∫ 2+3x______√ 1 − 2x (3+5x)3∕2 dx [2501]

Optimal. Leaf size=52 \[ -\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {3}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]

3/25*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-2/55*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {79, 56, 222} \begin {gather*} \frac {3}{5} \sqrt {\frac {2}{5}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )-\frac {2 \sqrt {1-2 x}}{55 \sqrt {5 x+3}} \end {gather*}

(-2*Sqrt[1 - 2*x])/(55*Sqrt[3 + 5*x]) + (3*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/5

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -

a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f

))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1

) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,

f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || L

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}

\begin {align*} \int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {3}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {6 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}+\frac {3}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 56, normalized size = 1.08 \begin {gather*} -\frac {2 \sqrt {1-2 x}}{55 \sqrt {3+5 x}}-\frac {3}{5} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right ) \end {gather*}

(-2*Sqrt[1 - 2*x])/(55*Sqrt[3 + 5*x]) - (3*Sqrt[2/5]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/5

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method	result	size

default	\(\frac {\left (165 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x +99 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-20 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{550 \sqrt {-10 x^{2}-x +3}\, \sqrt {3+5 x}}\)	\(67\)

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

1/550*(165*10^(1/2)*arcsin(20/11*x+1/11)*x+99*10^(1/2)*arcsin(20/11*x+1/11)-20*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1

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Maxima [A]
time = 0.53, size = 36, normalized size = 0.69 \begin {gather*} \frac {3}{50} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{55 \, {\left (5 \, x + 3\right )}} \end {gather*}

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

3/50*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/55*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (35) = 70\).
time = 0.65, size = 77, normalized size = 1.48 \begin {gather*} -\frac {33 \, \sqrt {5} \sqrt {2} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 20 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{550 \, {\left (5 \, x + 3\right )}} \end {gather*}

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

-1/550*(33*sqrt(5)*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x

^2 + x - 3)) + 20*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (35) = 70\).
time = 0.55, size = 79, normalized size = 1.52 \begin {gather*} \frac {3}{25} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {10} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{550 \, \sqrt {5 \, x + 3}} + \frac {2 \, \sqrt {10} \sqrt {5 \, x + 3}}{275 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}} \end {gather*}

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

3/25*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/550*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5

*x + 3) + 2/275*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {3\,x+2}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

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3.26.1 \(\int \frac {2+3 x}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\) [2501]