\(\int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx\) [4]

Optimal result

Integrand size = 29, antiderivative size = 90 \[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=-\frac {2 \sqrt {2} \sqrt {c d+e} \sqrt {\frac {c (d+e x)}{c d+e}} E\left (\arcsin \left (\frac {\sqrt {e} \sqrt {1-c x}}{\sqrt {c d+e}}\right )|\frac {c d+e}{2 e}\right )}{c \sqrt {e} \sqrt {d+e x}} \] Output:

-2*2^(1/2)*(c*d+e)^(1/2)*(c*(e*x+d)/(c*d+e))^(1/2)*EllipticE(e^(1/2)*(-c*x 
+1)^(1/2)/(c*d+e)^(1/2),1/2*2^(1/2)*((c*d+e)/e)^(1/2))/c/e^(1/2)/(e*x+d)^( 
1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.15 (sec) , antiderivative size = 282, normalized size of antiderivative = 3.13 \[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=-\frac {2 \left (e^2 \sqrt {-\frac {c d+e}{c}} \left (1-c^2 x^2\right )+i c (c d+e) \sqrt {\frac {e (-1+c x)}{c (d+e x)}} (d+e x)^{3/2} \sqrt {\frac {e+c e x}{c d+c e x}} E\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {c d+e}{c}}}{\sqrt {d+e x}}\right )|\frac {c d-e}{c d+e}\right )-2 i c e \sqrt {\frac {e (-1+c x)}{c (d+e x)}} (d+e x)^{3/2} \sqrt {\frac {e+c e x}{c d+c e x}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {c d+e}{c}}}{\sqrt {d+e x}}\right ),\frac {c d-e}{c d+e}\right )\right )}{c e^2 \sqrt {-\frac {c d+e}{c}} \sqrt {d+e x} \sqrt {1-c^2 x^2}} \] Input:

Integrate[(1 + c*x)/(Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]),x]
 

Output:

(-2*(e^2*Sqrt[-((c*d + e)/c)]*(1 - c^2*x^2) + I*c*(c*d + e)*Sqrt[(e*(-1 + 
c*x))/(c*(d + e*x))]*(d + e*x)^(3/2)*Sqrt[(e + c*e*x)/(c*d + c*e*x)]*Ellip 
ticE[I*ArcSinh[Sqrt[-((c*d + e)/c)]/Sqrt[d + e*x]], (c*d - e)/(c*d + e)] - 
 (2*I)*c*e*Sqrt[(e*(-1 + c*x))/(c*(d + e*x))]*(d + e*x)^(3/2)*Sqrt[(e + c* 
e*x)/(c*d + c*e*x)]*EllipticF[I*ArcSinh[Sqrt[-((c*d + e)/c)]/Sqrt[d + e*x] 
], (c*d - e)/(c*d + e)]))/(c*e^2*Sqrt[-((c*d + e)/c)]*Sqrt[d + e*x]*Sqrt[1 
 - c^2*x^2])
 

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {600, 508, 327, 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.

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

\(\Big \downarrow \) 600

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

\(\Big \downarrow \) 508

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 511

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

\(\Big \downarrow \) 321

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

Input:

Int[(1 + c*x)/(Sqrt[d + e*x]*Sqrt[1 - c^2*x^2]),x]
 

Output:

(-2*Sqrt[d + e*x]*EllipticE[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e) 
])/(e*Sqrt[(c*(d + e*x))/(c*d + e)]) + (2*(c*d - e)*Sqrt[(c*(d + e*x))/(c* 
d + e)]*EllipticF[ArcSin[Sqrt[1 - c*x]/Sqrt[2]], (2*e)/(c*d + e)])/(c*e*Sq 
rt[d + e*x])
 

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 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 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. \(274\) vs. \(2(77)=154\).

Time = 1.93 (sec) , antiderivative size = 275, normalized size of antiderivative = 3.06

method	result	size
default	\(\frac {2 \left (\operatorname {EllipticE}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c^{2} d^{2}-2 \operatorname {EllipticF}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) c d e -\operatorname {EllipticE}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}+2 \operatorname {EllipticF}\left (\sqrt {\frac {c \left (e x +d \right )}{c d -e}}, \sqrt {\frac {c d -e}{c d +e}}\right ) e^{2}\right ) \sqrt {-\frac {e \left (c x +1\right )}{c d -e}}\, \sqrt {-\frac {e \left (c x -1\right )}{c d +e}}\, \sqrt {\frac {c \left (e x +d \right )}{c d -e}}\, \sqrt {e x +d}\, \sqrt {-c^{2} x^{2}+1}}{e^{2} c \left (e \,x^{3} c^{2}+c^{2} d \,x^{2}-e x -d \right )}\)	\(275\)
elliptic	\(\frac {\sqrt {-\left (e x +d \right ) \left (c^{2} x^{2}-1\right )}\, \left (\frac {2 \left (\frac {d}{e}-\frac {1}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x +\frac {1}{c}}{-\frac {d}{e}+\frac {1}{c}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\right )}{\sqrt {-e \,x^{3} c^{2}-c^{2} d \,x^{2}+e x +d}}+\frac {2 c \left (\frac {d}{e}-\frac {1}{c}\right ) \sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x -\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\, \sqrt {\frac {x +\frac {1}{c}}{-\frac {d}{e}+\frac {1}{c}}}\, \left (\left (-\frac {d}{e}-\frac {1}{c}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\right )+\frac {\operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {d}{e}}{\frac {d}{e}-\frac {1}{c}}}, \sqrt {\frac {-\frac {d}{e}+\frac {1}{c}}{-\frac {d}{e}-\frac {1}{c}}}\right )}{c}\right )}{\sqrt {-e \,x^{3} c^{2}-c^{2} d \,x^{2}+e x +d}}\right )}{\sqrt {e x +d}\, \sqrt {-c^{2} x^{2}+1}}\)	\(422\)

Input:

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

Output:

2*(EllipticE((c*(e*x+d)/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c^2*d^2-2* 
EllipticF((c*(e*x+d)/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*c*d*e-Ellipti 
cE((c*(e*x+d)/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e^2+2*EllipticF((c*( 
e*x+d)/(c*d-e))^(1/2),((c*d-e)/(c*d+e))^(1/2))*e^2)*(-e*(c*x+1)/(c*d-e))^( 
1/2)*(-e*(c*x-1)/(c*d+e))^(1/2)*(c*(e*x+d)/(c*d-e))^(1/2)*(e*x+d)^(1/2)*(- 
c^2*x^2+1)^(1/2)/e^2/c/(c^2*e*x^3+c^2*d*x^2-e*x-d)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (77) = 154\).

Time = 0.07 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.12 \[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=\frac {2 \, {\left (3 \, \sqrt {-c^{2} e} c e {\rm weierstrassZeta}\left (\frac {4 \, {\left (c^{2} d^{2} + 3 \, e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {8 \, {\left (c^{2} d^{3} - 9 \, d e^{2}\right )}}{27 \, c^{2} e^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + 3 \, e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {8 \, {\left (c^{2} d^{3} - 9 \, d e^{2}\right )}}{27 \, c^{2} e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right ) + \sqrt {-c^{2} e} {\left (c d - 3 \, e\right )} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (c^{2} d^{2} + 3 \, e^{2}\right )}}{3 \, c^{2} e^{2}}, -\frac {8 \, {\left (c^{2} d^{3} - 9 \, d e^{2}\right )}}{27 \, c^{2} e^{3}}, \frac {3 \, e x + d}{3 \, e}\right )\right )}}{3 \, c^{2} e^{2}} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=\int \frac {c x + 1}{\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )} \sqrt {d + e x}}\, dx \] Input:

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

Output:

Integral((c*x + 1)/(sqrt(-(c*x - 1)*(c*x + 1))*sqrt(d + e*x)), x)
 

Maxima [F]

\[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=\int { \frac {c x + 1}{\sqrt {-c^{2} x^{2} + 1} \sqrt {e x + d}} \,d x } \] Input:

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

Output:

integrate((c*x + 1)/(sqrt(-c^2*x^2 + 1)*sqrt(e*x + d)), x)
 

Giac [F]

\[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=\int { \frac {c x + 1}{\sqrt {-c^{2} x^{2} + 1} \sqrt {e x + d}} \,d x } \] Input:

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

Output:

integrate((c*x + 1)/(sqrt(-c^2*x^2 + 1)*sqrt(e*x + d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=\int \frac {c\,x+1}{\sqrt {1-c^2\,x^2}\,\sqrt {d+e\,x}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1+c x}{\sqrt {d+e x} \sqrt {1-c^2 x^2}} \, dx=-\left (\int \frac {\sqrt {e x +d}\, \sqrt {-c^{2} x^{2}+1}}{c e \,x^{2}+c d x -e x -d}d x \right ) \] Input:

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

Output:

 - int((sqrt(d + e*x)*sqrt( - c**2*x**2 + 1))/(c*d*x + c*e*x**2 - d - e*x) 
,x)