3.2500 \(\int \frac{(2+3 x)^2}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{9}{50} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{2 \sqrt{1-2 x}}{275 \sqrt{5 x+3}}+\frac{123 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(275*Sqrt[3 + 5*x]) - (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 + (123*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(50*Sqrt[10])

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Rubi [A]  time = 0.016797, 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} \[ -\frac{9}{50} \sqrt{1-2 x} \sqrt{5 x+3}-\frac{2 \sqrt{1-2 x}}{275 \sqrt{5 x+3}}+\frac{123 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{50 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(275*Sqrt[3 + 5*x]) - (9*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/50 + (123*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(50*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}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{275 \sqrt{3+5 x}}+\frac{2}{275} \int \frac{\frac{363}{2}+\frac{495 x}{2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{275 \sqrt{3+5 x}}-\frac{9}{50} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{123}{100} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{275 \sqrt{3+5 x}}-\frac{9}{50} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{123 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{50 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x}}{275 \sqrt{3+5 x}}-\frac{9}{50} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{123 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{50 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0242314, size = 59, normalized size = 0.82 \[ \frac{-10 \sqrt{1-2 x} (495 x+301)-1353 \sqrt{50 x+30} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5500 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*(301 + 495*x) - 1353*Sqrt[30 + 50*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5500*Sqrt[3 + 5*x])

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Maple [A]  time = 0.012, size = 82, normalized size = 1.1 \begin{align*}{\frac{1}{11000} \left ( 6765\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+4059\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -9900\,x\sqrt{-10\,{x}^{2}-x+3}-6020\,\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)^2/(3+5*x)^(3/2)/(1-2*x)^(1/2),x)

[Out]

1/11000*(6765*10^(1/2)*arcsin(20/11*x+1/11)*x+4059*10^(1/2)*arcsin(20/11*x+1/11)-9900*x*(-10*x^2-x+3)^(1/2)-60
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 = 2.07248, size = 68, normalized size = 0.94 \begin{align*} \frac{123}{1000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9}{50} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{275 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

123/1000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/50*sqrt(-10*x^2 - x + 3) - 2/275*sqrt(-10*x^2 - x + 3)/(5*
x + 3)

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Fricas [A]  time = 1.86523, size = 234, normalized size = 3.25 \begin{align*} -\frac{1353 \, \sqrt{10}{\left (5 \, x + 3\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 (495 \, x + 301\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{11000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/11000*(1353*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3
)) + 20*(495*x + 301)*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{\left (3 x + 2\right )^{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)**2/(3+5*x)**(3/2)/(1-2*x)**(1/2),x)

[Out]

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

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Giac [A]  time = 2.32409, size = 132, normalized size = 1.83 \begin{align*} -\frac{9}{250} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{123}{500} \, \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 )}}{2750 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{1375 \,{\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)^2/(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-9/250*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 123/500*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/2750*s
qrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/1375*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))