3.29.19 \(\int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx\) [2819]

Optimal. Leaf size=81 \[ -\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{3 \sqrt {3+5 x}} \]

[Out]

2/21*EllipticE(5^(1/2)*(2+3*x)^(1/2),1/35*70^(1/2))*35^(1/2)*(-3-5*x)^(1/2)/(3+5*x)^(1/2)-2/7*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {101, 21, 115, 114} \begin {gather*} \frac {2 \sqrt {\frac {5}{7}} \sqrt {-5 x-3} E\left (\text {ArcSin}\left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{3 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (2*Sqrt[5/7]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt
[2 + 3*x]], 2/35])/(3*Sqrt[3 + 5*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {2}{7} \int \frac {\frac {5}{2}-5 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {5}{7} \int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {\left (5 \sqrt {-3-5 x}\right ) \int \frac {\sqrt {\frac {3}{7}-\frac {6 x}{7}}}{\sqrt {-9-15 x} \sqrt {2+3 x}} \, dx}{\sqrt {7} \sqrt {3+5 x}}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{3 \sqrt {3+5 x}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.76, size = 70, normalized size = 0.86 \begin {gather*} \frac {-6 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-2 i \sqrt {33} (2+3 x) E\left (i \sinh ^{-1}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )}{42+63 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (2*I)*Sqrt[33]*(2 + 3*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]],
-2/33])/(42 + 63*x)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(132\) vs. \(2(60)=120\).
time = 0.10, size = 133, normalized size = 1.64

method	result	size
default	\(-\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {1-2 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+60 x^{2}+6 x -18\right )}{21 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-30 x^{2}-3 x +9\right )}{21 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {5 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {10 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{147 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(33*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipti
cF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+2*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42
*x)^(1/2),1/2*70^(1/2))+60*x^2+6*x-18)/(30*x^3+23*x^2-7*x-6)

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

[Out]

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

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Fricas [A]
time = 0.16, size = 23, normalized size = 0.28 \begin {gather*} -\frac {2 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{7 \, \sqrt {3 \, x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/7*sqrt(5*x + 3)*sqrt(-2*x + 1)/sqrt(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {5\,x+3}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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