\(\int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx\) [71]

Optimal result

Integrand size = 33, antiderivative size = 869 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\frac {2 e \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right )|\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}\right )}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} (e f-d g) \sqrt {\frac {(e f-d g) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {a+b x+c x^2}}-\frac {4 c \left (b-\sqrt {b^2-4 a c}+2 c x\right ) \sqrt {\frac {(e f-d g) \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2 c d-b e+\sqrt {b^2-4 a c} e} \sqrt {f+g x}}{\sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {d+e x}}\right ),\frac {\left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e\right ) \left (2 c f-\left (b+\sqrt {b^2-4 a c}\right ) g\right )}\right )}{\sqrt {2 c d-\left (b-\sqrt {b^2-4 a c}\right ) e} \left (2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e\right ) \sqrt {2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g} \sqrt {\frac {(e f-d g) \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b-\sqrt {b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt {a+b x+c x^2}} \] Output:

2*e*(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(((-4*a*c+b^2)^(1/2)*e-b*e+2*c*d)^(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))/(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)/((-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)-4*c*(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(((-4*a*c+b^2)^(1/2)*e-b*e+2*c*d)^(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))/(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)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5981\) vs. \(2(869)=1738\).

Time = 6.40 (sec) , antiderivative size = 5981, normalized size of antiderivative = 6.88 \[ \int \frac {1}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[1/((d + e*x)^(3/2)*Sqrt[f + g*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.

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

\(\Big \downarrow \) 1281

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

\(\Big \downarrow \) 1280

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

\(\Big \downarrow \) 1292

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

\(\Big \downarrow \) 1416

\(\displaystyle \frac {e \int \frac {\sqrt {f+g x}}{(d+e x)^{3/2} \sqrt {c x^2+b x+a}}dx}{e f-d g}+\frac {g (d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt {\frac {\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right ) \sqrt {\frac {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac {(f+g x) \sqrt {a e^2-b d e+c d^2}}{(d+e x) \sqrt {c f^2-g (b f-a g)}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c d^2-b e d+a e^2} \sqrt {f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt {d+e x}}\right ),\frac {1}{4} \left (\frac {2 c d f+2 a e g-b (e f+d g)}{\sqrt {c d^2-e (b d-a e)} \sqrt {c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt {a+b x+c x^2} (e f-d g)^2 \sqrt [4]{a e^2-b d e+c d^2} \sqrt {\frac {(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac {(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}}\)

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1280

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.) 
*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-2*(d + e*x)*(Sqrt[(e*f - d*g)^2* 
((a + b*x + c*x^2)/((c*f^2 - b*f*g + a*g^2)*(d + e*x)^2))]/((e*f - d*g)*Sqr 
t[a + b*x + c*x^2]))   Subst[Int[1/Sqrt[1 - (2*c*d*f - b*e*f - b*d*g + 2*a* 
e*g)*(x^2/(c*f^2 - b*f*g + a*g^2)) + (c*d^2 - b*d*e + a*e^2)*(x^4/(c*f^2 - 
b*f*g + a*g^2))], x], x, Sqrt[f + g*x]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c 
, d, e, f, g}, x]
 

rule 1281

Int[1/(((d_.) + (e_.)*(x_))^(3/2)*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_ 
.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Simp[-g/(e*f - d*g)   Int[1/(Sqrt[d 
+ e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]), x], x] + Simp[e/(e*f - d*g) 
Int[Sqrt[f + g*x]/((d + e*x)^(3/2)*Sqrt[a + b*x + c*x^2]), x], 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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3090\) vs. \(2(777)=1554\).

Time = 14.47 (sec) , antiderivative size = 3091, normalized size of antiderivative = 3.56

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(3091\)
default	\(\text {Expression too large to display}\)	\(18084\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + f)/(c*e^2*g*x^5 + 
(c*e^2*f + (2*c*d*e + b*e^2)*g)*x^4 + a*d^2*f + ((2*c*d*e + b*e^2)*f + (c* 
d^2 + 2*b*d*e + a*e^2)*g)*x^3 + ((c*d^2 + 2*b*d*e + a*e^2)*f + (b*d^2 + 2* 
a*d*e)*g)*x^2 + (a*d^2*g + (b*d^2 + 2*a*d*e)*f)*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Giac [F]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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