\(\int \frac {(A+B x) \sqrt {a-c x^2}}{\sqrt {d+e x}} \, dx\) [263]

Optimal result

Integrand size = 27, antiderivative size = 359 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 \sqrt {d+e x} (4 B d-5 A e-3 B e x) \sqrt {a-c x^2}}{15 e^2}+\frac {4 \sqrt {a} \left (4 B c d^2-5 A c d e-3 a B e^2\right ) \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 \sqrt {c} e^3 \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}-\frac {4 \sqrt {a} (4 B d-5 A e) \left (c d^2-a e^2\right ) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{15 \sqrt {c} e^3 \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

-2/15*(e*x+d)^(1/2)*(-3*B*e*x-5*A*e+4*B*d)*(-c*x^2+a)^(1/2)/e^2+4/15*a^(1/ 
2)*(-5*A*c*d*e-3*B*a*e^2+4*B*c*d^2)*(e*x+d)^(1/2)*(1-c*x^2/a)^(1/2)*Ellipt 
icE(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^3/(c^(1/2)*(e*x+d)/(c^(1/2)*d+a^(1/2)*e))^(1/ 
2)/(-c*x^2+a)^(1/2)-4/15*a^(1/2)*(-5*A*e+4*B*d)*(-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^(1/2)/e^3/(e*x+d)^(1/2)/(-c*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 26.05 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.45 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{\sqrt {d+e x}} \, dx=\frac {\sqrt {a-c x^2} \left (\frac {2 (d+e x) (-4 B d+5 A e+3 B e x)}{e^2}+\frac {4 \left (e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (4 B c d^2-5 A c d e-3 a B e^2\right ) \left (-a+c x^2\right )+i \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-4 B c d^2+5 A c d e+3 a B e^2\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \sqrt {a} \sqrt {c} e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (-4 B \sqrt {c} d-3 \sqrt {a} B e+5 A \sqrt {c} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{c e^4 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (-a+c x^2\right )}\right )}{15 \sqrt {d+e x}} \] Input:

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

Output:

(Sqrt[a - c*x^2]*((2*(d + e*x)*(-4*B*d + 5*A*e + 3*B*e*x))/e^2 + (4*(e^2*S 
qrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(4*B*c*d^2 - 5*A*c*d*e - 3*a*B*e^2)*(-a + c* 
x^2) + I*Sqrt[c]*(Sqrt[c]*d - Sqrt[a]*e)*(-4*B*c*d^2 + 5*A*c*d*e + 3*a*B*e 
^2)*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]*Sqrt[c]*e*(Sqrt[c]*d - Sqrt[a]*e)*(-4*B*Sqrt[c]*d - 3*Sqrt[ 
a]*B*e + 5*A*Sqrt[c]*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)*EllipticF[I*ArcSinh 
[Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]/Sqrt[d + e*x]], (Sqrt[c]*d + Sqrt[a]*e)/(S 
qrt[c]*d - Sqrt[a]*e)]))/(c*e^4*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(-a + c*x^2 
))))/(15*Sqrt[d + e*x])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 351, 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 = {682, 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.

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

\(\Big \downarrow \) 682

\(\displaystyle -\frac {4 \int \frac {c \left (a e (B d-5 A e)+\left (4 B c d^2-5 A c e d-3 a B e^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 c e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \int \frac {a e (B d-5 A e)+\left (4 B c d^2-5 A c e d-3 a B e^2\right ) x}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 600

\(\displaystyle -\frac {2 \left (\frac {\left (-3 a B e^2-5 A c d e+4 B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {a-c x^2}}dx}{e}-\frac {\left (c d^2-a e^2\right ) (4 B d-5 A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 509

\(\displaystyle -\frac {2 \left (\frac {\sqrt {1-\frac {c x^2}{a}} \left (-3 a B e^2-5 A c d e+4 B c d^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {\left (c d^2-a e^2\right ) (4 B d-5 A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 508

\(\displaystyle -\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) (4 B d-5 A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a B e^2-5 A c d e+4 B c d^2\right ) \int \frac {\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}}}{\sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {2 \left (-\frac {\left (c d^2-a e^2\right ) (4 B d-5 A e) \int \frac {1}{\sqrt {d+e x} \sqrt {a-c x^2}}dx}{e}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a B e^2-5 A c d e+4 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 512

\(\displaystyle -\frac {2 \left (-\frac {\sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (4 B d-5 A e) \int \frac {1}{\sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}}}dx}{e \sqrt {a-c x^2}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a B e^2-5 A c d e+4 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 511

\(\displaystyle -\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (4 B d-5 A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \int \frac {1}{\sqrt {1-\frac {e \left (1-\frac {\sqrt {c} x}{\sqrt {a}}\right )}{\frac {\sqrt {c} d}{\sqrt {a}}+e}} \sqrt {\frac {1}{2} \left (\frac {\sqrt {c} x}{\sqrt {a}}-1\right )+1}}d\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a B e^2-5 A c d e+4 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

\(\Big \downarrow \) 321

\(\displaystyle -\frac {2 \left (\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \left (c d^2-a e^2\right ) (4 B d-5 A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {d+e x}}-\frac {2 \sqrt {a} \sqrt {1-\frac {c x^2}{a}} \sqrt {d+e x} \left (-3 a B e^2-5 A c d e+4 B c d^2\right ) E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 e}{\frac {\sqrt {c} d}{\sqrt {a}}+e}\right )}{\sqrt {c} e \sqrt {a-c x^2} \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {a} e+\sqrt {c} d}}}\right )}{15 e^2}-\frac {2 \sqrt {a-c x^2} \sqrt {d+e x} (-5 A e+4 B d-3 B e x)}{15 e^2}\)

Input:

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

Output:

(-2*Sqrt[d + e*x]*(4*B*d - 5*A*e - 3*B*e*x)*Sqrt[a - c*x^2])/(15*e^2) - (2 
*((-2*Sqrt[a]*(4*B*c*d^2 - 5*A*c*d*e - 3*a*B*e^2)*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)])/(Sqrt[c]*e*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d 
+ Sqrt[a]*e)]*Sqrt[a - c*x^2]) + (2*Sqrt[a]*(4*B*d - 5*A*e)*(c*d^2 - a*e^2 
)*Sqrt[(Sqrt[c]*(d + e*x))/(Sqrt[c]*d + Sqrt[a]*e)]*Sqrt[1 - (c*x^2)/a]*El 
lipticF[ArcSin[Sqrt[1 - (Sqrt[c]*x)/Sqrt[a]]/Sqrt[2]], (2*e)/((Sqrt[c]*d)/ 
Sqrt[a] + e)])/(Sqrt[c]*e*Sqrt[d + e*x]*Sqrt[a - c*x^2])))/(15*e^2)
 

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 682

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(667\) vs. \(2(295)=590\).

Time = 3.58 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.86

method	result	size
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {2 B x \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{5 e}-\frac {2 \left (-A c +\frac {4 c d B}{5 e}\right ) \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}{3 c e}+\frac {2 \left (A a -\frac {2 a d B}{5 e}+\frac {a \left (-A c +\frac {4 c d B}{5 e}\right )}{3 c}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 \left (\frac {2 B a}{5}-\frac {2 d \left (-A c +\frac {4 c d B}{5 e}\right )}{3 e}\right ) \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(668\)
risch	\(\frac {2 \left (3 B e x +5 A e -4 B d \right ) \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{15 e^{2}}+\frac {2 \left (\frac {\left (5 A c d e +3 B a \,e^{2}-4 B c \,d^{2}\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 )}{c \sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {5 A a \,e^{2} \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 {B a d 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}}\right ) \sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}}{15 e^{2} \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(675\)
default	\(\text {Expression too large to display}\)	\(1738\)

Input:

int((B*x+A)*(-c*x^2+a)^(1/2)/(e*x+d)^(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)*(2/5*B/e*x*(-c*e 
*x^3-c*d*x^2+a*e*x+a*d)^(1/2)-2/3*(-A*c+4/5*c*d/e*B)/c/e*(-c*e*x^3-c*d*x^2 
+a*e*x+a*d)^(1/2)+2*(A*a-2/5*a*d/e*B+1/3*a/c*(-A*c+4/5*c*d/e*B))*(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*(2/ 
5*B*a-2/3*d/e*(-A*c+4/5*c*d/e*B))*(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))*EllipticE(((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 = 271, normalized size of antiderivative = 0.75 \[ \int \frac {(A+B x) \sqrt {a-c x^2}}{\sqrt {d+e x}} \, dx=-\frac {2 \, {\left (2 \, {\left (4 \, B c d^{3} - 5 \, A c d^{2} e - 6 \, B a d e^{2} + 15 \, A a e^{3}\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right ) + 6 \, {\left (4 \, B c d^{2} e - 5 \, A c d e^{2} - 3 \, B a e^{3}\right )} \sqrt {-c e} {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) - 3 \, {\left (3 \, B c e^{3} x - 4 \, B c d e^{2} + 5 \, A c e^{3}\right )} \sqrt {-c x^{2} + a} \sqrt {e x + d}\right )}}{45 \, c e^{4}} \] Input:

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

Output:

-2/45*(2*(4*B*c*d^3 - 5*A*c*d^2*e - 6*B*a*d*e^2 + 15*A*a*e^3)*sqrt(-c*e)*w 
eierstrassPInverse(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) + 6*(4*B*c*d^2*e - 5*A*c*d*e^2 - 3*B*a*e^3)* 
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), -8/2 
7*(c*d^3 - 9*a*d*e^2)/(c*e^3), 1/3*(3*e*x + d)/e)) - 3*(3*B*c*e^3*x - 4*B* 
c*d*e^2 + 5*A*c*e^3)*sqrt(-c*x^2 + a)*sqrt(e*x + d))/(c*e^4)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(A+B x) \sqrt {a-c x^2}}{\sqrt {d+e x}} \, dx=\frac {-2 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, a b e +2 \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, b c d x -3 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a b c \,e^{2}-5 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a \,c^{2} d e +4 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}\, x^{2}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) b \,c^{2} d^{2}+\left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} b \,e^{2}+5 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a^{2} c d e -2 \left (\int \frac {\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{-c e \,x^{3}-c d \,x^{2}+a e x +a d}d x \right ) a b c \,d^{2}}{5 c d e} \] Input:

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

Output:

( - 2*sqrt(d + e*x)*sqrt(a - c*x**2)*a*b*e + 2*sqrt(d + e*x)*sqrt(a - c*x* 
*2)*b*c*d*x - 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*b*c*e**2 - 5*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**2*d*e + 4*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**2* 
d**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**2*b*e**2 + 5*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*c*d*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*c*d**2)/(5*c*d*e)