3.2554 \(\int \frac{2+3 x}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0101105, antiderivative size = 48, 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} \[ \frac{7 \sqrt{5 x+3}}{11 \sqrt{1-2 x}}-\frac{3 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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}{(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 \operatorname{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*}

Mathematica [A]  time = 0.0543381, size = 48, normalized size = 1. \[ \frac{7 \sqrt{5 x+3}}{11 \sqrt{1-2 x}}+\frac{3 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.01, size = 74, normalized size = 1.5 \begin{align*} -{\frac{1}{440\,x-220} \left ( 66\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-33\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +140\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.78745, size = 49, normalized size = 1.02 \begin{align*} -\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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]  time = 1.78605, size = 209, normalized size = 4.35 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 2.52988, size = 61, normalized size = 1.27 \begin{align*} -\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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)