\(\int \frac {A+B x}{\sqrt {e x} (c+d x) \sqrt {a-b x^2}} \, dx\) [250]

Optimal result

Integrand size = 32, antiderivative size = 177 \[ \int \frac {A+B x}{\sqrt {e x} (c+d x) \sqrt {a-b x^2}} \, dx=\frac {2 \sqrt [4]{a} B \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {e} \sqrt {a-b x^2}}-\frac {2 \sqrt [4]{a} (B c-A d) \sqrt {\frac {a-b x^2}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c d \sqrt {e} \sqrt {a-b x^2}} \] Output:

2*a^(1/4)*B*((-b*x^2+a)/a)^(1/2)*EllipticF(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^( 
1/2),I)/b^(1/4)/d/e^(1/2)/(-b*x^2+a)^(1/2)-2*a^(1/4)*(-A*d+B*c)*((-b*x^2+a 
)/a)^(1/2)*EllipticPi(b^(1/4)*(e*x)^(1/2)/a^(1/4)/e^(1/2),-a^(1/2)*d/b^(1/ 
2)/c,I)/b^(1/4)/c/d/e^(1/2)/(-b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.90 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.86 \[ \int \frac {A+B x}{\sqrt {e x} (c+d x) \sqrt {a-b x^2}} \, dx=\frac {2 i \sqrt {1-\frac {a}{b x^2}} x^{3/2} \left (A d \operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )+(B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} c}{\sqrt {a} d},i \text {arcsinh}\left (\frac {\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}}}{\sqrt {x}}\right ),-1\right )\right )}{\sqrt {-\frac {\sqrt {a}}{\sqrt {b}}} c d \sqrt {e x} \sqrt {a-b x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 1.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.01, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2354, 2229, 765, 762, 1543, 1542}

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.

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

\(\Big \downarrow \) 2354

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

\(\Big \downarrow \) 2229

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {2 \left (\frac {\sqrt [4]{a} B \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}-\frac {e (B c-A d) \int \frac {1}{(c e+d x e) \sqrt {a-b x^2}}d\sqrt {e x}}{d}\right )}{e}\)

\(\Big \downarrow \) 1543

\(\displaystyle \frac {2 \left (\frac {\sqrt [4]{a} B \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}-\frac {e \sqrt {1-\frac {b x^2}{a}} (B c-A d) \int \frac {1}{(c e+d x e) \sqrt {1-\frac {b x^2}{a}}}d\sqrt {e x}}{d \sqrt {a-b x^2}}\right )}{e}\)

\(\Big \downarrow \) 1542

\(\displaystyle \frac {2 \left (\frac {\sqrt [4]{a} B \sqrt {e} \sqrt {1-\frac {b x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} d \sqrt {a-b x^2}}-\frac {\sqrt [4]{a} \sqrt {e} \sqrt {1-\frac {b x^2}{a}} (B c-A d) \operatorname {EllipticPi}\left (-\frac {\sqrt {a} d}{\sqrt {b} c},\arcsin \left (\frac {\sqrt [4]{b} \sqrt {e x}}{\sqrt [4]{a} \sqrt {e}}\right ),-1\right )}{\sqrt [4]{b} c d \sqrt {a-b x^2}}\right )}{e}\)

Input:

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

Output:

(2*((a^(1/4)*B*Sqrt[e]*Sqrt[1 - (b*x^2)/a]*EllipticF[ArcSin[(b^(1/4)*Sqrt[ 
e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*d*Sqrt[a - b*x^2]) - (a^(1/4)*(B*c 
 - A*d)*Sqrt[e]*Sqrt[1 - (b*x^2)/a]*EllipticPi[-((Sqrt[a]*d)/(Sqrt[b]*c)), 
 ArcSin[(b^(1/4)*Sqrt[e*x])/(a^(1/4)*Sqrt[e])], -1])/(b^(1/4)*c*d*Sqrt[a - 
 b*x^2])))/e
 

Defintions of rubi rules used

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1542

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[ 
{q = Rt[-c/a, 4]}, Simp[(1/(d*Sqrt[a]*q))*EllipticPi[-e/(d*q^2), ArcSin[q*x 
], -1], x]] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && GtQ[a, 0]
 

rule 1543

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 
Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4]   Int[1/((d + e*x^2)*Sqrt[1 + c*(x^4/a) 
]), x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] &&  !GtQ[a, 0]
 

rule 2229

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) 
, x_Symbol] :> Simp[B/e   Int[1/Sqrt[a + c*x^4], x], x] + Simp[(e*A - d*B)/ 
e   Int[1/((d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e, A, B} 
, x] && NeQ[c*d^2 - a*e^2, 0] && NegQ[c/a]
 

rule 2354

Int[(Px_)*((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2) 
^(p_.), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/e   Subst[Int[(Px /. 
 x -> x^k/e)*x^(k*(m + 1) - 1)*(c + d*(x^k/e))^n*(a + b*(x^(2*k)/e^2))^p, x 
], x, (e*x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, n, p}, x] && PolyQ[Px, x] 
&& FractionQ[m]
 

Maple [A] (verified)

Time = 1.30 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.40

method	result	size
default	\(\frac {\left (A \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{\sqrt {a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) b d -B \sqrt {a b}\, \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) d +B \operatorname {EllipticF}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right ) b c -B \operatorname {EllipticPi}\left (\sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}, \frac {\sqrt {a b}\, d}{\sqrt {a b}\, d -b c}, \frac {\sqrt {2}}{2}\right ) b c \right ) \sqrt {-\frac {b x}{\sqrt {a b}}}\, \sqrt {\frac {-b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {2}\, \sqrt {\frac {b x +\sqrt {a b}}{\sqrt {a b}}}\, \sqrt {a b}}{\sqrt {-b \,x^{2}+a}\, d b \left (-\sqrt {a b}\, d +b c \right ) \sqrt {e x}}\)	\(248\)
elliptic	\(\frac {\sqrt {x e \left (-b \,x^{2}+a \right )}\, \left (\frac {B \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, \frac {\sqrt {2}}{2}\right )}{d b \sqrt {-b e \,x^{3}+a e x}}+\frac {\left (A d -B c \right ) \sqrt {a b}\, \sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}\, \sqrt {-\frac {b x}{\sqrt {a b}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {\left (x +\frac {\sqrt {a b}}{b}\right ) b}{\sqrt {a b}}}, -\frac {\sqrt {a b}}{b \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{d^{2} b \sqrt {-b e \,x^{3}+a e x}\, \left (\frac {c}{d}-\frac {\sqrt {a b}}{b}\right )}\right )}{\sqrt {e x}\, \sqrt {-b \,x^{2}+a}}\)	\(296\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(e)*(int((sqrt(x)*sqrt(a - b*x**2))/(a*c*x + a*d*x**2 - b*c*x**3 - b* 
d*x**4),x)*a + int((sqrt(x)*sqrt(a - b*x**2))/(a*c + a*d*x - b*c*x**2 - b* 
d*x**3),x)*b))/e