3.2638 \(\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\)

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

[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 = 0.07, 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 = {121, 119} \[ \frac {2 \sqrt {a} \sqrt {\frac {b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {c+d x}} \]

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 119

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*Rt[-(b/d
), 2]*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))])/(
b*Sqrt[(b*e - a*f)/b]), x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] &
& PosQ[-(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 121

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 = 0.53, size = 126, normalized size = 1.31 \[ -\frac {2 \sqrt {c+d x} \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {a}{e-1}}}{\sqrt {a+b x}}\right ),\frac {(e-1) (b c-a d)}{a d}\right )}{d \sqrt {-\frac {a}{e-1}} \sqrt {\frac {b (e-1) x}{a}+e} \sqrt {\frac {b (c+d x)}{d (a+b x)}}} \]

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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fricas [F]  time = 1.11, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} a \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{2} c e + {\left (b^{2} d e - b^{2} d\right )} x^{3} - {\left (b^{2} c + a b d - {\left (b^{2} c + 2 \, a b d\right )} e\right )} x^{2} - {\left (a b c - {\left (2 \, a b c + a^{2} d\right )} e\right )} x}, x\right ) \]

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]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]

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*(e - 1)*x/a + e)), x)

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maple [B]  time = 0.12, size = 207, normalized size = 2.16 \[ \frac {2 \sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (e -1\right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) \left (e -1\right ) b}{a d e -b c e +b c}}\, \left (a d e -b c e +b c \right ) \EllipticF \left (\sqrt {\frac {\left (b e x +a e -b x \right ) d}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{a d}}\right )}{\sqrt {\frac {b e x +a e -b x}{a}}\, \left (b d \,x^{2}+a d x +b c x +a c \right ) \left (e -1\right ) b d} \]

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*(e-1)*x/a)^(1/2),x)

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b {\left (e - 1\right )} x}{a} + e}}\,{d x} \]

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*(e - 1)*x/a + e)), x)

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

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

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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