∫ 2+3x______ (1−2x)3∕2√ ___

Optimal. Leaf size=48 \[ \frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{\sqrt {10}} \]

-3/10*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+7/11*(3+5*x)^(1/2)/(1-2*x)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 48, 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 {7 \sqrt {5 x+3}}{11 \sqrt {1-2 x}}-\frac {3 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{\sqrt {10}} \end {gather*}

(7*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) - (3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/Sqrt[10]

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}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx &=\frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3}{2} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{\sqrt {5}}\\ &=\frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}-\frac {3 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{\sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.08, size = 52, normalized size = 1.08 \begin {gather*} \frac {7 \sqrt {3+5 x}}{11 \sqrt {1-2 x}}+\frac {3 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{\sqrt {10}} \end {gather*}

(7*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/Sqrt[10]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(73\) vs. \(2(35)=70\).
time = 0.09, size = 74, normalized size = 1.54


method	result	size

default	\(-\frac {\left (66 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) x -33 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+140 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{220 \left (-1+2 x \right ) \sqrt {-10 x^{2}-x +3}}\)	\(74\)

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

-1/220*(66*10^(1/2)*arcsin(20/11*x+1/11)*x-33*10^(1/2)*arcsin(20/11*x+1/11)+140*(-10*x^2-x+3)^(1/2))*(3+5*x)^(

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Maxima [A]
time = 0.65, size = 36, normalized size = 0.75 \begin {gather*} -\frac {3}{20} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {7 \, \sqrt {-10 \, x^{2} - x + 3}}{11 \, {\left (2 \, x - 1\right )}} \end {gather*}

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

-3/20*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 7/11*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (35) = 70\).
time = 0.78, size = 71, normalized size = 1.48 \begin {gather*} \frac {33 \, \sqrt {10} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 140 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{220 \, {\left (2 \, x - 1\right )}} \end {gather*}

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

1/220*(33*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) -

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

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Giac [A]
time = 0.64, size = 45, normalized size = 0.94 \begin {gather*} -\frac {3}{10} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {7 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{55 \, {\left (2 \, x - 1\right )}} \end {gather*}

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

-3/10*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 7/55*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)

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

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3.26.54 \(\int \frac {2+3 x}{(1-2 x)^{3/2} \sqrt {3+5 x}} \, dx\) [2554]