3.2491 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=77 \[ -\frac{3}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)-\frac{333}{400} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{3827 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

[Out]

(-333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x])/20 + (3827*ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

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Rubi [A]  time = 0.0176384, antiderivative size = 77, 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 = {90, 80, 54, 216} \[ -\frac{3}{20} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)-\frac{333}{400} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{3827 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-333*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400 - (3*Sqrt[1 - 2*x]*(2 + 3*x)*Sqrt[3 + 5*x])/20 + (3827*ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]])/(400*Sqrt[10])

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

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}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx &=-\frac{3}{20} \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}-\frac{1}{20} \int \frac{-104-\frac{333 x}{2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{333}{400} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}+\frac{3827}{800} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{333}{400} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}+\frac{3827 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{400 \sqrt{5}}\\ &=-\frac{333}{400} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}+\frac{3827 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.02322, size = 55, normalized size = 0.71 \[ \frac{-30 \sqrt{1-2 x} \sqrt{5 x+3} (60 x+151)-3827 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{4000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(151 + 60*x) - 3827*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/4000

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Maple [A]  time = 0.011, size = 70, normalized size = 0.9 \begin{align*}{\frac{1}{8000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3827\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -3600\,x\sqrt{-10\,{x}^{2}-x+3}-9060\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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)^(1/2)/(3+5*x)^(1/2),x)

[Out]

1/8000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(3827*10^(1/2)*arcsin(20/11*x+1/11)-3600*x*(-10*x^2-x+3)^(1/2)-9060*(-10*x^
2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.55899, size = 55, normalized size = 0.71 \begin{align*} -\frac{9}{20} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{3827}{8000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{453}{400} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/20*sqrt(-10*x^2 - x + 3)*x - 3827/8000*sqrt(10)*arcsin(-20/11*x - 1/11) - 453/400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.65764, size = 203, normalized size = 2.64 \begin{align*} -\frac{3}{400} \,{\left (60 \, x + 151\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{3827}{8000} \, \sqrt{10} \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/400*(60*x + 151)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 3827/8000*sqrt(10)*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., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (3 x + 2\right )^{2}}{\sqrt{1 - 2 x} \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)**(1/2)/(3+5*x)**(1/2),x)

[Out]

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

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Giac [A]  time = 1.73106, size = 61, normalized size = 0.79 \begin{align*} -\frac{1}{4000} \, \sqrt{5}{\left (6 \,{\left (60 \, x + 151\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 3827 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4000*sqrt(5)*(6*(60*x + 151)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 3827*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3
)))