\(\int \frac {\sqrt {d+e x}}{x^2 (a+b x+c x^2)^{3/2}} \, dx\) [87]

Optimal result

Integrand size = 27, antiderivative size = 723 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\frac {2 \left (b^2-2 a c+b c x\right ) \sqrt {d+e x}}{a \left (b^2-4 a c\right ) x \sqrt {a+b x+c x^2}}-\frac {\left (3 b^2-8 a c\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}}{a^2 \left (b^2-4 a c\right ) x}+\frac {\left (3 b^2-8 a c\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {2} a^2 \sqrt {b^2-4 a c} \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {\sqrt {2} \left (3 b^2 d-8 a c d-2 a b e\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a^2 \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {a+b x+c x^2}}+\frac {2 \sqrt {2} \sqrt {b^2-4 a c} (3 b d-a e) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (\frac {2 \sqrt {b^2-4 a c}}{b+\sqrt {b^2-4 a c}},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{a^2 \left (b+\sqrt {b^2-4 a c}\right ) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

2*(b*c*x-2*a*c+b^2)*(e*x+d)^(1/2)/a/(-4*a*c+b^2)/x/(c*x^2+b*x+a)^(1/2)-(-8 
*a*c+3*b^2)*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/a^2/(-4*a*c+b^2)/x+1/2*(-8*a 
*c+3*b^2)*(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))*2^(1/2)/a^2/(-4*a*c+b^2)^(1/2)/(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^(1/ 
2)*(-2*a*b*e-8*a*c*d+3*b^2*d)*(c*(e*x+d)/(2*c*d-(b+(-4*a*c+b^2)^(1/2))*e)) 
^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticF(1/2*(1+(2*c*x+b)/(- 
4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c 
+b^2)^(1/2))*e))^(1/2))/a^2/(-4*a*c+b^2)^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a) 
^(1/2)+2*2^(1/2)*(-4*a*c+b^2)^(1/2)*(-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)*EllipticPi 
(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)/( 
b+(-4*a*c+b^2)^(1/2)),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^2)^(1/2 
))*e))^(1/2))/a^2/(b+(-4*a*c+b^2)^(1/2))/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 35.38 (sec) , antiderivative size = 7762, normalized size of antiderivative = 10.74 \[ \int \frac {\sqrt {d+e x}}{x^2 \left (a+b x+c x^2\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

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

\(\Big \downarrow \) 1292

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1292

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x 
_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Unintegrable[(d + e*x)^m*(f + g*x)^n* 
(a + b*x + c*x^2)^p, x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1343\) vs. \(2(644)=1288\).

Time = 11.75 (sec) , antiderivative size = 1344, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1344\)
risch	\(\text {Expression too large to display}\)	\(1731\)
default	\(\text {Expression too large to display}\)	\(5827\)

Input:

int((e*x+d)^(1/2)/x^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)*(-1/a^2*(c 
*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)/x-2*(c*e*x+c*d)*((2*a*c-b^2) 
/a^2/(4*a*c-b^2)*x+(3*a*c-b^2)*b/a^2/(4*a*c-b^2)/c)/((a/c+b/c*x+x^2)*(c*e* 
x+c*d))^(1/2)+2*(-(4*a*b*c*e+4*a*c^2*d-b^3*e-2*b^2*c*d)/a^2/(4*a*c-b^2)+e* 
(3*a*c-b^2)*b/a^2/(4*a*c-b^2)+2*c*d*(2*a*c-b^2)/a^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*(1/2*c*e/a^2+(2*a* 
c-b^2)*c*e/a^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)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*EllipticE(((x+ 
d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2),((-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))+1/2/c*(-b+(-4*a*c+b 
^2)^(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)))...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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