3.34 \(\int \frac{A+B x}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\)
Optimal. Leaf size=221 \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) \sqrt{\frac{b (c+d x)}{b c-a d}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right ),-\frac{a d}{(1-e) (b c-a d)}\right )}{b^2 (1-e)^{3/2} \sqrt{c+d x}}-\frac{2 a B \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|-\frac{(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt{d} (1-e) \sqrt{c+d x}} \]
[Out]
(-2*a*B*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c
) + a*d]], -(((b*c - a*d)*(1 - e))/(a*d))])/(b^2*Sqrt[d]*(1 - e)*Sqrt[c + d*x]) + (2*Sqrt[a]*(a*B*e + A*(b - b
*e))*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^2*(1 - e)^(3/2)*Sqrt[c + d*x])
________________________________________________________________________________________
Rubi [A] time = 0.199431, antiderivative size = 221, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.128, Rules used =
{158, 114, 113, 121, 119} \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b^2 (1-e)^{3/2} \sqrt{c+d x}}-\frac{2 a B \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|-\frac{(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt{d} (1-e) \sqrt{c+d x}} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]
[Out]
(-2*a*B*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticE[ArcSin[(Sqrt[d]*Sqrt[a + b*x])/Sqrt[-(b*c
) + a*d]], -(((b*c - a*d)*(1 - e))/(a*d))])/(b^2*Sqrt[d]*(1 - e)*Sqrt[c + d*x]) + (2*Sqrt[a]*(a*B*e + A*(b - b
*e))*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^2*(1 - e)^(3/2)*Sqrt[c + d*x])
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 114
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], 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]
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 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]
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+d x} \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+d x}} \, dx}{b (1-e)}+\left (A+\frac{a B e}{b-b e}\right ) \int \frac{1}{\sqrt{a+b x} \sqrt{c+d x} \sqrt{e+\frac{b (-1+e) x}{a}}} \, dx\\ &=\frac{\left (\left (A+\frac{a B e}{b-b e}\right ) \sqrt{\frac{b (c+d x)}{b c-a d}}\right ) \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{\left (a B \sqrt{\frac{b (c+d x)}{b c-a d}} \sqrt{e+\frac{b (-1+e) x}{a}}\right ) \int \frac{\sqrt{\frac{b e}{-b (-1+e)+b e}+\frac{b^2 (-1+e) x}{a (-b (-1+e)+b e)}}}{\sqrt{a+b x} \sqrt{\frac{b c}{b c-a d}+\frac{b d x}{b c-a d}}} \, dx}{b (1-e) \sqrt{c+d x} \sqrt{\frac{b \left (e+\frac{b (-1+e) x}{a}\right )}{-b (-1+e)+b e}}}\\ &=-\frac{2 a B \sqrt{-b c+a d} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{-b c+a d}}\right )|-\frac{(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt{d} (1-e) \sqrt{c+d x}}+\frac{2 \sqrt{a} (a B e+A (b-b e)) \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^2 (1-e)^{3/2} \sqrt{c+d x}}\\ \end{align*}
Mathematica [C] time = 2.17453, size = 312, normalized size = 1.41 \[ -\frac{2 \sqrt{\frac{a}{e-1}} (a+b x)^{3/2} \left (\frac{i d \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} (a B e+A (b-b e)) \sqrt{\frac{b (c+d x)}{d (a+b x)}} \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{e-1}}}{\sqrt{a+b x}}\right ),\frac{(e-1) (b c-a d)}{a d}\right )}{\sqrt{a+b x}}-\frac{b B \sqrt{\frac{a}{e-1}} (c+d x) (a e+b (e-1) x)}{(a+b x)^2}-\frac{i a B d \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{(b c-a d) (e-1)}{a d}\right )}{\sqrt{a+b x}}\right )}{a b^2 d \sqrt{c+d x} \sqrt{\frac{b (e-1) x}{a}+e}} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*x)/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a]),x]
[Out]
(-2*Sqrt[a/(-1 + e)]*(a + b*x)^(3/2)*(-((b*B*Sqrt[a/(-1 + e)]*(c + d*x)*(a*e + b*(-1 + e)*x))/(a + b*x)^2) - (
I*a*B*d*Sqrt[(b*(c + d*x))/(d*(a + b*x))]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticE[I*ArcSinh[Sqrt[a/(-1
+ e)]/Sqrt[a + b*x]], ((b*c - a*d)*(-1 + e))/(a*d)])/Sqrt[a + b*x] + (I*d*(a*B*e + A*(b - b*e))*Sqrt[(b*(c +
d*x))/(d*(a + b*x))]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticF[I*ArcSinh[Sqrt[a/(-1 + e)]/Sqrt[a + b*x]]
, ((b*c - a*d)*(-1 + e))/(a*d)])/Sqrt[a + b*x]))/(a*b^2*d*Sqrt[c + d*x]*Sqrt[e + (b*(-1 + e)*x)/a])
________________________________________________________________________________________
Maple [B] time = 0.039, size = 940, normalized size = 4.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(e+b*(-1+e)*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)*(-1+e)/a)^(1/2)*(-(d*x+c)*
b*(-1+e)/(a*d*e-b*c*e+b*c))^(1/2)*(A*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*b*d*e^2-A*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))*b
^2*c*e^2-B*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^2*d*e^2+B*El
lipticF((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*b*c*e^2-A*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*b*d*e+2*A*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))*b^2*c*e+B*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^2*d*e-2*B*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*b*c*e-B*EllipticE((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^2*d*e+B*EllipticE((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
*b*c*e-A*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))*b^2*c+B*Elliptic
F((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*b*c-B*EllipticE((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*b*c)/((b*e*x+a*e-b*x)/a)^(1/2)/(b*d*x^2+a*d*x+b*
c*x+a*c)/(-1+e)^2/b^2/d
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{b x + a} \sqrt{d x + 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)/(d*x+c)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="maxima")
[Out]
integrate((B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)), x)
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B a x + A a\right )} \sqrt{b x + a} \sqrt{d x + c} \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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((B*x+A)/(b*x+a)^(1/2)/(d*x+c)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="fricas")
[Out]
integral((B*a*x + A*a)*sqrt(b*x + a)*sqrt(d*x + c)*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)
________________________________________________________________________________________
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)/(d*x+c)**(1/2)/(e+b*(-1+e)*x/a)**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B x + A}{\sqrt{b x + a} \sqrt{d x + 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)/(d*x+c)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="giac")
[Out]
integrate((B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)), x)