3.27.42 \(\int \frac {\sqrt {a+b x}}{\sqrt {c+\frac {b (-1+c) x}{a}} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [2642]

Optimal. Leaf size=162 \[ -\frac {2 a \sqrt {c-e} \sqrt {a+b x} \sqrt {-\frac {(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\sin ^{-1}\left (\frac {\sqrt {1-e} \sqrt {c-\frac {b (1-c) x}{a}}}{\sqrt {c-e}}\right )|\frac {c-e}{1-e}\right )}{b (1-c) \sqrt {1-e} \sqrt {\frac {(1-c) (a+b x)}{a}} \sqrt {e-\frac {b (1-e) x}{a}}} \]

[Out]

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

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Rubi [A]
time = 0.11, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {115, 114} \begin {gather*} -\frac {2 a \sqrt {c-e} \sqrt {a+b x} \sqrt {-\frac {(1-c) (a e-b (1-e) x)}{a (c-e)}} E\left (\text {ArcSin}\left (\frac {\sqrt {1-e} \sqrt {c-\frac {b (1-c) x}{a}}}{\sqrt {c-e}}\right )|\frac {c-e}{1-e}\right )}{b (1-c) \sqrt {1-e} \sqrt {\frac {(1-c) (a+b x)}{a}} \sqrt {e-\frac {b (1-e) x}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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])

Rule 115

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

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Mathematica [C] Result contains complex when optimal does not.
time = 11.83, size = 103, normalized size = 0.64 \begin {gather*} -\frac {2 i a \sqrt {a+b x} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {(-1+c) (a+b x)}{a}}\right )|\frac {-1+e}{-1+c}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {(-1+c) (a+b x)}{a}}\right )|\frac {-1+e}{-1+c}\right )\right )}{b (-1+e) \sqrt {\frac {(-1+c) (a+b x)}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*I)*a*Sqrt[a + b*x]*(EllipticE[I*ArcSinh[Sqrt[((-1 + c)*(a + b*x))/a]], (-1 + e)/(-1 + c)] - EllipticF[I*A
rcSinh[Sqrt[((-1 + c)*(a + b*x))/a]], (-1 + e)/(-1 + c)]))/(b*(-1 + e)*Sqrt[((-1 + c)*(a + b*x))/a])

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Maple [A]
time = 0.09, size = 181, normalized size = 1.12

method	result	size
default	\(-\frac {2 a^{2} \left (c -e \right ) \EllipticE \left (\sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}, \sqrt {\frac {c -e}{-1+c}}\right ) \sqrt {-\frac {\left (-1+e \right ) \left (b c x +a c -b x \right )}{a \left (c -e \right )}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {\frac {\left (-1+c \right ) \left (b e x +a e -b x \right )}{a \left (c -e \right )}}}{\sqrt {b x +a}\, \sqrt {\frac {b e x +a e -b x}{a}}\, \sqrt {\frac {b c x +a c -b x}{a}}\, \left (-1+e \right )^{2} b \left (-1+c \right )}\)	\(181\)
elliptic	\(\frac {\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 a \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 (-\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 )}{\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}}\, \sqrt {\frac {b e x +a e -b x}{a}}}\)	\(891\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 1179, normalized size = 7.28 \begin {gather*} -\frac {2 \, {\left ({\left (a^{3} c + a^{3} e - 2 \, a^{3}\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^{2} b c - a^{2} b - {\left (a^{2} b c - a^{2} 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^{2} - 2 \, {\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((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]

-2/3*((a^3*c + a^3*e - 2*a^3)*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^2*b*c - a^2*b - (a^2*b*c - a^2*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^2 - 2*
(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 {a + b x}}{\sqrt {c + \frac {b c x}{a} - \frac {b x}{a}} \sqrt {e + \frac {b e x}{a} - \frac {b x}{a}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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