\(\int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a-c x^2}} \, dx\) [274]

Optimal result

Integrand size = 27, antiderivative size = 325 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a-c x^2}} \, dx=-\frac {2 B \sqrt {d+e x} \sqrt {a-c x^2}}{3 c}-\frac {2 \sqrt {a} (B d+3 A e) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{3 \sqrt {c} e \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {2 \sqrt {a} B \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{3 c^{3/2} e \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 24.35 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.39 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a-c x^2}} \, dx=\frac {2 \sqrt {a-c x^2} \left (-B (d+e x)+\frac {e^2 (B d+3 A e) \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) (B d+3 A e) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \left (\sqrt {a} B-3 A \sqrt {c}\right ) e \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )}{e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{3 c \sqrt {d+e x}} \] Input:

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

Output:

(2*Sqrt[a - c*x^2]*(-(B*(d + e*x)) + (e^2*(B*d + 3*A*e)*Sqrt[-d + (Sqrt[a] 
*e)/Sqrt[c]]*(a - c*x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(B*d + 3*A*e) 
*Sqrt[(e*(Sqrt[a]/Sqrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e 
*x)/(d + e*x))]*(d + e*x)^(3/2)*EllipticE[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/ 
Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)] 
+ I*(Sqrt[a]*B - 3*A*Sqrt[c])*e*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[(e*(Sqrt[a]/S 
qrt[c] + x))/(d + e*x)]*Sqrt[-(((Sqrt[a]*e)/Sqrt[c] - e*x)/(d + e*x))]*(d 
+ e*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e 
*x]], (Sqrt[c]*d + Sqrt[a]*e)/(Sqrt[c]*d - Sqrt[a]*e)])/(e^2*Sqrt[-d + (Sq 
rt[a]*e)/Sqrt[c]]*(-a + c*x^2))))/(3*c*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {687, 27, 600, 509, 508, 327, 512, 511, 321}

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.

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

\(\Big \downarrow \) 687

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

\(\displaystyle \frac {\frac {2 \sqrt {a} B \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 A e+B d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{3 c}-\frac {2 B \sqrt {a-c x^2} \sqrt {d+e x}}{3 c}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {2 \sqrt {a} B \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {c} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} (3 A e+B d) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}}{3 c}-\frac {2 B \sqrt {a-c x^2} \sqrt {d+e x}}{3 c}\)

Input:

Int[((A + B*x)*Sqrt[d + e*x])/Sqrt[a - c*x^2],x]
 

Output:

(-2*B*Sqrt[d + e*x]*Sqrt[a - c*x^2])/(3*c) + ((-2*Sqrt[a]*Sqrt[c]*(B*d + 3 
*A*e)*Sqrt[d + e*x]*Sqrt[1 - (c*x^2)/a]*EllipticE[ArcSin[Sqrt[1 - (Sqrt[c] 
*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/Sqrt[a] + e)])/(e*Sqrt[(Sqrt[c]* 
(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]*B*(c*d^2 
 - a*e^2)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^ 
2)/a]*EllipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqr 
t[c]*d)/Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2]))/(3*c)
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 508

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> With[{q 
 = Rt[-b/a, 2]}, Simp[-2*(Sqrt[c + d*x]/(Sqrt[a]*q*Sqrt[q*((c + d*x)/(d + c 
*q))]))   Subst[Int[Sqrt[1 - 2*d*(x^2/(d + c*q))]/Sqrt[1 - x^2], x], x, Sqr 
t[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[a, 0]
 

rule 509

Int[Sqrt[(c_) + (d_.)*(x_)]/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[Sq 
rt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[Sqrt[c + d*x]/Sqrt[1 + b*(x^2/a)], 
x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 511

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Wit 
h[{q = Rt[-b/a, 2]}, Simp[-2*(Sqrt[q*((c + d*x)/(d + c*q))]/(Sqrt[a]*q*Sqrt 
[c + d*x]))   Subst[Int[1/(Sqrt[1 - 2*d*(x^2/(d + c*q))]*Sqrt[1 - x^2]), x] 
, x, Sqrt[(1 - q*x)/2]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] && GtQ[ 
a, 0]
 

rule 512

Int[1/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Sim 
p[Sqrt[1 + b*(x^2/a)]/Sqrt[a + b*x^2]   Int[1/(Sqrt[c + d*x]*Sqrt[1 + b*(x^ 
2/a)]), x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[b/a] &&  !GtQ[a, 0]
 

rule 600

Int[((A_.) + (B_.)*(x_))/(Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(a_) + (b_.)*(x_)^2] 
), x_Symbol] :> Simp[B/d   Int[Sqrt[c + d*x]/Sqrt[a + b*x^2], x], x] - Simp 
[(B*c - A*d)/d   Int[1/(Sqrt[c + d*x]*Sqrt[a + b*x^2]), x], x] /; FreeQ[{a, 
 b, c, d, A, B}, x] && NegQ[b/a]
 

rule 687

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g*(d + e*x)^m*((a + c*x^2)^(p + 1)/(c*(m + 2*p + 2)) 
), x] + Simp[1/(c*(m + 2*p + 2))   Int[(d + e*x)^(m - 1)*(a + c*x^2)^p*Simp 
[c*d*f*(m + 2*p + 2) - a*e*g*m + c*(e*f*(m + 2*p + 2) + d*g*m)*x, x], x], x 
] /; FreeQ[{a, c, d, e, f, g, p}, x] && GtQ[m, 0] && NeQ[m + 2*p + 2, 0] && 
 (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p]) &&  !(IGtQ[m, 0] && Eq 
Q[f, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(586\) vs. \(2(261)=522\).

Time = 2.81 (sec) , antiderivative size = 587, normalized size of antiderivative = 1.81

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (-\frac {2 B \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 c}+\frac {2 \left (A d +\frac {B a e}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (A e +\frac {B d}{3}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(587\)
risch	\(-\frac {2 B \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{3 c}+\frac {\left (\frac {\left (3 A e +B d \right ) \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \left (\left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )-\frac {d \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{e}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {B a e \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {3 A d \sqrt {a c}\, \sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}\, \operatorname {EllipticF}\left (\frac {\sqrt {2}\, \sqrt {\frac {\left (x +\frac {\sqrt {a c}}{c}\right ) c}{\sqrt {a c}}}}{2}, \sqrt {-\frac {2 \sqrt {a c}}{c \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right )}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right ) \sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}}{3 c \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(638\)
default	\(\text {Expression too large to display}\)	\(1225\)

Input:

int((B*x+A)*(e*x+d)^(1/2)/(-c*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

((e*x+d)*(-c*x^2+a))^(1/2)/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)*(-2/3*B/c*(-c*e* 
x^3-c*d*x^2+a*e*x+a*d)^(1/2)+2*(A*d+1/3*B/c*a*e)*(d/e-1/c*(a*c)^(1/2))*((x 
+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1 
/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2)))^(1/2)/(-c*e*x^3-c 
*d*x^2+a*e*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),(( 
-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))+2*(A*e+1/3*B*d)*(d/e- 
1/c*(a*c)^(1/2))*((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2)*((x-1/c*(a*c)^(1/2) 
)/(-d/e-1/c*(a*c)^(1/2)))^(1/2)*((x+1/c*(a*c)^(1/2))/(-d/e+1/c*(a*c)^(1/2) 
))^(1/2)/(-c*e*x^3-c*d*x^2+a*e*x+a*d)^(1/2)*((-d/e-1/c*(a*c)^(1/2))*Ellipt 
icE(((x+d/e)/(d/e-1/c*(a*c)^(1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/ 
c*(a*c)^(1/2)))^(1/2))+1/c*(a*c)^(1/2)*EllipticF(((x+d/e)/(d/e-1/c*(a*c)^( 
1/2)))^(1/2),((-d/e+1/c*(a*c)^(1/2))/(-d/e-1/c*(a*c)^(1/2)))^(1/2))))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.71 \[ \int \frac {(A+B x) \sqrt {d+e x}}{\sqrt {a-c x^2}} \, dx=-\frac {2 \, {\left (3 \, \sqrt {-c x^{2} + a} \sqrt {e x + d} B c e^{2} - {\left (B c d^{2} - 6 \, A c d e - 3 \, B a e^{2}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) - 3 \, {\left (B c d e + 3 \, A c e^{2}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )\right )}}{9 \, c^{2} e^{2}} \] Input:

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

Output:

-2/9*(3*sqrt(-c*x^2 + a)*sqrt(e*x + d)*B*c*e^2 - (B*c*d^2 - 6*A*c*d*e - 3* 
B*a*e^2)*sqrt(-c*e)*weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/ 
27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e) - 3*(B*c*d*e + 3*A*c*e^ 
2)*sqrt(-c*e)*weierstrassZeta(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), -8/27*(c*d^3 
- 9*a*d*e^2)/(c*e^3), weierstrassPInverse(4/3*(c*d^2 + 3*a*e^2)/(c*e^2), - 
8/27*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)))/(c^2*e^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 2*sqrt(d + e*x)*sqrt(a - c*x**2)*a*e - 2*sqrt(d + e*x)*sqrt(a - c*x**2 
)*b*d - 3*int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d + a*e*x - c*d*x** 
2 - c*e*x**3),x)*a*c*e**2 - int((sqrt(d + e*x)*sqrt(a - c*x**2)*x**2)/(a*d 
 + a*e*x - c*d*x**2 - c*e*x**3),x)*b*c*d*e + int((sqrt(d + e*x)*sqrt(a - c 
*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a**2*e**2 + int((sqrt(d + e 
*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x)*a*b*d*e + 2*i 
nt((sqrt(d + e*x)*sqrt(a - c*x**2))/(a*d + a*e*x - c*d*x**2 - c*e*x**3),x) 
*a*c*d**2)/(2*c*d)