\(\int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx\) [5]

Optimal result

Integrand size = 43, antiderivative size = 1265 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(29355\) vs. \(2(1265)=2530\).

Time = 38.68 (sec) , antiderivative size = 29355, normalized size of antiderivative = 23.21 \[ \int \frac {A+B x+C x^2}{(d+e x)^{3/2} (f+g x)^{3/2} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[(A + B*x + C*x^2)/((d + e*x)^(3/2)*(f + g*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.

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 1292

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

\(\Big \downarrow \) 2154

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1281

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

\(\Big \downarrow \) 1280

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

\(\Big \downarrow \) 1292

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

\(\Big \downarrow \) 1416

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

Input:

Int[(A + B*x + C*x^2)/((d + e*x)^(3/2)*(f + g*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 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]
 

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. \(7327\) vs. \(2(1117)=2234\).

Time = 17.79 (sec) , antiderivative size = 7328, normalized size of antiderivative = 5.79

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(7328\)
default	\(\text {Expression too large to display}\)	\(571362\)

Input:

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

Output:

result too large to display
 

Fricas [F]

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

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

Output:

integral((C*x^2 + B*x + A)*sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + 
f)/(c*e^2*g^2*x^6 + (2*c*e^2*f*g + (2*c*d*e + b*e^2)*g^2)*x^5 + a*d^2*f^2 
+ (c*e^2*f^2 + 2*(2*c*d*e + b*e^2)*f*g + (c*d^2 + 2*b*d*e + a*e^2)*g^2)*x^ 
4 + ((2*c*d*e + b*e^2)*f^2 + 2*(c*d^2 + 2*b*d*e + a*e^2)*f*g + (b*d^2 + 2* 
a*d*e)*g^2)*x^3 + (a*d^2*g^2 + (c*d^2 + 2*b*d*e + a*e^2)*f^2 + 2*(b*d^2 + 
2*a*d*e)*f*g)*x^2 + (2*a*d^2*f*g + (b*d^2 + 2*a*d*e)*f^2)*x), x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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