3.27.41 \(\int \frac {\sqrt {c+d x}}{\sqrt {a+b x} \sqrt {e+\frac {b (-1+e) x}{a}}} \, dx\) [2641]

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 = 0.04, 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 = {115, 114} \begin {gather*} \frac {2 \sqrt {a} \sqrt {c+d x} E\left (\text {ArcSin}\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 {gather*}

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 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 {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] Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(96)=192\).
time = 12.31, size = 200, normalized size = 2.08 \begin {gather*} \frac {2 \sqrt {\frac {-1+e+\frac {a}{a+b x}}{-1+e}} \left (b \sqrt {a-\frac {b c}{d}} \sqrt {a+b x} (c+d x) \sqrt {\frac {a e+b (-1+e) x}{(-1+e) (a+b x)}}-(b c-a d) (a+b x) \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) (-1+e)}\right )\right )}{b^2 \sqrt {a-\frac {b c}{d}} \sqrt {c+d x} \sqrt {e+\frac {b (-1+e) x}{a}}} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(821\) vs. \(2(83)=166\).
time = 0.09, size = 822, normalized size = 8.56

method	result	size
elliptic	\(\frac {\sqrt {\frac {\left (d x +c \right ) \left (b x +a \right ) \left (b e x +a e -b x \right )}{a}}\, \left (\frac {2 c \left (-\frac {c}{d}+\frac {a e}{b \left (-1+e \right )}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {c}{d}+\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 {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {c}{d}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{\sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {x^{3} d \,b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}+\frac {2 d \left (-\frac {c}{d}+\frac {a e}{b \left (-1+e \right )}\right ) \sqrt {\frac {x +\frac {a e}{b \left (-1+e \right )}}{-\frac {c}{d}+\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 {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}}\, \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 {c}{d}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\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 {c}{d}+\frac {a e}{b \left (-1+e \right )}}}, \sqrt {\frac {-\frac {a e}{b \left (-1+e \right )}+\frac {c}{d}}{-\frac {a e}{b \left (-1+e \right )}+\frac {a}{b}}}\right )}{b}\right )}{\sqrt {\frac {b^{2} d e \,x^{3}}{a}+2 b d e \,x^{2}+\frac {b^{2} c e \,x^{2}}{a}-\frac {x^{3} d \,b^{2}}{a}+a d e x +2 b c e x -b d \,x^{2}-\frac {b^{2} c \,x^{2}}{a}+a c e -b c x}}\right )}{\sqrt {b x +a}\, \sqrt {d x +c}\, \sqrt {\frac {b e x +a e -b x}{a}}}\)	\(729\)
default	\(-\frac {2 \sqrt {d x +c}\, \sqrt {b x +a}\, \sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}\, \sqrt {-\frac {\left (b x +a \right ) \left (-1+e \right )}{a}}\, \sqrt {-\frac {\left (d x +c \right ) b \left (-1+e \right )}{a d e -b c e +b c}}\, \left (\EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d^{2} e^{2}-2 \EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c d \,e^{2}+\EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c^{2} e^{2}-\EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d^{2} e +3 \EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c d e -2 \EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c^{2} e +\EllipticE \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a^{2} d^{2} e -\EllipticE \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c d e -\EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c d +\EllipticF \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) b^{2} c^{2}+\EllipticE \left (\sqrt {\frac {d \left (b e x +a e -b x \right )}{a d e -b c e +b c}}, \sqrt {\frac {a d e -b c e +b c}{d a}}\right ) a b c d \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 (-1+e \right )^{2} b^{2} d}\)	\(822\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d*x+c)^(1/2)*(b*x+a)^(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)*(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^2*e^2-2*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*d*e^2+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^2*e^2-E
llipticF((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^2*e+3*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*d*e-2*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^2*e+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^2*e-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*d*e-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*d+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^2+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*d)/((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

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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((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*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 = 0.20, size = 1100, normalized size = 11.46 \begin {gather*} -\frac {2 \, {\left ({\left (2 \, a b c - a^{2} d - 2 \, {\left (a b c - a^{2} d\right )} e\right )} \sqrt {\frac {b^{2} d e - b^{2} d}{a}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} - {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2}\right )} e\right )}}{3 \, {\left (b^{2} d^{2} e^{2} - 2 \, b^{2} d^{2} e + b^{2} d^{2}\right )}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (2 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{2} - 3 \, {\left (2 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} e\right )}}{27 \, {\left (b^{3} d^{3} e^{3} - 3 \, b^{3} d^{3} e^{2} + 3 \, b^{3} d^{3} e - b^{3} d^{3}\right )}}, -\frac {3 \, b d x + b c + a d - {\left (3 \, b d x + b c + 2 \, a d\right )} e}{3 \, {\left (b d e - b d\right )}}\right ) + 3 \, {\left (a b d e - a b d\right )} \sqrt {\frac {b^{2} d e - b^{2} d}{a}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} - {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2}\right )} e\right )}}{3 \, {\left (b^{2} d^{2} e^{2} - 2 \, b^{2} d^{2} e + b^{2} d^{2}\right )}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (2 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{2} - 3 \, {\left (2 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} e\right )}}{27 \, {\left (b^{3} d^{3} e^{3} - 3 \, b^{3} d^{3} e^{2} + 3 \, b^{3} d^{3} e - b^{3} d^{3}\right )}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (b^{2} c^{2} - a b c d + a^{2} d^{2} + {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} e^{2} - {\left (2 \, b^{2} c^{2} - 3 \, a b c d + a^{2} d^{2}\right )} e\right )}}{3 \, {\left (b^{2} d^{2} e^{2} - 2 \, b^{2} d^{2} e + b^{2} d^{2}\right )}}, \frac {4 \, {\left (2 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 3 \, a^{2} b c d^{2} + 2 \, a^{3} d^{3} - 2 \, {\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{3} + 3 \, {\left (2 \, b^{3} c^{3} - 5 \, a b^{2} c^{2} d + 4 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} e^{2} - 3 \, {\left (2 \, b^{3} c^{3} - 4 \, a b^{2} c^{2} d + a^{2} b c d^{2} + a^{3} d^{3}\right )} e\right )}}{27 \, {\left (b^{3} d^{3} e^{3} - 3 \, b^{3} d^{3} e^{2} + 3 \, b^{3} d^{3} e - b^{3} d^{3}\right )}}, -\frac {3 \, b d x + b c + a d - {\left (3 \, b d x + b c + 2 \, a d\right )} e}{3 \, {\left (b d e - b d\right )}}\right )\right )\right )}}{3 \, {\left (b^{3} d e^{2} - 2 \, b^{3} d e + b^{3} d\right )}} \end {gather*}

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]

-2/3*((2*a*b*c - a^2*d - 2*(a*b*c - a^2*d)*e)*sqrt((b^2*d*e - b^2*d)/a)*weierstrassPInverse(4/3*(b^2*c^2 - a*b
*c*d + a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2 - (2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*e)/(b^2*d^2*e^2 - 2*b
^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3 - 2*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*e^3 + 3*(2*b^3*c^3 - 5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*e^2 - 3*(2*b^3*c^3 -
4*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*e)/(b^3*d^3*e^3 - 3*b^3*d^3*e^2 + 3*b^3*d^3*e - b^3*d^3), -1/3*(3*b*d*x
+ b*c + a*d - (3*b*d*x + b*c + 2*a*d)*e)/(b*d*e - b*d)) + 3*(a*b*d*e - a*b*d)*sqrt((b^2*d*e - b^2*d)/a)*weier
strassZeta(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2 - (2*b^2*c^2 - 3*a*b*c*d + a
^2*d^2)*e)/(b^2*d^2*e^2 - 2*b^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + 2*a^3*d^3
- 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3 + 3*(2*b^3*c^3 - 5*a*b^2*c^2*d + 4*a^2*b*c*d^2 - a
^3*d^3)*e^2 - 3*(2*b^3*c^3 - 4*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*e)/(b^3*d^3*e^3 - 3*b^3*d^3*e^2 + 3*b^3*d^
3*e - b^3*d^3), weierstrassPInverse(4/3*(b^2*c^2 - a*b*c*d + a^2*d^2 + (b^2*c^2 - 2*a*b*c*d + a^2*d^2)*e^2 - (
2*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*e)/(b^2*d^2*e^2 - 2*b^2*d^2*e + b^2*d^2), 4/27*(2*b^3*c^3 - 3*a*b^2*c^2*d - 3
*a^2*b*c*d^2 + 2*a^3*d^3 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*e^3 + 3*(2*b^3*c^3 - 5*a*b^2*
c^2*d + 4*a^2*b*c*d^2 - a^3*d^3)*e^2 - 3*(2*b^3*c^3 - 4*a*b^2*c^2*d + a^2*b*c*d^2 + a^3*d^3)*e)/(b^3*d^3*e^3 -
3*b^3*d^3*e^2 + 3*b^3*d^3*e - b^3*d^3), -1/3*(3*b*d*x + b*c + a*d - (3*b*d*x + b*c + 2*a*d)*e)/(b*d*e - b*d))
))/(b^3*d*e^2 - 2*b^3*d*e + b^3*d)

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

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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