3.2641 \(\int \frac {\sqrt {c+d x}}{\sqrt {a+b x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.06, 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 = {114, 113} \[ \frac {2 \sqrt {a} \sqrt {c+d x} E\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {a+b x}}{\sqrt {a}}\right )|-\frac {a d}{(b c-a d) (1-e)}\right )}{b \sqrt {1-e} \sqrt {\frac {b (c+d x)}{b c-a d}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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]

Rubi steps

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

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Mathematica [B]  time = 1.38, size = 200, normalized size = 2.08 \[ \frac {2 \sqrt {\frac {\frac {a}{a+b x}+e-1}{e-1}} \left (b \sqrt {a+b x} (c+d x) \sqrt {a-\frac {b c}{d}} \sqrt {\frac {a e+b (e-1) x}{(e-1) (a+b x)}}-(a+b x) (b c-a d) \sqrt {\frac {b (c+d x)}{d (a+b x)}} E\left (\sin ^{-1}\left (\frac {\sqrt {a-\frac {b c}{d}}}{\sqrt {a+b x}}\right )|\frac {a d}{(b c-a d) (e-1)}\right )\right )}{b^2 \sqrt {c+d x} \sqrt {a-\frac {b c}{d}} \sqrt {\frac {b (e-1) x}{a}+e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 1.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b x + a} \sqrt {d x + c} a \sqrt {\frac {a e + {\left (b e - b\right )} x}{a}}}{a^{2} e + {\left (b^{2} e - b^{2}\right )} x^{2} + {\left (2 \, a b e - a b\right )} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.03, size = 822, normalized size = 8.56 \[ -\frac {2 \sqrt {d x +c}\, \sqrt {b x +a}\, \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^{2} d^{2} e^{2} \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 )-2 a b c d \,e^{2} \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 )+b^{2} c^{2} e^{2} \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 )+a^{2} d^{2} e \EllipticE \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 )-a^{2} d^{2} e \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 )-a b c d e \EllipticE \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 )+3 a b c d e \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 )-2 b^{2} c^{2} e \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 )+a b c d \EllipticE \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 )-a b c d \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 )+b^{2} c^{2} \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 )\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 )^{2} b^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d*x+c)^(1/2)*(b*x+a)^(1/2)*((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2)*(-(b*x+a)*(e-1)/a)^(1/2)*(-(d*x+c)*
(e-1)/(a*d*e-b*c*e+b*c)*b)^(1/2)*(EllipticF(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d
)^(1/2))*a^2*d^2*e^2-2*EllipticF(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*a*
b*c*d*e^2+EllipticF(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*b^2*c^2*e^2-Ell
ipticF(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*a^2*d^2*e+3*EllipticF(((b*e*
x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*a*b*c*d*e-2*EllipticF(((b*e*x+a*e-b*x)/(a
*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*b^2*c^2*e+EllipticE(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)
*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*a^2*d^2*e-EllipticE(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d
*e-b*c*e+b*c)/a/d)^(1/2))*a*b*c*d*e-EllipticF(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a
/d)^(1/2))*a*b*c*d+EllipticF(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*b^2*c^
2+EllipticE(((b*e*x+a*e-b*x)/(a*d*e-b*c*e+b*c)*d)^(1/2),((a*d*e-b*c*e+b*c)/a/d)^(1/2))*a*b*c*d)/((b*e*x+a*e-b*
x)/a)^(1/2)/(b*d*x^2+a*d*x+b*c*x+a*c)/(e-1)^2/b^2/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(b*x+a)^(1/2)/(e+b*(-1+e)*x/a)^(1/2),x, algorithm="maxima")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(c + d*x)/(sqrt(a + b*x)*sqrt(e + b*e*x/a - b*x/a)), x)

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