3.4.97 \(\int \frac {\sqrt {a x}}{\sqrt {d+e x} \sqrt {e+f x}} \, dx\) [397]

Optimal. Leaf size=114 \[ \frac {2 \sqrt {-e^2+d f} \sqrt {a x} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\sin ^{-1}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {-e^2+d f}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \]

[Out]

2*EllipticE(f^(1/2)*(e*x+d)^(1/2)/(d*f-e^2)^(1/2),(1-e^2/d/f)^(1/2))*(d*f-e^2)^(1/2)*(a*x)^(1/2)*(e*(f*x+e)/(-
d*f+e^2))^(1/2)/e/f^(1/2)/(-e*x/d)^(1/2)/(f*x+e)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {115, 114} \begin {gather*} \frac {2 \sqrt {a x} \sqrt {d f-e^2} \sqrt {\frac {e (e+f x)}{e^2-d f}} E\left (\text {ArcSin}\left (\frac {\sqrt {f} \sqrt {d+e x}}{\sqrt {d f-e^2}}\right )|1-\frac {e^2}{d f}\right )}{e \sqrt {f} \sqrt {-\frac {e x}{d}} \sqrt {e+f x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x]/(Sqrt[d + e*x]*Sqrt[e + f*x]),x]

[Out]

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

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

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Mathematica [C] Result contains complex when optimal does not.
time = 1.91, size = 106, normalized size = 0.93 \begin {gather*} -\frac {2 i e \sqrt {a x} \sqrt {1+\frac {f x}{e}} \left (E\left (i \sinh ^{-1}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )-F\left (i \sinh ^{-1}\left (\sqrt {\frac {e x}{d}}\right )|\frac {d f}{e^2}\right )\right )}{f \sqrt {\frac {e x}{d+e x}} \sqrt {d+e x} \sqrt {e+f x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x]/(Sqrt[d + e*x]*Sqrt[e + f*x]),x]

[Out]

((-2*I)*e*Sqrt[a*x]*Sqrt[1 + (f*x)/e]*(EllipticE[I*ArcSinh[Sqrt[(e*x)/d]], (d*f)/e^2] - EllipticF[I*ArcSinh[Sq
rt[(e*x)/d]], (d*f)/e^2]))/(f*Sqrt[(e*x)/(d + e*x)]*Sqrt[d + e*x]*Sqrt[e + f*x])

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Maple [A]
time = 0.23, size = 191, normalized size = 1.68

method	result	size
default	\(-\frac {2 \left (d \EllipticF \left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) f -\EllipticE \left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) d f +\EllipticE \left (\sqrt {\frac {f x +e}{e}}, \sqrt {-\frac {e^{2}}{d f -e^{2}}}\right ) e^{2}\right ) \sqrt {-\frac {f x}{e}}\, \sqrt {\frac {\left (e x +d \right ) f}{d f -e^{2}}}\, \sqrt {\frac {f x +e}{e}}\, \sqrt {a x}\, \sqrt {e x +d}\, \sqrt {f x +e}}{f^{2} x \left (e f \,x^{2}+d f x +e^{2} x +d e \right )}\)	\(191\)
elliptic	\(\frac {2 \sqrt {a x}\, \sqrt {\left (e x +d \right ) \left (f x +e \right ) a x}\, e \sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {e}{f}+\frac {d}{e}}}\, \sqrt {-\frac {f x}{e}}\, \left (\left (-\frac {e}{f}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}, \sqrt {-\frac {e}{f \left (-\frac {e}{f}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (x +\frac {e}{f}\right ) f}{e}}, \sqrt {-\frac {e}{f \left (-\frac {e}{f}+\frac {d}{e}\right )}}\right )}{e}\right )}{\sqrt {e x +d}\, \sqrt {f x +e}\, x f \sqrt {a e f \,x^{3}+a d f \,x^{2}+a \,e^{2} x^{2}+a d e x}}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(d*EllipticF(((f*x+e)/e)^(1/2),(-e^2/(d*f-e^2))^(1/2))*f-EllipticE(((f*x+e)/e)^(1/2),(-e^2/(d*f-e^2))^(1/2)
)*d*f+EllipticE(((f*x+e)/e)^(1/2),(-e^2/(d*f-e^2))^(1/2))*e^2)*(-f*x/e)^(1/2)*((e*x+d)*f/(d*f-e^2))^(1/2)*((f*
x+e)/e)^(1/2)*(a*x)^(1/2)*(e*x+d)^(1/2)*(f*x+e)^(1/2)/f^2/x/(e*f*x^2+d*f*x+e^2*x+d*e)

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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((a*x)^(1/2)/(e*x+d)^(1/2)/(f*x+e)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x)/(sqrt(f*x + e)*sqrt(x*e + d)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 250, normalized size = 2.19 \begin {gather*} -\frac {2 \, {\left (\sqrt {a f} {\left (d f + e^{2}\right )} e^{\frac {1}{2}} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} - d f e^{2} + e^{4}\right )} e^{\left (-2\right )}}{3 \, f^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} - 3 \, d^{2} f^{2} e^{2} - 3 \, d f e^{4} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{27 \, f^{3}}, \frac {{\left (3 \, f x e + d f + e^{2}\right )} e^{\left (-1\right )}}{3 \, f}\right ) + 3 \, \sqrt {a f} f e^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (d^{2} f^{2} - d f e^{2} + e^{4}\right )} e^{\left (-2\right )}}{3 \, f^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} - 3 \, d^{2} f^{2} e^{2} - 3 \, d f e^{4} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{27 \, f^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (d^{2} f^{2} - d f e^{2} + e^{4}\right )} e^{\left (-2\right )}}{3 \, f^{2}}, -\frac {4 \, {\left (2 \, d^{3} f^{3} - 3 \, d^{2} f^{2} e^{2} - 3 \, d f e^{4} + 2 \, e^{6}\right )} e^{\left (-3\right )}}{27 \, f^{3}}, \frac {{\left (3 \, f x e + d f + e^{2}\right )} e^{\left (-1\right )}}{3 \, f}\right )\right )\right )} e^{\left (-2\right )}}{3 \, f^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x)^(1/2)/(e*x+d)^(1/2)/(f*x+e)^(1/2),x, algorithm="fricas")

[Out]

-2/3*(sqrt(a*f)*(d*f + e^2)*e^(1/2)*weierstrassPInverse(4/3*(d^2*f^2 - d*f*e^2 + e^4)*e^(-2)/f^2, -4/27*(2*d^3
*f^3 - 3*d^2*f^2*e^2 - 3*d*f*e^4 + 2*e^6)*e^(-3)/f^3, 1/3*(3*f*x*e + d*f + e^2)*e^(-1)/f) + 3*sqrt(a*f)*f*e^(3
/2)*weierstrassZeta(4/3*(d^2*f^2 - d*f*e^2 + e^4)*e^(-2)/f^2, -4/27*(2*d^3*f^3 - 3*d^2*f^2*e^2 - 3*d*f*e^4 + 2
*e^6)*e^(-3)/f^3, weierstrassPInverse(4/3*(d^2*f^2 - d*f*e^2 + e^4)*e^(-2)/f^2, -4/27*(2*d^3*f^3 - 3*d^2*f^2*e
^2 - 3*d*f*e^4 + 2*e^6)*e^(-3)/f^3, 1/3*(3*f*x*e + d*f + e^2)*e^(-1)/f)))*e^(-2)/f^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x)**(1/2)/(e*x+d)**(1/2)/(f*x+e)**(1/2),x)

[Out]

Integral(sqrt(a*x)/(sqrt(d + e*x)*sqrt(e + f*x)), 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((a*x)^(1/2)/(e*x+d)^(1/2)/(f*x+e)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*x)/(sqrt(f*x + e)*sqrt(x*e + d)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x)^(1/2)/((e + f*x)^(1/2)*(d + e*x)^(1/2)),x)

[Out]

int((a*x)^(1/2)/((e + f*x)^(1/2)*(d + e*x)^(1/2)), x)

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