\(\int \frac {(A+B x) \sqrt {f+g x}}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx\) [1]

Optimal result

Integrand size = 35, antiderivative size = 1 \[ \int \frac {(A+B x) \sqrt {f+g x}}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=0 \] Output:

0
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 1 in optimal.

Time = 27.71 (sec) , antiderivative size = 7130, normalized size of antiderivative = 7130.00 \[ \int \frac {(A+B x) \sqrt {f+g x}}{\sqrt {d+e x} \sqrt {a+c x^2}} \, dx=\text {Result too large to show} \] Input:

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

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 726

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

\(\Big \downarrow \) 744

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

Input:

Int[((A + B*x)*Sqrt[f + g*x])/(Sqrt[d + e*x]*Sqrt[a + 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 726

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

rule 744

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

rule 2349

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 4 vs. order 1.

Time = 6.73 (sec) , antiderivative size = 1903, normalized size of antiderivative = 1903.00

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1903\)
default	\(\text {Expression too large to display}\)	\(22386\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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