\(\int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx\) [577]

Optimal result

Integrand size = 31, antiderivative size = 511 \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=-\frac {2 e^2 \sqrt {a+b x+c x^2}}{\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {d+e x}}+\frac {\sqrt {2} \sqrt {b^2-4 a c} e \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (c d^2-b d e+a e^2\right ) (e f-d g) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {a+b x+c x^2}}-\frac {4 \sqrt {2} \sqrt {b^2-4 a c} g \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} \operatorname {EllipticPi}\left (-\frac {2 \sqrt {b^2-4 a c} g}{2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g},\arcsin \left (\frac {\sqrt {1+\frac {b+2 c x}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (e f-d g) \sqrt {d+e x} \sqrt {a+b x+c x^2}} \] Output:

-2*e^2*(c*x^2+b*x+a)^(1/2)/(a*e^2-b*d*e+c*d^2)/(-d*g+e*f)/(e*x+d)^(1/2)+2^ 
(1/2)*(-4*a*c+b^2)^(1/2)*e*(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))/(a*e^2-b*d*e+c*d 
^2)/(-d*g+e*f)/(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*2^(1/2)*(-4*a*c+b^2)^(1/2)*g*(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)*g/(2*c 
*f-(b+(-4*a*c+b^2)^(1/2))*g),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c*d-(b+(-4*a*c+b^ 
2)^(1/2))*e))^(1/2))/(2*c*f-(b+(-4*a*c+b^2)^(1/2))*g)/(-d*g+e*f)/(e*x+d)^( 
1/2)/(c*x^2+b*x+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 28.00 (sec) , antiderivative size = 949, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\frac {2 \left (-\frac {e^2 (a+x (b+c x))}{e f-d g}-\frac {(d+e x)^2 \left (c+\frac {c d^2}{(d+e x)^2}-\frac {b d e}{(d+e x)^2}+\frac {a e^2}{(d+e x)^2}-\frac {2 c d}{d+e x}+\frac {b e}{d+e x}-\frac {i \sqrt {1-\frac {2 \left (c d^2+e (-b d+a e)\right )}{\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \sqrt {2+\frac {4 \left (c d^2+e (-b d+a e)\right )}{\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (d+e x)}} \left (\left (2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (e f-d g) \left (E\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right )|-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )+2 \left (c d^2+e (-b d+a e)\right ) g \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )-2 \left (c d^2+e (-b d+a e)\right ) g \operatorname {EllipticPi}\left (\frac {\left (-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}\right ) (e f-d g)}{2 \left (c d^2+e (-b d+a e)\right ) g},i \text {arcsinh}\left (\frac {\sqrt {2} \sqrt {\frac {c d^2-b d e+a e^2}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}}}{\sqrt {d+e x}}\right ),-\frac {-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt {\left (b^2-4 a c\right ) e^2}}\right )\right )}{4 \sqrt {\frac {c d^2+e (-b d+a e)}{-2 c d+b e+\sqrt {\left (b^2-4 a c\right ) e^2}}} (e f-d g) \sqrt {d+e x}}\right )}{-e f+d g}\right )}{\left (c d^2+e (-b d+a e)\right ) \sqrt {d+e x} \sqrt {a+x (b+c x)}} \] Input:

Integrate[1/((d + e*x)^(3/2)*(f + g*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(2*(-((e^2*(a + x*(b + c*x)))/(e*f - d*g)) - ((d + e*x)^2*(c + (c*d^2)/(d 
+ e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d + e*x) + 
 (b*e)/(d + e*x) - ((I/4)*Sqrt[1 - (2*(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d 
- b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*Sqrt[2 + (4*(c*d^2 + e*(-(b*d 
) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((2*c*d - 
 b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(e*f - d*g)*(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])])/Sq 
rt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sq 
rt[(b^2 - 4*a*c)*e^2]))] - EllipticF[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]))]) + 2*(c*d^2 + e*(-(b*d) + a*e))*g*EllipticF[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]))] - 2*(c*d^2 + e*(-(b*d) + a*e))*g*EllipticPi[((-2*c*d 
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(e*f - d*g))/(2*(c*d^2 + e*(-(b*d) + a*e) 
)*g), 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[(c*d^2 + e*(-(b* 
d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])]*(e*f - d*g)*Sqrt[d...
 

Rubi [A] (warning: unable to verify)

Time = 2.40 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.08, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {1288, 2009}

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.

\(\displaystyle \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 1288

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {2} e \sqrt {b^2-4 a c} \sqrt {d+e x} \sqrt {-\frac {c \left (a+b x+c x^2\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{\sqrt {a+b x+c x^2} (e f-d g) \left (a e^2-b d e+c d^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}}}-\frac {\sqrt {2} g \sqrt {2 c d-e \left (b-\sqrt {b^2-4 a c}\right )} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (b-\sqrt {b^2-4 a c}\right )}} \sqrt {1-\frac {2 c (d+e x)}{2 c d-e \left (\sqrt {b^2-4 a c}+b\right )}} \operatorname {EllipticPi}\left (-\frac {\left (2 c d-b e+\sqrt {b^2-4 a c} e\right ) g}{2 c (e f-d g)},\arcsin \left (\frac {\sqrt {2} \sqrt {c} \sqrt {d+e x}}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e}}\right ),\frac {b-\sqrt {b^2-4 a c}-\frac {2 c d}{e}}{b+\sqrt {b^2-4 a c}-\frac {2 c d}{e}}\right )}{\sqrt {c} \sqrt {a+b x+c x^2} (e f-d g)^2}-\frac {2 e^2 \sqrt {a+b x+c x^2}}{\sqrt {d+e x} (e f-d g) \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[1/((d + e*x)^(3/2)*(f + g*x)*Sqrt[a + b*x + c*x^2]),x]
 

Output:

(-2*e^2*Sqrt[a + b*x + c*x^2])/((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*Sqrt[d 
 + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*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 + S 
qrt[b^2 - 4*a*c])*e)])/((c*d^2 - b*d*e + a*e^2)*(e*f - d*g)*Sqrt[(c*(d + e 
*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) - (Sqrt[2 
]*Sqrt[2*c*d - (b - Sqrt[b^2 - 4*a*c])*e]*g*Sqrt[1 - (2*c*(d + e*x))/(2*c* 
d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*c*(d + e*x))/(2*c*d - (b + Sqr 
t[b^2 - 4*a*c])*e)]*EllipticPi[-1/2*((2*c*d - b*e + Sqrt[b^2 - 4*a*c]*e)*g 
)/(c*(e*f - d*g)), ArcSin[(Sqrt[2]*Sqrt[c]*Sqrt[d + e*x])/Sqrt[2*c*d - (b 
- Sqrt[b^2 - 4*a*c])*e]], (b - Sqrt[b^2 - 4*a*c] - (2*c*d)/e)/(b + Sqrt[b^ 
2 - 4*a*c] - (2*c*d)/e)])/(Sqrt[c]*(e*f - d*g)^2*Sqrt[a + b*x + c*x^2])
 

Defintions of rubi rules used

rule 1288

Int[((f_.) + (g_.)*(x_))^(n_)/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) 
 + (c_.)*(x_)^2]), x_Symbol] :> Int[ExpandIntegrand[1/(Sqrt[f + g*x]*Sqrt[a 
 + b*x + c*x^2]), (f + g*x)^(n + 1/2)/(d + e*x), x], x] /; FreeQ[{a, b, c, 
d, e, f, g}, x] && IntegerQ[n + 1/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1263\) vs. \(2(457)=914\).

Time = 7.96 (sec) , antiderivative size = 1264, normalized size of antiderivative = 2.47

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1264\)
default	\(\text {Expression too large to display}\)	\(4757\)

Input:

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

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right )^{\frac {3}{2}} \left (f + g x\right ) \sqrt {a + b x + c x^{2}}}\, dx \] Input:

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

Output:

Integral(1/((d + e*x)**(3/2)*(f + g*x)*sqrt(a + b*x + c*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + b x + a} {\left (e x + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (f+g\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{(d+e x)^{3/2} (f+g x) \sqrt {a+b x+c x^2}} \, dx=\int \frac {1}{\left (e x +d \right )^{\frac {3}{2}} \left (g x +f \right ) \sqrt {c \,x^{2}+b x +a}}d x \] Input:

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

Output:

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