\(\int \frac {A+B x}{\sqrt {d+e x} \sqrt {a-c x^2}} \, dx\) [275]

Optimal result

Integrand size = 27, antiderivative size = 280 \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a-c x^2}} \, dx=-\frac {2 \sqrt {a} B \sqrt {d+e x} \sqrt {1-\frac {c x^2}{a}} E\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right )|\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {c} e \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {a-c x^2}}+\frac {2 \sqrt {a} (B d-A e) \sqrt {\frac {\sqrt {c} (d+e x)}{\sqrt {c} d+\sqrt {a} e}} \sqrt {1-\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {1-\frac {\sqrt {c} x}{\sqrt {a}}}}{\sqrt {2}}\right ),\frac {2 \sqrt {a} e}{\sqrt {c} d+\sqrt {a} e}\right )}{\sqrt {c} e \sqrt {d+e x} \sqrt {a-c x^2}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.69 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.45 \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a-c x^2}} \, dx=-\frac {2 \left (B e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \left (a-c x^2\right )+i B \sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right )|\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )+i \left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt {c} e \sqrt {\frac {e \left (\frac {\sqrt {a}}{\sqrt {c}}+x\right )}{d+e x}} \sqrt {-\frac {\frac {\sqrt {a} e}{\sqrt {c}}-e x}{d+e x}} (d+e x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}}}{\sqrt {d+e x}}\right ),\frac {\sqrt {c} d+\sqrt {a} e}{\sqrt {c} d-\sqrt {a} e}\right )\right )}{c e^2 \sqrt {-d+\frac {\sqrt {a} e}{\sqrt {c}}} \sqrt {d+e x} \sqrt {a-c x^2}} \] Input:

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

Output:

(-2*(B*e^2*Sqrt[-d + (Sqrt[a]*e)/Sqrt[c]]*(a - c*x^2) + I*B*Sqrt[c]*(Sqrt[ 
c]*d - Sqrt[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 - A*Sqrt[c])*Sqrt[c]*e*Sqrt[(e*(Sqrt[a]/Sq 
rt[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)]))/(c*e^2*Sqrt[-d + ( 
Sqrt[a]*e)/Sqrt[c]]*Sqrt[d + e*x]*Sqrt[a - c*x^2])
 

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {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.

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 509

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 512

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

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

Output:

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

Defintions of rubi rules used

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(497\) vs. \(2(226)=452\).

Time = 4.23 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.78

method	result	size
default	\(\frac {2 \left (A \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) c d e -A \sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) e^{2}-B \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) a \,e^{2}+B \sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) d e +B \operatorname {EllipticE}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) a \,e^{2}-B \operatorname {EllipticE}\left (\sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}, \sqrt {-\frac {\sqrt {a c}\, e -c d}{\sqrt {a c}\, e +c d}}\right ) c \,d^{2}\right ) \sqrt {\frac {\left (c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e -c d}}\, \sqrt {\frac {\left (-c x +\sqrt {a c}\right ) e}{\sqrt {a c}\, e +c d}}\, \sqrt {-\frac {c \left (e x +d \right )}{\sqrt {a c}\, e -c d}}\, \sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}{c \,e^{2} \left (-c e \,x^{3}-c d \,x^{2}+a e x +a d \right )}\)	\(498\)
elliptic	\(\frac {\sqrt {\left (e x +d \right ) \left (-c \,x^{2}+a \right )}\, \left (\frac {2 A \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}+\frac {2 B \left (\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x -\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\, \sqrt {\frac {x +\frac {\sqrt {a c}}{c}}{-\frac {d}{e}+\frac {\sqrt {a c}}{c}}}\, \left (\left (-\frac {d}{e}-\frac {\sqrt {a c}}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )+\frac {\sqrt {a c}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {\sqrt {a c}}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {\sqrt {a c}}{c}}{-\frac {d}{e}-\frac {\sqrt {a c}}{c}}}\right )}{c}\right )}{\sqrt {-c e \,x^{3}-c d \,x^{2}+a e x +a d}}\right )}{\sqrt {e x +d}\, \sqrt {-c \,x^{2}+a}}\)	\(539\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.65 \[ \int \frac {A+B x}{\sqrt {d+e x} \sqrt {a-c x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-c e} B e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + {\left (B d - 3 \, A e\right )} \sqrt {-c e} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c d^{2} + 3 \, a e^{2}\right )}}{3 \, c e^{2}}, -\frac {8 \, {\left (c d^{3} - 9 \, a d e^{2}\right )}}{27 \, c e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )}}{3 \, c e^{2}} \] Input:

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

Output:

2/3*(3*sqrt(-c*e)*B*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)) + (B*d - 3*A 
*e)*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))/(c*e^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

Giac [F]

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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