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

Optimal result

Integrand size = 38, antiderivative size = 1474 \[ \int \frac {(A+B x) \sqrt {f+g x}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(13559\) vs. \(2(1474)=2948\).

Time = 28.35 (sec) , antiderivative size = 13559, normalized size of antiderivative = 9.20 \[ \int \frac {(A+B x) \sqrt {f+g x}}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[((A + B*x)*Sqrt[f + g*x])/(Sqrt[d + e*x]*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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1276

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

\(\Big \downarrow \) 1292

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

Input:

Int[((A + B*x)*Sqrt[f + g*x])/(Sqrt[d + e*x]*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. \(2658\) vs. \(2(1325)=2650\).

Time = 7.41 (sec) , antiderivative size = 2659, normalized size of antiderivative = 1.80

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2659\)
default	\(\text {Expression too large to display}\)	\(85419\)

Input:

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

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*A*f*(-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*(A*g+B*f)*(-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)...
 

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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