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 \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right ),\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)}-\frac{2 a^{3/2} B E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)} \]

[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[ArcSin[(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 = 0.108071, 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, Rules used = {158, 113, 119} \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) F\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)}-\frac{2 a^{3/2} B E\left (\sin ^{-1}\left (\frac{\sqrt{1-c} \sqrt{a+b x}}{\sqrt{a}}\right )|\frac{1-e}{1-c}\right )}{b^2 \sqrt{1-c} (1-e)} \]

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

Rule 158

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

Rule 113

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

Rubi steps

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

Mathematica [C]  time = 1.44704, size = 309, normalized size = 2.13 \[ -\frac{2 \sqrt{\frac{a}{c-1}} (a+b x)^{3/2} \left (\frac{i (e-1) \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} (a B c+A (b-b c)) \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{c-1}}}{\sqrt{a+b x}}\right ),\frac{c-1}{e-1}\right )}{\sqrt{a+b x}}-B \sqrt{\frac{a}{c-1}} \left (\frac{a}{a+b x}+c-1\right ) \left (\frac{a}{a+b x}+e-1\right )-\frac{i a B (e-1) \sqrt{\frac{\frac{a}{a+b x}+c-1}{c-1}} \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{c-1}}}{\sqrt{a+b x}}\right )|\frac{c-1}{e-1}\right )}{\sqrt{a+b x}}\right )}{a b^2 (e-1) \sqrt{\frac{b (c-1) x}{a}+c} \sqrt{\frac{b (e-1) x}{a}+e}} \]

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)]/Sqrt[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)]*EllipticF[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 = 0.047, size = 624, normalized size = 4.3 \begin{align*} -2\,{\frac{a}{\sqrt{bx+a} \left ( -1+e \right ) \left ( c-1 \right ) ^{2}{b}^{2}} \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 ( c-1 \right ) \left ( bxe+ae-bx \right ) }{a \left ( c-e \right ) }}}\sqrt{-{\frac{ \left ( bx+a \right ) \left ( c-1 \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{bxe+ae-bx}{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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. \begin{align*}{\rm integral}\left (\frac{{\left (B a^{2} x + A a^{2}\right )} \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}}}{a^{3} c e -{\left (b^{3} c - b^{3} -{\left (b^{3} c - b^{3}\right )} e\right )} x^{3} -{\left (2 \, a b^{2} c - a b^{2} -{\left (3 \, a b^{2} c - 2 \, a b^{2}\right )} e\right )} x^{2} -{\left (a^{2} b c -{\left (3 \, a^{2} b c - a^{2} b\right )} e\right )} x}, x\right ) \end{align*}

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, algorithm="fricas")

[Out]

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

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

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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, 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)