3.2553 \(\int \frac{(2+3 x)^2}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ \frac{9}{20} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{49 \sqrt{5 x+3}}{22 \sqrt{1-2 x}}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

[Out]

(49*Sqrt[3 + 5*x])/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 - (321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/(20*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0174986, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {89, 80, 54, 216} \[ \frac{9}{20} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{49 \sqrt{5 x+3}}{22 \sqrt{1-2 x}}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(49*Sqrt[3 + 5*x])/(22*Sqrt[1 - 2*x]) + (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 - (321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/(20*Sqrt[10])

Rule 89

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c - a*
d)^2*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d^2*(d*e - c*f)*(n + 1)), x] - Dist[1/(d^2*(d*e - c*f)*(n + 1)), In
t[(c + d*x)^(n + 1)*(e + f*x)^p*Simp[a^2*d^2*f*(n + p + 2) + b^2*c*(d*e*(n + 1) + c*f*(p + 1)) - 2*a*b*d*(d*e*
(n + 1) + c*f*(p + 1)) - b^2*d*(d*e - c*f)*(n + 1)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && (LtQ
[n, -1] || (EqQ[n + p + 3, 0] && NeQ[n, -1] && (SumSimplerQ[n, 1] ||  !SumSimplerQ[p, 1])))

Rule 80

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

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)^2}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}-\frac{1}{22} \int \frac{\frac{363}{2}+99 x}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}+\frac{9}{20} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{321}{40} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}+\frac{9}{20} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{321 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{20 \sqrt{5}}\\ &=\frac{49 \sqrt{3+5 x}}{22 \sqrt{1-2 x}}+\frac{9}{20} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{321 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{20 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0233546, size = 59, normalized size = 0.82 \[ \frac{10 \sqrt{5 x+3} (589-198 x)+3531 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2200 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(589 - 198*x)*Sqrt[3 + 5*x] + 3531*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(2200*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 89, normalized size = 1.2 \begin{align*} -{\frac{1}{8800\,x-4400} \left ( 7062\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-3531\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3960\,x\sqrt{-10\,{x}^{2}-x+3}+11780\,\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)^2/(1-2*x)^(3/2)/(3+5*x)^(1/2),x)

[Out]

-1/4400*(7062*10^(1/2)*arcsin(20/11*x+1/11)*x-3531*10^(1/2)*arcsin(20/11*x+1/11)-3960*x*(-10*x^2-x+3)^(1/2)+11
780*(-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)

________________________________________________________________________________________

Maxima [A]  time = 3.10343, size = 68, normalized size = 0.94 \begin{align*} -\frac{321}{400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{20} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{22 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-321/400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 9/20*sqrt(-10*x^2 - x + 3) - 49/22*sqrt(-10*x^2 - x + 3)/(2*
x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.76242, size = 231, normalized size = 3.21 \begin{align*} \frac{3531 \, \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 ) + 20 \,{\left (198 \, x - 589\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4400 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4400*(3531*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))
+ 20*(198*x - 589)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 2.01226, size = 78, normalized size = 1.08 \begin{align*} -\frac{321}{200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (198 \, \sqrt{5}{\left (5 \, x + 3\right )} - 3539 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{5500 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-321/200*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/5500*(198*sqrt(5)*(5*x + 3) - 3539*sqrt(5))*sqrt(5*x
+ 3)*sqrt(-10*x + 5)/(2*x - 1)