\(\int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx\) [47]

Optimal result

Integrand size = 36, antiderivative size = 677 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=-\frac {2 (B d-A e) \sqrt {\sqrt {c} f+\sqrt {a} g} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} E\left (\arcsin \left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {a} g} \sqrt {d+e x}}\right )|\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} (e f-d g) \sqrt {\frac {(e f-d g) \left (\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} \sqrt {a-c x^2}}+\frac {2 \left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {\sqrt {c} f+\sqrt {a} g} \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {-\frac {(e f-d g) \left (\sqrt {a}-\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \sqrt {f+g x}}{\sqrt {\sqrt {c} f+\sqrt {a} g} \sqrt {d+e x}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )}{\left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\sqrt {c} d+\sqrt {a} e} \left (\sqrt {c} f-\sqrt {a} g\right ) \sqrt {\frac {(e f-d g) \left (\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} \sqrt {a-c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 26.04 (sec) , antiderivative size = 734, normalized size of antiderivative = 1.08 \[ \int \frac {A+B x}{(d+e x)^{3/2} \sqrt {f+g x} \sqrt {a-c x^2}} \, dx=\frac {\sqrt {d+e x} \sqrt {\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (f+g x)}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \sqrt {a-c x^2} \left (2 e (-B d+A e) \left (\sqrt {c} f-\sqrt {a} g\right ) \sqrt {\frac {(e f-d g) \left (\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} E\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (f+g x)}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}}\right )|\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )-(e f-d g) \left (-\sqrt {2} B \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\frac {d+\frac {\sqrt {a} e}{\sqrt {c}}+\frac {\sqrt {c} d x}{\sqrt {a}}+e x}{d+e x}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {(e f-d g) \left (-\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}}\right ),-\frac {\frac {\sqrt {c} d f}{\sqrt {a}}-e f+d g-\frac {\sqrt {a} e g}{\sqrt {c}}}{2 e f-2 d g}\right )-2 \sqrt {c} (B d-A e) \sqrt {\frac {(e f-d g) \left (\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f-\sqrt {a} g\right ) (d+e x)}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) (f+g x)}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}}\right ),\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f+\sqrt {a} g\right )}{\left (\sqrt {c} d+\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right )}\right )\right )\right )}{e \left (-c d^2+a e^2\right ) (e f-d g) \left (\sqrt {a}+\sqrt {c} x\right ) \sqrt {\frac {(e f-d g) \left (-\sqrt {a}+\sqrt {c} x\right )}{\left (\sqrt {c} f+\sqrt {a} g\right ) (d+e x)}} \sqrt {f+g x}} \] Input:

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

Output:

(Sqrt[d + e*x]*Sqrt[((Sqrt[c]*d + Sqrt[a]*e)*(f + g*x))/((Sqrt[c]*f + Sqrt 
[a]*g)*(d + e*x))]*Sqrt[a - c*x^2]*(2*e*(-(B*d) + A*e)*(Sqrt[c]*f - Sqrt[a 
]*g)*Sqrt[((e*f - d*g)*(Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - Sqrt[a]*g)*(d 
+ e*x))]*EllipticE[ArcSin[Sqrt[((Sqrt[c]*d + Sqrt[a]*e)*(f + g*x))/((Sqrt[ 
c]*f + Sqrt[a]*g)*(d + e*x))]], ((Sqrt[c]*d - Sqrt[a]*e)*(Sqrt[c]*f + Sqrt 
[a]*g))/((Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]*f - Sqrt[a]*g))] - (e*f - d*g)*( 
-(Sqrt[2]*B*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[(d + (Sqrt[a]*e)/Sqrt[c] + (Sqrt[ 
c]*d*x)/Sqrt[a] + e*x)/(d + e*x)]*EllipticF[ArcSin[Sqrt[((e*f - d*g)*(-Sqr 
t[a] + Sqrt[c]*x))/((Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]], -(((Sqrt[c]*d*f) 
/Sqrt[a] - e*f + d*g - (Sqrt[a]*e*g)/Sqrt[c])/(2*e*f - 2*d*g))]) - 2*Sqrt[ 
c]*(B*d - A*e)*Sqrt[((e*f - d*g)*(Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f - Sqrt 
[a]*g)*(d + e*x))]*EllipticF[ArcSin[Sqrt[((Sqrt[c]*d + Sqrt[a]*e)*(f + g*x 
))/((Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]], ((Sqrt[c]*d - Sqrt[a]*e)*(Sqrt[c 
]*f + Sqrt[a]*g))/((Sqrt[c]*d + Sqrt[a]*e)*(Sqrt[c]*f - Sqrt[a]*g))])))/(e 
*(-(c*d^2) + a*e^2)*(e*f - d*g)*(Sqrt[a] + Sqrt[c]*x)*Sqrt[((e*f - d*g)*(- 
Sqrt[a] + Sqrt[c]*x))/((Sqrt[c]*f + Sqrt[a]*g)*(d + e*x))]*Sqrt[f + g*x])
 

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.

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

\(\Big \downarrow \) 2349

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 732

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

\(\Big \downarrow \) 733

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

\(\Big \downarrow \) 732

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

\(\Big \downarrow \) 744

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

\(\Big \downarrow \) 1416

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

Input:

Int[(A + B*x)/((d + e*x)^(3/2)*Sqrt[f + g*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 732

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

rule 733

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(3/2)*Sqrt[(a_) + (b_. 
)*(x_)^2]), x_Symbol] :> Simp[d/(d*e - c*f)   Int[1/(Sqrt[c + d*x]*Sqrt[e + 
 f*x]*Sqrt[a + b*x^2]), x], x] - Simp[f/(d*e - c*f)   Int[Sqrt[c + d*x]/((e 
 + f*x)^(3/2)*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, 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 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 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2186\) vs. \(2(523)=1046\).

Time = 7.49 (sec) , antiderivative size = 2187, normalized size of antiderivative = 3.23

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(2187\)
default	\(\text {Expression too large to display}\)	\(15225\)

Input:

int((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x,method=_RETURNV 
ERBOSE)
 

Output:

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

Fricas [F]

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

integrate((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algorith 
m="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algorith 
m="maxima")
 

Output:

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

Giac [F]

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

integrate((B*x+A)/(e*x+d)^(3/2)/(g*x+f)^(1/2)/(-c*x^2+a)^(1/2),x, algorith 
m="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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