3.33 \(\int \frac{A+B x}{\sqrt{a+b x} \sqrt{c+\frac{b (-1+c) x}{a}} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)

Optimal. Leaf size=145 \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) F\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)}-\frac{2 a^{3/2} B E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)} \]

[Out]

(-2*a^(3/2)*B*EllipticE[ArcSin[(Sqrt[1 - c]*Sqrt[a + b*x])/Sqrt[a]], (1 - e)/(1
- c)])/(b^2*Sqrt[1 - c]*(1 - e)) + (2*Sqrt[a]*(a*B*e + A*(b - b*e))*EllipticF[Ar
cSin[(Sqrt[1 - c]*Sqrt[a + b*x])/Sqrt[a]], (1 - e)/(1 - c)])/(b^2*Sqrt[1 - c]*(1
- e))

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Rubi [A]  time = 0.68064, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) F\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)}-\frac{2 a^{3/2} B E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^(3/2)*B*EllipticE[ArcSin[(Sqrt[1 - c]*Sqrt[a + b*x])/Sqrt[a]], (1 - e)/(1
- c)])/(b^2*Sqrt[1 - c]*(1 - e)) + (2*Sqrt[a]*(a*B*e + A*(b - b*e))*EllipticF[Ar
cSin[(Sqrt[1 - c]*Sqrt[a + b*x])/Sqrt[a]], (1 - e)/(1 - c)])/(b^2*Sqrt[1 - c]*(1
- e))

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Rubi in Sympy [A]  time = 84.3629, size = 107, normalized size = 0.74 \[ - \frac{2 B a^{\frac{3}{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{a + b x} \sqrt{- c + 1}}{\sqrt{a}} \right )}\middle | \frac{e - 1}{c - 1}\right )}{b^{2} \sqrt{- c + 1} \left (- e + 1\right )} + \frac{2 \sqrt{a} \left (A b \left (- e + 1\right ) + B a e\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{a + b x} \sqrt{- e + 1}}{\sqrt{a}} \right )}\middle | \frac{c - 1}{e - 1}\right )}{b^{2} \left (- e + 1\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/(b*x+a)**(1/2)/(c+b*(-1+c)*x/a)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)

[Out]

-2*B*a**(3/2)*elliptic_e(asin(sqrt(a + b*x)*sqrt(-c + 1)/sqrt(a)), (e - 1)/(c -
1))/(b**2*sqrt(-c + 1)*(-e + 1)) + 2*sqrt(a)*(A*b*(-e + 1) + B*a*e)*elliptic_f(a
sin(sqrt(a + b*x)*sqrt(-e + 1)/sqrt(a)), (c - 1)/(e - 1))/(b**2*(-e + 1)**(3/2))

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Mathematica [C]  time = 2.3075, size = 309, normalized size = 2.13 \[ -\frac{2 \sqrt{\frac{a}{c-1}} (a+b x)^{3/2} \left (\frac{i (e-1) \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} (a B c+A (b-b c)) F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{c-1}}}{\sqrt{a+b x}}\right )|\frac{c-1}{e-1}\right )}{\sqrt{a+b x}}-B \sqrt{\frac{a}{c-1}} \left (\frac{a}{a+b x}+c-1\right ) \left (\frac{a}{a+b x}+e-1\right )-\frac{i a B (e-1) \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{c-1}}}{\sqrt{a+b x}}\right )|\frac{c-1}{e-1}\right )}{\sqrt{a+b x}}\right )}{a b^2 (e-1) \sqrt{\frac{b (c-1) x}{a}+c} \sqrt{\frac{b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.245, size = 624, normalized size = 4.3 \[ -2\,{\frac{a}{\sqrt{bx+a} \left ( -1+c \right ) ^{2}{b}^{2} \left ( -1+e \right ) } \left ( A{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) b{c}^{2}-A{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) bce-B{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) a{c}^{2}+B{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ace-A{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) bc+A{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) be+B{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ac-B{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ae-B{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ac+B{\it EllipticE} \left ( \sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}},\sqrt{-{\frac{c-e}{-1+e}}} \right ) ae \right ) \sqrt{{\frac{ \left ( -1+c \right ) \left ( bex+ae-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( -1+c \right ) }{a}}}\sqrt{-{\frac{ \left ( -1+e \right ) \left ( bcx+ac-bx \right ) }{a \left ( c-e \right ) }}}{\frac{1}{\sqrt{{\frac{bcx+ac-bx}{a}}}}}{\frac{1}{\sqrt{{\frac{bex+ae-bx}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x + A}{\sqrt{b x + a} \sqrt{\frac{a c +{\left (b c - b\right )} x}{a}} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="fricas")

[Out]

integral((B*x + A)/(sqrt(b*x + a)*sqrt((a*c + (b*c - b)*x)/a)*sqrt((a*e + (b*e -
b)*x)/a)), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{B x + A}{\sqrt{b x + a} \sqrt{\frac{b{\left (c - 1\right )} x}{a} + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*sqrt(b*(c - 1)*x/a + c)*sqrt(b*(e - 1)*x/a + e)),x, algorithm="giac")

[Out]

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