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

Optimal result

Integrand size = 43, antiderivative size = 1585 \[ \int \frac {\sqrt {f+g x} \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:

(3*c*C*d^2-C*e*(-a*e+b*d)-2*c*e*(-A*e+B*d))*g*(e*x+d)^(1/2)*(c*x^2+b*x+a)^ 
(1/2)/c/e^2/(a*e^2-b*d*e+c*d^2)/(g*x+f)^(1/2)+C*(g*x+f)^(1/2)*(c*x^2+b*x+a 
)^(1/2)/c/e/(e*x+d)^(1/2)-(3*c*C*d^2-C*e*(-a*e+b*d)-2*c*e*(-A*e+B*d))*(g*x 
+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/c/e/(a*e^2-b*d*e+c*d^2)/(e*x+d)^(1/2)+(-4*a* 
c+b^2)^(1/4)*(3*c*C*d^2-C*e*(-a*e+b*d)-2*c*e*(-A*e+B*d))*(c*f^2-g*(-a*g+b* 
f))*(b-(-4*a*c+b^2)^(1/2)+2*c*x)^(1/2)*(e*x+d)^(1/2)*((2*c*f-(b-(-4*a*c+b^ 
2)^(1/2))*g)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/c/(-4*a*c+b^2)^(1/2)/(g*x+f))^(1 
/2)*EllipticE(1/2*(-2*c*f+(b+(-4*a*c+b^2)^(1/2))*g)^(1/2)*(b-(-4*a*c+b^2)^ 
(1/2)+2*c*x)^(1/2)/c^(1/2)/(-4*a*c+b^2)^(1/4)/(g*x+f)^(1/2),2*(c*(-4*a*c+b 
^2)^(1/2)*(-d*g+e*f)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(2*c*f-(b+(-4*a*c+b^ 
2)^(1/2))*g))^(1/2))/c^(1/2)/e^2/(a*e^2-b*d*e+c*d^2)/(2*c*f-(b-(-4*a*c+b^2 
)^(1/2))*g)/(-2*c*f+(b+(-4*a*c+b^2)^(1/2))*g)^(1/2)/((2*c*f-(b-(-4*a*c+b^2 
)^(1/2))*g)*(e*x+d)/(2*c*d-(b-(-4*a*c+b^2)^(1/2))*e)/(g*x+f))^(1/2)/(c*x^2 
+b*x+a)^(1/2)+1/2*(C*e*(-a*e+b*d)*g-c*C*d*(-3*d*g+4*e*f)+2*B*c*e*(-d*g+e*f 
))*(b-(-4*a*c+b^2)^(1/2)+2*c*x)^(1/2)*(((b+(-4*a*c+b^2)^(1/2))*d-2*a*e+(2* 
c*d-(b-(-4*a*c+b^2)^(1/2))*e)*x)/(-4*a*c+b^2)^(1/2)/(e*x+d))^(1/2)*(g*x+f) 
^(1/2)*EllipticF((-d*g+e*f)^(1/2)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)^(1/2)/(2*c* 
f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(e*x+d)^(1/2),1/2*(-2*(2*c*d*f+2*a*e*g-( 
-4*a*c+b^2)^(1/2)*(-d*g+e*f)-b*(d*g+e*f))/(-4*a*c+b^2)^(1/2)/(-d*g+e*f))^( 
1/2))*2^(1/2)/c/e^3/(2*c*f-(b-(-4*a*c+b^2)^(1/2))*g)^(1/2)/(-d*g+e*f)^(...
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(19147\) vs. \(2(1585)=3170\).

Time = 36.28 (sec) , antiderivative size = 19147, normalized size of antiderivative = 12.08 \[ \int \frac {\sqrt {f+g x} \left (A+B x+C x^2\right )}{(d+e x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(Sqrt[f + g*x]*(A + B*x + C*x^2))/((d + e*x)^(3/2)*Sqrt[a + b*x 
+ c*x^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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1292

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1276

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx+\frac {C \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c x^2+b x+a}}dx}{e^2}-\frac {\sqrt {2} (f+g x) \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} (2 C d-B e) \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \sqrt {-\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(f+g x) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}} \sqrt {-\frac {\left (x \left (\sqrt {b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(f+g x) \left (d \sqrt {b^2-4 a c}-2 a e+b d\right )}} \operatorname {EllipticPi}\left (\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g}{e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )},\arcsin \left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right ),\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}{\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}\right )}{e^3 \sqrt {\frac {2 a c}{\sqrt {b^2-4 a c}+b}+c x} \sqrt {a+b x+c x^2} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}\)

\(\Big \downarrow \) 1292

\(\displaystyle \left (A+\frac {d (C d-B e)}{e^2}\right ) \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx+\frac {C \int \frac {\sqrt {d+e x} \sqrt {f+g x}}{\sqrt {c x^2+b x+a}}dx}{e^2}-\frac {\sqrt {2} (f+g x) \sqrt {-\sqrt {b^2-4 a c}+b+2 c x} (2 C d-B e) \sqrt {2 c d-e \left (\sqrt {b^2-4 a c}+b\right )} \sqrt {-\frac {\left (\sqrt {b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(f+g x) \left (2 c d-e \left (\sqrt {b^2-4 a c}+b\right )\right )}} \sqrt {-\frac {\left (x \left (\sqrt {b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(f+g x) \left (d \sqrt {b^2-4 a c}-2 a e+b d\right )}} \operatorname {EllipticPi}\left (\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) g}{e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )},\arcsin \left (\frac {\sqrt {2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}{\sqrt {2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e} \sqrt {f+g x}}\right ),\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (b f+\sqrt {b^2-4 a c} f-2 a g\right )}{\left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}\right )}{e^3 \sqrt {\frac {2 a c}{\sqrt {b^2-4 a c}+b}+c x} \sqrt {a+b x+c x^2} \sqrt {2 c f-g \left (\sqrt {b^2-4 a c}+b\right )}}\)

Input:

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

Output:

$Aborted
 

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 1276

Int[Sqrt[(d_.) + (e_.)*(x_)]/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*( 
x_) + (c_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[Sqrt 
[2]*Sqrt[2*c*f - g*(b + q)]*Sqrt[b - q + 2*c*x]*(d + e*x)*Sqrt[(e*f - d*g)* 
((b + q + 2*c*x)/((2*c*f - g*(b + q))*(d + e*x)))]*(Sqrt[(e*f - d*g)*((2*a 
+ (b + q)*x)/((b*f + q*f - 2*a*g)*(d + e*x)))]/(g*Sqrt[2*c*d - e*(b + q)]*S 
qrt[2*a*(c/(b + q)) + c*x]*Sqrt[a + b*x + c*x^2]))*EllipticPi[e*((2*c*f - g 
*(b + q))/(g*(2*c*d - e*(b + q)))), ArcSin[Sqrt[2*c*d - e*(b + q)]*(Sqrt[f 
+ g*x]/(Sqrt[2*c*f - g*(b + q)]*Sqrt[d + e*x]))], (b*d + q*d - 2*a*e)*((2*c 
*f - g*(b + q))/((b*f + q*f - 2*a*g)*(2*c*d - e*(b + q))))], x]] /; FreeQ[{ 
a, b, c, d, e, f, g}, x]
 

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]
 

rule 2154

Int[(Px_)*((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b 
_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[PolynomialQuotient[Px, d + 
 e*x, x]*(d + e*x)^(m + 1)*(f + g*x)^n*(a + b*x + c*x^2)^p, x] + Simp[Polyn 
omialRemainder[Px, d + e*x, x]   Int[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x 
^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && PolynomialQ[Px, x 
] && LtQ[m, 0] &&  !IntegerQ[n] && IntegersQ[2*m, 2*n, 2*p]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3090\) vs. \(2(1426)=2852\).

Time = 9.78 (sec) , antiderivative size = 3091, normalized size of antiderivative = 1.95

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3091\)
default	\(\text {Expression too large to display}\)	\(261664\)

Input:

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

Output:

((g*x+f)*(c*x^2+b*x+a)*(e*x+d))^(1/2)/(e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b 
*x+a)^(1/2)*(-2*(c*e*g*x^3+b*e*g*x^2+c*e*f*x^2+a*e*g*x+b*e*f*x+a*e*f)*(A*e 
^2-B*d*e+C*d^2)/e^2/(a*e^2-b*d*e+c*d^2)/((x+d/e)*(c*e*g*x^3+b*e*g*x^2+c*e* 
f*x^2+a*e*g*x+b*e*f*x+a*e*f))^(1/2)+2*((A*e^2*g-B*d*e*g+B*e^2*f+C*d^2*g-C* 
d*e*f)/e^3-(A*e^2-B*d*e+C*d^2)/e^3*(a*e^2*g-b*d*e*g+b*e^2*f+c*d^2*g-c*d*e* 
f)/(a*e^2-b*d*e+c*d^2)+(a*e*g+b*e*f)*(A*e^2-B*d*e+C*d^2)/e^2/(a*e^2-b*d*e+ 
c*d^2))*(-f/g+d/e)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+f/g)/(f/g-d/e) 
/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)) 
)^2*((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)*(x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/ 
(-1/2*(b+(-4*a*c+b^2)^(1/2))/c+f/g)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/ 
2)*((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+f/g)*(x+d/e)/(f/g-d/e)/(x-1/2/c*(-b+(-4 
*a*c+b^2)^(1/2))))^(1/2)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))/(1/2/c*(-b+( 
-4*a*c+b^2)^(1/2))+f/g)/(c*e*g*(x+f/g)*(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*( 
x+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(x+d/e))^(1/2)*EllipticF(((-d/e-1/2/c*(-b+ 
(-4*a*c+b^2)^(1/2)))*(x+f/g)/(f/g-d/e)/(x-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^ 
(1/2),((1/2/c*(-b+(-4*a*c+b^2)^(1/2))+1/2*(b+(-4*a*c+b^2)^(1/2))/c)*(-f/g+ 
d/e)/(1/2*(b+(-4*a*c+b^2)^(1/2))/c-f/g)/(d/e+1/2/c*(-b+(-4*a*c+b^2)^(1/2)) 
))^(1/2))+2*(1/e^2*(B*e*g-C*d*g+C*e*f)-(A*e^2-B*d*e+C*d^2)/e^2*(b*e*g-c*d* 
g+c*e*f)/(a*e^2-b*d*e+c*d^2)+(2*b*e*g+2*c*e*f)*(A*e^2-B*d*e+C*d^2)/e^2/(a* 
e^2-b*d*e+c*d^2))*(-f/g+d/e)*((-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2)))*(x+f...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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