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

[Out]

(-2*a*B*Sqrt[-(b*c) + a*d]*Sqrt[(b*(c + d*x))/(b*c - a*d)]*EllipticE[ArcSin[(Sqr
t[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], -(((b*c - a*d)*(1 - e))/(a*d))])/(b^2*S
qrt[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 = 1.06753, 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 \[ \frac{2 \sqrt{a} (a B e+A (b-b e)) \sqrt{\frac{b (c+d x)}{b c-a d}} F\left (\sin ^{-1}\left (\frac{\sqrt{1-e} \sqrt{a+b x}}{\sqrt{a}}\right )|-\frac{a d}{(b c-a d) (1-e)}\right )}{b^2 (1-e)^{3/2} \sqrt{c+d x}}-\frac{2 a B \sqrt{a d-b c} \sqrt{\frac{b (c+d x)}{b c-a d}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x}}{\sqrt{a d-b c}}\right )|-\frac{(b c-a d) (1-e)}{a d}\right )}{b^2 \sqrt{d} (1-e) \sqrt{c+d x}} \]

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[(Sqr
t[d]*Sqrt[a + b*x])/Sqrt[-(b*c) + a*d]], -(((b*c - a*d)*(1 - e))/(a*d))])/(b^2*S
qrt[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 in Sympy [A]  time = 134.127, size = 194, normalized size = 0.88 \[ - \frac{2 B a \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{a d - b c} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}\middle | \frac{\left (e - 1\right ) \left (- a d + b c\right )}{a d}\right )}{b^{2} \sqrt{d} \sqrt{c + d x} \left (- e + 1\right )} + \frac{2 \sqrt{\frac{b \left (- c - d x\right )}{a d - b c}} \sqrt{a d - b c} \left (A b \left (- e + 1\right ) + B a e\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{d} \sqrt{a + b x}}{\sqrt{a d - b c}} \right )}\middle | \frac{\left (e - 1\right ) \left (- a d + b c\right )}{a d}\right )}{b^{2} \sqrt{d} \sqrt{c + d x} \left (- e + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((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*a*sqrt(b*(-c - d*x)/(a*d - b*c))*sqrt(a*d - b*c)*elliptic_e(asin(sqrt(d)*sq
rt(a + b*x)/sqrt(a*d - b*c)), (e - 1)*(-a*d + b*c)/(a*d))/(b**2*sqrt(d)*sqrt(c +
d*x)*(-e + 1)) + 2*sqrt(b*(-c - d*x)/(a*d - b*c))*sqrt(a*d - b*c)*(A*b*(-e + 1)
+ B*a*e)*elliptic_f(asin(sqrt(d)*sqrt(a + b*x)/sqrt(a*d - b*c)), (e - 1)*(-a*d
+ b*c)/(a*d))/(b**2*sqrt(d)*sqrt(c + d*x)*(-e + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 3.82881, size = 312, normalized size = 1.41 \[ -\frac{2 \sqrt{\frac{a}{e-1}} (a+b x)^{3/2} \left (\frac{i d \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} (a B e+A (b-b e)) \sqrt{\frac{b (c+d x)}{d (a+b x)}} F\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{(b c-a d) (e-1)}{a d}\right )}{\sqrt{a+b x}}-\frac{b B \sqrt{\frac{a}{e-1}} (c+d x) (a e+b (e-1) x)}{(a+b x)^2}-\frac{i a B d \sqrt{\frac{\frac{a}{a+b x}+e-1}{e-1}} \sqrt{\frac{b (c+d x)}{d (a+b x)}} E\left (i \sinh ^{-1}\left (\frac{\sqrt{\frac{a}{e-1}}}{\sqrt{a+b x}}\right )|\frac{(b c-a d) (e-1)}{a d}\right )}{\sqrt{a+b x}}\right )}{a b^2 d \sqrt{c+d x} \sqrt{\frac{b (e-1) x}{a}+e}} \]

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))*Sqr
t[(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 = 0.174, size = 940, normalized size = 4.3 \[ \text{result too large to display} \]

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*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^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*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))*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*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-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. \[ \int \frac{B x + A}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{b{\left (e - 1\right )} x}{a} + e}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(b*x + a)*sqrt(d*x + c)*sqrt(b*(e - 1)*x/a + e)),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. \[{\rm integral}\left (\frac{B x + A}{\sqrt{b x + a} \sqrt{d x + c} \sqrt{\frac{a e +{\left (b e - b\right )} x}{a}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

[Out]

Timed out

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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