\(\int \frac {a+\frac {a b x}{2}}{\sqrt {2-b x} \sqrt {2+b x} \sqrt {c+d x}} \, dx\) [213]

Optimal result

Integrand size = 38, antiderivative size = 92 \[ \int \frac {a+\frac {a b x}{2}}{\sqrt {2-b x} \sqrt {2+b x} \sqrt {c+d x}} \, dx=-\frac {2 a \sqrt {b c+2 d} \sqrt {\frac {b (c+d x)}{b c+2 d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {2-b x}}{\sqrt {b c+2 d}}\right )|\frac {1}{4} \left (2+\frac {b c}{d}\right )\right )}{b \sqrt {d} \sqrt {c+d x}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.67 (sec) , antiderivative size = 289, normalized size of antiderivative = 3.14 \[ \int \frac {a+\frac {a b x}{2}}{\sqrt {2-b x} \sqrt {2+b x} \sqrt {c+d x}} \, dx=-\frac {a \left (d^2 \sqrt {-c-\frac {2 d}{b}} \left (4-b^2 x^2\right )+i b (b c+2 d) \sqrt {\frac {d (-2+b x)}{b (c+d x)}} \sqrt {\frac {d (2+b x)}{b (c+d x)}} (c+d x)^{3/2} E\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\frac {2 d}{b}}}{\sqrt {c+d x}}\right )|\frac {b c-2 d}{b c+2 d}\right )-4 i b d \sqrt {\frac {d (-2+b x)}{b (c+d x)}} \sqrt {\frac {d (2+b x)}{b (c+d x)}} (c+d x)^{3/2} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-c-\frac {2 d}{b}}}{\sqrt {c+d x}}\right ),\frac {b c-2 d}{b c+2 d}\right )\right )}{b d^2 \sqrt {-c-\frac {2 d}{b}} \sqrt {c+d x} \sqrt {4-b^2 x^2}} \] Input:

Integrate[(a + (a*b*x)/2)/(Sqrt[2 - b*x]*Sqrt[2 + b*x]*Sqrt[c + d*x]),x]
 

Output:

-((a*(d^2*Sqrt[-c - (2*d)/b]*(4 - b^2*x^2) + I*b*(b*c + 2*d)*Sqrt[(d*(-2 + 
 b*x))/(b*(c + d*x))]*Sqrt[(d*(2 + b*x))/(b*(c + d*x))]*(c + d*x)^(3/2)*El 
lipticE[I*ArcSinh[Sqrt[-c - (2*d)/b]/Sqrt[c + d*x]], (b*c - 2*d)/(b*c + 2* 
d)] - (4*I)*b*d*Sqrt[(d*(-2 + b*x))/(b*(c + d*x))]*Sqrt[(d*(2 + b*x))/(b*( 
c + d*x))]*(c + d*x)^(3/2)*EllipticF[I*ArcSinh[Sqrt[-c - (2*d)/b]/Sqrt[c + 
 d*x]], (b*c - 2*d)/(b*c + 2*d)]))/(b*d^2*Sqrt[-c - (2*d)/b]*Sqrt[c + d*x] 
*Sqrt[4 - b^2*x^2]))
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {35, 124, 27, 123}

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.

\(\displaystyle \int \frac {\frac {a b x}{2}+a}{\sqrt {2-b x} \sqrt {b x+2} \sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 35

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

\(\Big \downarrow \) 124

\(\displaystyle \frac {a \sqrt {\frac {b (c+d x)}{b c+2 d}} \int \frac {\sqrt {b x+2}}{2 \sqrt {2-b x} \sqrt {\frac {b c}{b c+2 d}+\frac {b d x}{b c+2 d}}}dx}{\sqrt {c+d x}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {a \sqrt {\frac {b (c+d x)}{b c+2 d}} \int \frac {\sqrt {b x+2}}{\sqrt {2-b x} \sqrt {\frac {b c}{b c+2 d}+\frac {b d x}{b c+2 d}}}dx}{2 \sqrt {c+d x}}\)

\(\Big \downarrow \) 123

\(\displaystyle -\frac {2 a \sqrt {b c+2 d} \sqrt {\frac {b (c+d x)}{b c+2 d}} E\left (\arcsin \left (\frac {\sqrt {d} \sqrt {2-b x}}{\sqrt {b c+2 d}}\right )|\frac {1}{4} \left (\frac {b c}{d}+2\right )\right )}{b \sqrt {d} \sqrt {c+d x}}\)

Input:

Int[(a + (a*b*x)/2)/(Sqrt[2 - b*x]*Sqrt[2 + b*x]*Sqrt[c + d*x]),x]
 

Output:

(-2*a*Sqrt[b*c + 2*d]*Sqrt[(b*(c + d*x))/(b*c + 2*d)]*EllipticE[ArcSin[(Sq 
rt[d]*Sqrt[2 - b*x])/Sqrt[b*c + 2*d]], (2 + (b*c)/d)/4])/(b*Sqrt[d]*Sqrt[c 
 + d*x])
 

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 35

Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> 
 Simp[(b/d)^m   Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} 
, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] &&  !(IntegerQ[n] && SimplerQ[a + 
b*x, c + d*x])
 

rule 123

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] 
/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, 
b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !L 
tQ[-(b*c - a*d)/d, 0] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d 
), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])
 

rule 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(287\) vs. \(2(77)=154\).

Time = 1.45 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.13

method	result	size
default	\(\frac {a \left (\operatorname {EllipticE}\left (\sqrt {\frac {\left (x d +c \right ) b}{b c -2 d}}, \sqrt {\frac {b c -2 d}{b c +2 d}}\right ) b^{2} c^{2}-4 \operatorname {EllipticF}\left (\sqrt {\frac {\left (x d +c \right ) b}{b c -2 d}}, \sqrt {\frac {b c -2 d}{b c +2 d}}\right ) b c d -4 \operatorname {EllipticE}\left (\sqrt {\frac {\left (x d +c \right ) b}{b c -2 d}}, \sqrt {\frac {b c -2 d}{b c +2 d}}\right ) d^{2}+8 \operatorname {EllipticF}\left (\sqrt {\frac {\left (x d +c \right ) b}{b c -2 d}}, \sqrt {\frac {b c -2 d}{b c +2 d}}\right ) d^{2}\right ) \sqrt {-\frac {d \left (b x +2\right )}{b c -2 d}}\, \sqrt {-\frac {d \left (b x -2\right )}{b c +2 d}}\, \sqrt {\frac {\left (x d +c \right ) b}{b c -2 d}}\, \sqrt {-b x +2}\, \sqrt {b x +2}\, \sqrt {x d +c}}{b \,d^{2} \left (x^{3} d \,b^{2}+b^{2} c \,x^{2}-4 x d -4 c \right )}\)	\(288\)
elliptic	\(\frac {\sqrt {-\left (x d +c \right ) \left (b^{2} x^{2}-4\right )}\, \left (\frac {2 a \left (\frac {c}{d}-\frac {2}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {2}{b}}}\, \sqrt {\frac {x -\frac {2}{b}}{-\frac {c}{d}-\frac {2}{b}}}\, \sqrt {\frac {x +\frac {2}{b}}{-\frac {c}{d}+\frac {2}{b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {2}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {2}{b}}{-\frac {c}{d}-\frac {2}{b}}}\right )}{\sqrt {-x^{3} d \,b^{2}-b^{2} c \,x^{2}+4 x d +4 c}}+\frac {a b \left (\frac {c}{d}-\frac {2}{b}\right ) \sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {2}{b}}}\, \sqrt {\frac {x -\frac {2}{b}}{-\frac {c}{d}-\frac {2}{b}}}\, \sqrt {\frac {x +\frac {2}{b}}{-\frac {c}{d}+\frac {2}{b}}}\, \left (\left (-\frac {c}{d}-\frac {2}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {2}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {2}{b}}{-\frac {c}{d}-\frac {2}{b}}}\right )+\frac {2 \operatorname {EllipticF}\left (\sqrt {\frac {x +\frac {c}{d}}{\frac {c}{d}-\frac {2}{b}}}, \sqrt {\frac {-\frac {c}{d}+\frac {2}{b}}{-\frac {c}{d}-\frac {2}{b}}}\right )}{b}\right )}{\sqrt {-x^{3} d \,b^{2}-b^{2} c \,x^{2}+4 x d +4 c}}\right )}{\sqrt {b x +2}\, \sqrt {-b x +2}\, \sqrt {x d +c}}\)	\(447\)

Input:

int((a+1/2*a*b*x)/(-b*x+2)^(1/2)/(b*x+2)^(1/2)/(d*x+c)^(1/2),x,method=_RET 
URNVERBOSE)
 

Output:

a*(EllipticE(((d*x+c)*b/(b*c-2*d))^(1/2),((b*c-2*d)/(b*c+2*d))^(1/2))*b^2* 
c^2-4*EllipticF(((d*x+c)*b/(b*c-2*d))^(1/2),((b*c-2*d)/(b*c+2*d))^(1/2))*b 
*c*d-4*EllipticE(((d*x+c)*b/(b*c-2*d))^(1/2),((b*c-2*d)/(b*c+2*d))^(1/2))* 
d^2+8*EllipticF(((d*x+c)*b/(b*c-2*d))^(1/2),((b*c-2*d)/(b*c+2*d))^(1/2))*d 
^2)*(-d*(b*x+2)/(b*c-2*d))^(1/2)*(-d*(b*x-2)/(b*c+2*d))^(1/2)*((d*x+c)*b/( 
b*c-2*d))^(1/2)*(-b*x+2)^(1/2)*(b*x+2)^(1/2)*(d*x+c)^(1/2)/b/d^2/(b^2*d*x^ 
3+b^2*c*x^2-4*d*x-4*c)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (77) = 154\).

Time = 0.07 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.11 \[ \int \frac {a+\frac {a b x}{2}}{\sqrt {2-b x} \sqrt {2+b x} \sqrt {c+d x}} \, dx=\frac {3 \, \sqrt {-b^{2} d} a b d {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} + 12 \, d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {8 \, {\left (b^{2} c^{3} - 36 \, c d^{2}\right )}}{27 \, b^{2} d^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} + 12 \, d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {8 \, {\left (b^{2} c^{3} - 36 \, c d^{2}\right )}}{27 \, b^{2} d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )\right ) + {\left (a b c - 6 \, a d\right )} \sqrt {-b^{2} d} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} + 12 \, d^{2}\right )}}{3 \, b^{2} d^{2}}, -\frac {8 \, {\left (b^{2} c^{3} - 36 \, c d^{2}\right )}}{27 \, b^{2} d^{3}}, \frac {3 \, d x + c}{3 \, d}\right )}{3 \, b^{2} d^{2}} \] Input:

integrate((a+1/2*a*b*x)/(-b*x+2)^(1/2)/(b*x+2)^(1/2)/(d*x+c)^(1/2),x, algo 
rithm="fricas")
 

Output:

1/3*(3*sqrt(-b^2*d)*a*b*d*weierstrassZeta(4/3*(b^2*c^2 + 12*d^2)/(b^2*d^2) 
, -8/27*(b^2*c^3 - 36*c*d^2)/(b^2*d^3), weierstrassPInverse(4/3*(b^2*c^2 + 
 12*d^2)/(b^2*d^2), -8/27*(b^2*c^3 - 36*c*d^2)/(b^2*d^3), 1/3*(3*d*x + c)/ 
d)) + (a*b*c - 6*a*d)*sqrt(-b^2*d)*weierstrassPInverse(4/3*(b^2*c^2 + 12*d 
^2)/(b^2*d^2), -8/27*(b^2*c^3 - 36*c*d^2)/(b^2*d^3), 1/3*(3*d*x + c)/d))/( 
b^2*d^2)
 

Sympy [F]

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

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

Output:

a*Integral(sqrt(b*x + 2)/(sqrt(c + d*x)*sqrt(-b*x + 2)), x)/2
 

Maxima [F]

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

integrate((a+1/2*a*b*x)/(-b*x+2)^(1/2)/(b*x+2)^(1/2)/(d*x+c)^(1/2),x, algo 
rithm="maxima")
 

Output:

1/2*integrate((a*b*x + 2*a)/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(d*x + c)), 
x)
 

Giac [F]

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

integrate((a+1/2*a*b*x)/(-b*x+2)^(1/2)/(b*x+2)^(1/2)/(d*x+c)^(1/2),x, algo 
rithm="giac")
 

Output:

integrate(1/2*(a*b*x + 2*a)/(sqrt(b*x + 2)*sqrt(-b*x + 2)*sqrt(d*x + c)), 
x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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