*c*arctan((e*x+d)^(1/2)*c*2 
^(1/2)/((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2))+8/5*(e*x+d)^(1/ 
2)*((b*e-2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2))*c)^(1/2)*(-4*e^2*(a*c-1/4*b^2 
))^(1/2)*(3/2*c^4*d^2*x^3+5/2*(1/2*a*e^2*x^2+1/10*d*x*(-6*b*x+a)*e+d^2*(9/ 
10*b*x+a))*x*c^3+1/4*((1/4*b^2*x^3+7*a*b*x^2+9*a^2*x)*e^2+d*(-37/4*b^2*x^2 
-9*a*b*x+a^2)*e+5*(2/5*b*x+a)*d^2*b)*c^2+((1/8*b^2*x^2+5/16*a*b*x+a^2)*e^2 
-21/16*d*(3/7*b*x+a)*b*e-1/8*b^2*d^2)*b*c-1/16*e*b^3*((-b*x+a)*e-2*b*d)))) 
*c^3/(-4*e^2*(a*c-1/4*b^2))^(1/2)/(-b*e+2*c*d+(-4*e^2*(a*c-1/4*b^2))^(1/2) 
)/((-4*e^2*(a*c-1/4*b^2))^(1/2)+e*(2*c*x+b))^2/(b*e-2*c*d+(-4*e^2*(a*c-1/4 
*b^2))^(1/2))/((-4*e^2*(a*c-1/4*b^2))^(1/2)-e*(2*c*x+b))^2/(a*c-1/4*b^2)^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30962 vs. \(2 (566) = 1132\).

Time = 19.32 (sec) , antiderivative size = 30962, normalized size of antiderivative = 48.84 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Timed out} \] Input:

integrate((e*x+d)**(1/2)/(c*x**2+b*x+a)**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=\int { \frac {\sqrt {e x + d}}{{\left (c x^{2} + b x + a\right )}^{3}} \,d x } \] Input:

integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="maxima")
 

Output:

integrate(sqrt(e*x + d)/(c*x^2 + b*x + a)^3, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5171 vs. \(2 (566) = 1132\).

Time = 1.44 (sec) , antiderivative size = 5171, normalized size of antiderivative = 8.16 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

integrate((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x, algorithm="giac")
 

Output:

1/32*((b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2*c^3*d^2*e - b^5*d*e^2 + 8* 
a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 8*a^2*b^2*c*e^3 + 16*a^3* 
c^2*e^3)^2*(24*c^2*d^2*e - 24*b*c*d*e^2 + (b^2 + 20*a*c)*e^3)*sqrt(-4*c^2* 
d + 2*(b*c - sqrt(b^2 - 4*a*c)*c)*e) + 2*(24*(b^2*c^4 - 4*a*c^5)*sqrt(b^2 
- 4*a*c)*d^5*e - 60*(b^3*c^3 - 4*a*b*c^4)*sqrt(b^2 - 4*a*c)*d^4*e^2 + 2*(2 
3*b^4*c^2 - 64*a*b^2*c^3 - 112*a^2*c^4)*sqrt(b^2 - 4*a*c)*d^3*e^3 - 3*(3*b 
^5*c + 16*a*b^3*c^2 - 112*a^2*b*c^3)*sqrt(b^2 - 4*a*c)*d^2*e^4 - (b^6 - 30 
*a*b^4*c + 72*a^2*b^2*c^2 + 128*a^3*c^3)*sqrt(b^2 - 4*a*c)*d*e^5 + (a*b^5 
- 20*a^2*b^3*c + 64*a^3*b*c^2)*sqrt(b^2 - 4*a*c)*e^6)*sqrt(-4*c^2*d + 2*(b 
*c - sqrt(b^2 - 4*a*c)*c)*e)*abs(b^4*c*d^2*e - 8*a*b^2*c^2*d^2*e + 16*a^2* 
c^3*d^2*e - b^5*d*e^2 + 8*a*b^3*c*d*e^2 - 16*a^2*b*c^2*d*e^2 + a*b^4*e^3 - 
 8*a^2*b^2*c*e^3 + 16*a^3*c^2*e^3) - (192*(b^6*c^6 - 12*a*b^4*c^7 + 48*a^2 
*b^2*c^8 - 64*a^3*c^9)*d^8*e - 768*(b^7*c^5 - 12*a*b^5*c^6 + 48*a^2*b^3*c^ 
7 - 64*a^3*b*c^8)*d^7*e^2 + 4*(299*b^8*c^4 - 3440*a*b^6*c^5 + 12576*a^2*b^ 
4*c^6 - 12032*a^3*b^2*c^7 - 9472*a^4*c^8)*d^6*e^3 - 12*(75*b^9*c^3 - 752*a 
*b^7*c^4 + 1824*a^2*b^5*c^5 + 2304*a^3*b^3*c^6 - 9472*a^4*b*c^7)*d^5*e^4 + 
 (323*b^10*c^2 - 1960*a*b^8*c^3 - 6880*a^2*b^6*c^4 + 64000*a^3*b^4*c^5 - 9 
3440*a^4*b^2*c^6 - 38912*a^5*c^7)*d^4*e^5 - 2*(21*b^11*c + 184*a*b^9*c^2 - 
 3616*a^2*b^7*c^3 + 12288*a^3*b^5*c^4 + 1280*a^4*b^3*c^5 - 38912*a^5*b*c^6 
)*d^3*e^6 - (b^12 - 150*a*b^10*c + 948*a^2*b^8*c^2 + 2176*a^3*b^6*c^3 -...
 

Mupad [B] (verification not implemented)

Time = 51.39 (sec) , antiderivative size = 23750, normalized size of antiderivative = 37.46 \[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=\text {Too large to display} \] Input:

int((d + e*x)^(1/2)/(a + b*x + c*x^2)^3,x)
 

Output:

log(- (2^(1/2)*((2^(1/2)*((c^2*e^3*(b*e - 2*c*d)*(b^2*e^2 - 12*c^2*d^2 - 1 
6*a*c*e^2 + 12*b*c*d*e))/((4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) - (2^(1/2 
)*c^2*e^2*(4*a*c - b^2)*(b*e - 2*c*d)*(d + e*x)^(1/2)*(-(b^17*e^7 + 471859 
2*a^5*c^12*d^7 - 4608*b^10*c^7*d^7 + b^2*e^7*(-(4*a*c - b^2)^15)^(1/2) + 9 
2160*a*b^8*c^8*d^7 - 1720320*a^8*b*c^8*e^7 + 3440640*a^8*c^9*d*e^6 + 16128 
*b^11*c^6*d^6*e - 737280*a^2*b^6*c^9*d^7 + 2949120*a^3*b^4*c^10*d^7 - 5898 
240*a^4*b^2*c^11*d^7 + 1140*a^2*b^13*c^2*e^7 - 10160*a^3*b^11*c^3*e^7 + 34 
880*a^4*b^9*c^4*e^7 + 43776*a^5*b^7*c^5*e^7 - 680960*a^6*b^5*c^6*e^7 + 186 
3680*a^7*b^3*c^7*e^7 + 13762560*a^6*c^11*d^5*e^2 + 12615680*a^7*c^10*d^3*e 
^4 - 20832*b^12*c^5*d^5*e^2 + 11760*b^13*c^4*d^4*e^3 - 2450*b^14*c^3*d^3*e 
^4 - 21*b^15*c^2*d^2*e^5 - 21*c^2*d^2*e^5*(-(4*a*c - b^2)^15)^(1/2) - 55*a 
*b^15*c*e^7 - 25*a*c*e^7*(-(4*a*c - b^2)^15)^(1/2) + 21*b^16*c*d*e^6 + 21* 
b*c*d*e^6*(-(4*a*c - b^2)^15)^(1/2) - 3064320*a^2*b^8*c^7*d^5*e^2 + 120960 
0*a^2*b^9*c^6*d^4*e^3 + 144480*a^2*b^10*c^5*d^3*e^4 - 136080*a^2*b^11*c^4* 
d^2*e^5 + 11182080*a^3*b^6*c^8*d^5*e^2 - 2150400*a^3*b^7*c^7*d^4*e^3 - 257 
6000*a^3*b^8*c^6*d^3*e^4 + 853440*a^3*b^9*c^5*d^2*e^5 - 18063360*a^4*b^4*c 
^9*d^5*e^2 - 6451200*a^4*b^5*c^8*d^4*e^3 + 12454400*a^4*b^6*c^7*d^3*e^4 - 
1908480*a^4*b^7*c^6*d^2*e^5 + 4128768*a^5*b^2*c^10*d^5*e^2 + 30965760*a^5* 
b^3*c^9*d^4*e^3 - 24729600*a^5*b^4*c^8*d^3*e^4 - 2128896*a^5*b^5*c^7*d^2*e 
^5 + 12328960*a^6*b^2*c^9*d^3*e^4 + 15912960*a^6*b^3*c^8*d^2*e^5 - 3225...
 

Reduce [F]

\[ \int \frac {\sqrt {d+e x}}{\left (a+b x+c x^2\right )^3} \, dx=\int \frac {\sqrt {e x +d}}{\left (c \,x^{2}+b x +a \right )^{3}}d x \] Input:

int((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x)
 

Output:

int((e*x+d)^(1/2)/(c*x^2+b*x+a)^3,x)