3.27.40 \(\int \frac {\sqrt {e+\frac {b (-1+e) x}{a}}}{\sqrt {a+b x} \sqrt {c+\frac {b (-1+c) x}{a}}} \, dx\) [2640]

Optimal. Leaf size=58 \[ \frac {2 \sqrt {a} E\left (\sin ^{-1}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.025, Rules used = {114} \begin {gather*} \frac {2 \sqrt {a} E\left (\text {ArcSin}\left (\frac {\sqrt {1-c} \sqrt {a+b x}}{\sqrt {a}}\right )|\frac {1-e}{1-c}\right )}{b \sqrt {1-c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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])

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(58)=116\).
time = 11.65, size = 191, normalized size = 3.29 \begin {gather*} -\frac {2 (a+b x)^{3/2} \left (-\frac {\sqrt {-\frac {a}{-1+e}} \left (-1+c+\frac {a}{a+b x}\right ) \left (-1+e+\frac {a}{a+b x}\right )}{-1+c}+\frac {a \sqrt {\frac {-1+c+\frac {a}{a+b x}}{-1+c}} \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} E\left (\sin ^{-1}\left (\frac {\sqrt {-\frac {a}{-1+e}}}{\sqrt {a+b x}}\right )|\frac {-1+e}{-1+c}\right )}{\sqrt {a+b x}}\right )}{a b \sqrt {-\frac {a}{-1+e}} \sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x)^(3/2)*(-((Sqrt[-(a/(-1 + e))]*(-1 + c + a/(a + b*x))*(-1 + e + a/(a + b*x)))/(-1 + c)) + (a*Sqrt
[(-1 + c + a/(a + b*x))/(-1 + c)]*Sqrt[(-1 + e + a/(a + b*x))/(-1 + e)]*EllipticE[ArcSin[Sqrt[-(a/(-1 + e))]/S
qrt[a + b*x]], (-1 + e)/(-1 + c)])/Sqrt[a + b*x]))/(a*b*Sqrt[-(a/(-1 + e))]*Sqrt[c + (b*(-1 + c)*x)/a]*Sqrt[e
+ (b*(-1 + e)*x)/a])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(49)=98\).
time = 0.09, size = 348, normalized size = 6.00

method	result	size
default	\(\frac {2 a^{2} \left (\EllipticF \left (\sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{-1+c}}\right ) c -\EllipticF \left (\sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{-1+c}}\right ) e -\EllipticE \left (\sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{-1+c}}\right ) c +\EllipticE \left (\sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{-1+c}}\right ) e \right ) \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{a \left (c -e \right )}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}\, \sqrt {b x +a}\, \sqrt {\frac {b e x +a e -b x}{a}}}{\sqrt {\frac {b c x +a c -b x}{a}}\, \left (b^{2} e \,x^{2}+2 a b e x -b^{2} x^{2}+a^{2} e -a b x \right ) b \left (-1+e \right ) \left (-1+c \right )}\)	\(348\)
elliptic	\(\frac {\sqrt {\frac {b e x +a e -b x}{a}}\, a \sqrt {\frac {\left (b x +a \right ) \left (b c x +a c -b x \right ) \left (b e x +a e -b x \right )}{a^{2}}}\, \left (\frac {2 e \left (-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{\sqrt {\frac {b^{3} c e \,x^{3}}{a^{2}}+\frac {3 b^{2} c e \,x^{2}}{a}-\frac {b^{3} c \,x^{3}}{a^{2}}-\frac {b^{3} e \,x^{3}}{a^{2}}+3 b c e x -\frac {2 b^{2} c \,x^{2}}{a}-\frac {2 b^{2} e \,x^{2}}{a}+\frac {b^{3} x^{3}}{a^{2}}+a c e -b c x -b e x +\frac {b^{2} x^{2}}{a}}}+\frac {2 b \left (-1+e \right ) \left (-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}\, \sqrt {\frac {x +\frac {a}{b}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\, \sqrt {\frac {x +\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}}\, \left (\left (-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}\right ) \EllipticE \left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )-\frac {a \EllipticF \left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {a c}{b \left (-1+c \right )}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {a c}{b \left (-1+c \right )}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{b}\right )}{a \sqrt {\frac {b^{3} c e \,x^{3}}{a^{2}}+\frac {3 b^{2} c e \,x^{2}}{a}-\frac {b^{3} c \,x^{3}}{a^{2}}-\frac {b^{3} e \,x^{3}}{a^{2}}+3 b c e x -\frac {2 b^{2} c \,x^{2}}{a}-\frac {2 b^{2} e \,x^{2}}{a}+\frac {b^{3} x^{3}}{a^{2}}+a c e -b c x -b e x +\frac {b^{2} x^{2}}{a}}}\right )}{\sqrt {b x +a}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \left (b e x +a e -b x \right )}\)	\(912\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.17, size = 1153, normalized size = 19.88 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, a^{2} c - a^{2} e - a^{2}\right )} \sqrt {-\frac {b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e}{a^{2}}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} c^{2} - a^{2} c + a^{2} e^{2} + a^{2} - {\left (a^{2} c + a^{2}\right )} e\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2} + {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e\right )}}, \frac {4 \, {\left (2 \, a^{3} c^{3} - 3 \, a^{3} c^{2} - 3 \, a^{3} c + 2 \, a^{3} e^{3} + 2 \, a^{3} - 3 \, {\left (a^{3} c + a^{3}\right )} e^{2} - 3 \, {\left (a^{3} c^{2} - 4 \, a^{3} c + a^{3}\right )} e\right )}}{27 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3} - {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{2} - 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e\right )}}, \frac {2 \, a c + 3 \, {\left (b c - b\right )} x - {\left (3 \, a c + 3 \, {\left (b c - b\right )} x - 2 \, a\right )} e - a}{3 \, {\left (b c - {\left (b c - b\right )} e - b\right )}}\right ) + 3 \, {\left (a b c - a b - {\left (a b c - a b\right )} e\right )} \sqrt {-\frac {b^{3} c - b^{3} - {\left (b^{3} c - b^{3}\right )} e}{a^{2}}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (a^{2} c^{2} - a^{2} c + a^{2} e^{2} + a^{2} - {\left (a^{2} c + a^{2}\right )} e\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2} + {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e\right )}}, \frac {4 \, {\left (2 \, a^{3} c^{3} - 3 \, a^{3} c^{2} - 3 \, a^{3} c + 2 \, a^{3} e^{3} + 2 \, a^{3} - 3 \, {\left (a^{3} c + a^{3}\right )} e^{2} - 3 \, {\left (a^{3} c^{2} - 4 \, a^{3} c + a^{3}\right )} e\right )}}{27 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3} - {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{2} - 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (a^{2} c^{2} - a^{2} c + a^{2} e^{2} + a^{2} - {\left (a^{2} c + a^{2}\right )} e\right )}}{3 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2} + {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e^{2} - 2 \, {\left (b^{2} c^{2} - 2 \, b^{2} c + b^{2}\right )} e\right )}}, \frac {4 \, {\left (2 \, a^{3} c^{3} - 3 \, a^{3} c^{2} - 3 \, a^{3} c + 2 \, a^{3} e^{3} + 2 \, a^{3} - 3 \, {\left (a^{3} c + a^{3}\right )} e^{2} - 3 \, {\left (a^{3} c^{2} - 4 \, a^{3} c + a^{3}\right )} e\right )}}{27 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3} - {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{3} + 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e^{2} - 3 \, {\left (b^{3} c^{3} - 3 \, b^{3} c^{2} + 3 \, b^{3} c - b^{3}\right )} e\right )}}, \frac {2 \, a c + 3 \, {\left (b c - b\right )} x - {\left (3 \, a c + 3 \, {\left (b c - b\right )} x - 2 \, a\right )} e - a}{3 \, {\left (b c - {\left (b c - b\right )} e - b\right )}}\right )\right )\right )}}{3 \, {\left (b^{3} c^{2} - 2 \, b^{3} c + b^{3} - {\left (b^{3} c^{2} - 2 \, b^{3} c + b^{3}\right )} e\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*((2*a^2*c - a^2*e - a^2)*sqrt(-(b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*weierstrassPInverse(4/3*(a^2*c^2 - a^
2*c + a^2*e^2 + a^2 - (a^2*c + a^2)*e)/(b^2*c^2 - 2*b^2*c + b^2 + (b^2*c^2 - 2*b^2*c + b^2)*e^2 - 2*(b^2*c^2 -
2*b^2*c + b^2)*e), 4/27*(2*a^3*c^3 - 3*a^3*c^2 - 3*a^3*c + 2*a^3*e^3 + 2*a^3 - 3*(a^3*c + a^3)*e^2 - 3*(a^3*c
^2 - 4*a^3*c + a^3)*e)/(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3 - (b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^3 + 3*(b
^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^2 - 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e), 1/3*(2*a*c + 3*(b*c - b)
*x - (3*a*c + 3*(b*c - b)*x - 2*a)*e - a)/(b*c - (b*c - b)*e - b)) + 3*(a*b*c - a*b - (a*b*c - a*b)*e)*sqrt(-(
b^3*c - b^3 - (b^3*c - b^3)*e)/a^2)*weierstrassZeta(4/3*(a^2*c^2 - a^2*c + a^2*e^2 + a^2 - (a^2*c + a^2)*e)/(b
^2*c^2 - 2*b^2*c + b^2 + (b^2*c^2 - 2*b^2*c + b^2)*e^2 - 2*(b^2*c^2 - 2*b^2*c + b^2)*e), 4/27*(2*a^3*c^3 - 3*a
^3*c^2 - 3*a^3*c + 2*a^3*e^3 + 2*a^3 - 3*(a^3*c + a^3)*e^2 - 3*(a^3*c^2 - 4*a^3*c + a^3)*e)/(b^3*c^3 - 3*b^3*c
^2 + 3*b^3*c - b^3 - (b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^3 + 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^2 -
3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e), weierstrassPInverse(4/3*(a^2*c^2 - a^2*c + a^2*e^2 + a^2 - (a^2*c
+ a^2)*e)/(b^2*c^2 - 2*b^2*c + b^2 + (b^2*c^2 - 2*b^2*c + b^2)*e^2 - 2*(b^2*c^2 - 2*b^2*c + b^2)*e), 4/27*(2*
a^3*c^3 - 3*a^3*c^2 - 3*a^3*c + 2*a^3*e^3 + 2*a^3 - 3*(a^3*c + a^3)*e^2 - 3*(a^3*c^2 - 4*a^3*c + a^3)*e)/(b^3*
c^3 - 3*b^3*c^2 + 3*b^3*c - b^3 - (b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e^3 + 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c
- b^3)*e^2 - 3*(b^3*c^3 - 3*b^3*c^2 + 3*b^3*c - b^3)*e), 1/3*(2*a*c + 3*(b*c - b)*x - (3*a*c + 3*(b*c - b)*x
- 2*a)*e - a)/(b*c - (b*c - b)*e - b))))/(b^3*c^2 - 2*b^3*c + b^3 - (b^3*c^2 - 2*b^3*c + b^3)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}{\sqrt {a + b x} \sqrt {c + \frac {b c x}{a} - \frac {b x}{a}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {e+\frac {b\,x\,\left (e-1\right )}{a}}}{\sqrt {c+\frac {b\,x\,\left (c-1\right )}{a}}\,\sqrt {a+b\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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