3.2501 \(\int \frac{2+3 x}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0103922, antiderivative size = 52, normalized size of antiderivative = 1., 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 = {78, 54, 216} \[ \frac{3}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{2 \sqrt{1-2 x}}{55 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 78

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] || LtQ
[p, n]))))

Rule 54

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]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\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 \operatorname{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*}

Mathematica [A]  time = 0.0512938, size = 52, normalized size = 1. \[ -\frac{2 \sqrt{1-2 x}}{55 \sqrt{5 x+3}}-\frac{3}{5} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.012, size = 67, normalized size = 1.3 \begin{align*}{\frac{1}{550} \left ( 165\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+99\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -20\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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
/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 2.22976, size = 49, normalized size = 0.94 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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]  time = 1.86831, size = 228, normalized size = 4.38 \begin{align*} -\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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., size = 0, normalized size = 0. \begin{align*} \int \frac{3 x + 2}{\sqrt{1 - 2 x} \left (5 x + 3\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.04578, size = 107, normalized size = 2.06 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))