3.27.38 \(\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [2638]

Optimal. Leaf size=96 \[ \frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {c+d x}} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {122, 120} \begin {gather*} \frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\text {ArcSin}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {c+d x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticF[ArcSin[(Sqrt[1 - e]*Sqrt[a + b*x])/Sqrt[a]], -((a*d)/((b*
c - a*d)*(1 - e)))])/(b*Sqrt[1 - e]*Sqrt[c + d*x])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e
+ f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx &=\frac {\sqrt {\frac {b (c+d x)}{b c-a d}} \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx}{\sqrt {c+d x}}\\ &=\frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {c+d x}}\\ \end {align*}

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Mathematica [A]
time = 11.74, size = 126, normalized size = 1.31 \begin {gather*} -\frac {2 \sqrt {c+d x} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} F\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {a}{-1+e}}}{\sqrt {a+b x}}\right )|\frac {(b c-a d) (-1+e)}{a d}\right )}{d \sqrt {-\frac {a}{-1+e}} \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {e+\frac {b (-1+e) x}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[c + d*x]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticF[ArcSin[Sqrt[-(a/(-1 + e))]/Sqrt[a + b*x]], (
(b*c - a*d)*(-1 + e))/(a*d)])/(d*Sqrt[-(a/(-1 + e))]*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[e + (b*(-1 + e)*x)
/a])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(206\) vs. \(2(83)=166\).
time = 0.10, size = 207, normalized size = 2.16

method	result	size
default	\(\frac {2 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) b \left (-1+e \right )}{a d e -b c e +b c}}\, \EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) \left (a d e -b c e +b c \right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) b d \left (-1+e \right )}\)	\(207\)
elliptic	\(\frac {2 \sqrt {\frac {\left (d x +c \right ) \left (b x +a \right ) \left (b e x +a e -b x \right )}{a}}\, \left (-\frac {c}{d}+\frac {a e}{b \left (-1+e \right )}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {c}{d}+\frac {a e}{b \left (-1+e \right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {c}{d}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {x^{3} d \,b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}\)	\(341\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.16, size = 384, normalized size = 4.00 \begin {gather*} \frac {2 \, a \sqrt {\frac {b^{2} d e - b^{2} d}{a}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} - {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2}\right )} e\right )}}{3 \, {\left (b^{2} d^{2} e^{2} - 2 \, b^{2} d^{2} e + b^{2} d^{2}\right )}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (2 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{2} - 3 \, {\left (2 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} e\right )}}{27 \, {\left (b^{3} d^{3} e^{3} - 3 \, b^{3} d^{3} e^{2} + 3 \, b^{3} d^{3} e - b^{3} d^{3}\right )}}, -\frac {3 \, b d x + b c + a d - {\left (3 \, b d x + b c + 2 \, a d\right )} e}{3 \, {\left (b d e - b d\right )}}\right )}{b^{2} d e - b^{2} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt((b^2*d*e - b^2*d)/a)*weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^
2*d^2)*e^2 - (2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*e)/(b^2*d^2*e^2 - 2*b^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^3 - 3*a
*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3 + 3*(2*b^3*
c^3 - 5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*e^2 - 3*(2*b^3*c^3 - 4*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*e)/
(b^3*d^3*e^3 - 3*b^3*d^3*e^2 + 3*b^3*d^3*e - b^3*d^3), -1/3*(3*b*d*x + b*c + a*d - (3*b*d*x + b*c + 2*a*d)*e)/
(b*d*e - b*d))/(b^2*d*e - b^2*d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a + b x} \sqrt {c + d x} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}\,\sqrt {a+b\,x}\,\sqrt {c+d\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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