3.29.62 \(\int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\) [2862]

Optimal. Leaf size=31 \[ -\sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \]

[Out]

-1/5*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {114} \begin {gather*} -\sqrt {\frac {7}{5}} E\left (\text {ArcSin}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + 3*x]/(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]),x]

[Out]

-(Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35])

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 {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.25, size = 129, normalized size = 4.16 \begin {gather*} \frac {\sqrt {2+3 x} \sqrt {\frac {-1+2 x}{3+5 x}} \left (5 \sqrt {\frac {-1+2 x}{3+5 x}} \sqrt {\frac {2+3 x}{3+5 x}} \sqrt {3+5 x}+i \sqrt {2} E\left (i \sinh ^{-1}\left (\frac {1}{\sqrt {9+15 x}}\right )|-\frac {33}{2}\right )\right )}{5 \sqrt {1-2 x} \sqrt {\frac {2+3 x}{3+5 x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + 3*x]/(Sqrt[1 - 2*x]*Sqrt[3 + 5*x]),x]

[Out]

(Sqrt[2 + 3*x]*Sqrt[(-1 + 2*x)/(3 + 5*x)]*(5*Sqrt[(-1 + 2*x)/(3 + 5*x)]*Sqrt[(2 + 3*x)/(3 + 5*x)]*Sqrt[3 + 5*x
] + I*Sqrt[2]*EllipticE[I*ArcSinh[1/Sqrt[9 + 15*x]], -33/2]))/(5*Sqrt[1 - 2*x]*Sqrt[(2 + 3*x)/(3 + 5*x)])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(52\) vs. \(2(23)=46\).
time = 0.10, size = 53, normalized size = 1.71

method	result	size
default	\(\frac {\left (\EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )\right ) \sqrt {-3-5 x}\, \sqrt {2}}{5 \sqrt {3+5 x}}\)	\(53\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {\sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(173\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5*(EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2)))*(-3-5*x)^(1/2)*2
^(1/2)/(3+5*x)^(1/2)

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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((2+3*x)^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(3*x + 2)/(sqrt(5*x + 3)*sqrt(-2*x + 1)), x)

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Fricas [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((2+3*x)^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

0

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {3 x + 2}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**(1/2)/(1-2*x)**(1/2)/(3+5*x)**(1/2),x)

[Out]

Integral(sqrt(3*x + 2)/(sqrt(1 - 2*x)*sqrt(5*x + 3)), 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((2+3*x)^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(3*x + 2)/(sqrt(5*x + 3)*sqrt(-2*x + 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {3\,x+2}}{\sqrt {1-2\,x}\,\sqrt {5\,x+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^(1/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(1/2)),x)

[Out]

int((3*x + 2)^(1/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(1/2)), x)

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