3.2268 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^2} \, dx\)

Optimal. Leaf size=91 \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)}-\frac{2}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)) - (2*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 - (37*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9*Sqrt[7])

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Rubi [A]  time = 0.0313506, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {97, 157, 54, 216, 93, 204} \[ -\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{3 (3 x+2)}-\frac{2}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{9 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)) - (2*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/9 - (37*ArcTan[Sq
rt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(9*Sqrt[7])

Rule 97

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

Rule 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

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]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^2} \, dx &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)}+\frac{1}{3} \int \frac{-\frac{1}{2}-10 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)}-\frac{10}{9} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{37}{18} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)}+\frac{37}{9} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )-\frac{1}{9} \left (4 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{3 (2+3 x)}-\frac{2}{9} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{37 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{9 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0877772, size = 112, normalized size = 1.23 \[ \frac{21 \sqrt{5 x+3} (2 x-1)+14 \sqrt{10-20 x} (3 x+2) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-37 \sqrt{7-14 x} (3 x+2) \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{63 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*(-1 + 2*x)*Sqrt[3 + 5*x] + 14*Sqrt[10 - 20*x]*(2 + 3*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] - 37*Sqrt[7 - 14*
x]*(2 + 3*x)*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(63*Sqrt[1 - 2*x]*(2 + 3*x))

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Maple [A]  time = 0.008, size = 131, normalized size = 1.4 \begin{align*} -{\frac{1}{252+378\,x}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 42\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-111\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+28\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -74\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +42\,\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((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2,x)

[Out]

-1/126*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(42*10^(1/2)*arcsin(20/11*x+1/11)*x-111*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/
2)/(-10*x^2-x+3)^(1/2))*x+28*10^(1/2)*arcsin(20/11*x+1/11)-74*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))+42*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)

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Maxima [A]  time = 1.77129, size = 82, normalized size = 0.9 \begin{align*} -\frac{1}{9} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{37}{126} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{\sqrt{-10 \, x^{2} - x + 3}}{3 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/9*sqrt(10)*arcsin(20/11*x + 1/11) + 37/126*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/3*
sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.75109, size = 347, normalized size = 3.81 \begin{align*} -\frac{37 \, \sqrt{7}{\left (3 \, x + 2\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, \sqrt{10}{\left (3 \, x + 2\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 ) + 42 \, \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{126 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/126*(37*sqrt(7)*(3*x + 2)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) -
14*sqrt(10)*(3*x + 2)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*sqrt
(5*x + 3)*sqrt(-2*x + 1))/(3*x + 2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 2.31798, size = 358, normalized size = 3.93 \begin{align*} \frac{1}{1260} \, \sqrt{5}{\left (37 \, \sqrt{70} \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - 140 \, \sqrt{2}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{9240 \, \sqrt{2}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1260*sqrt(5)*(37*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 140*sqrt(2)*(pi + 2*arctan(-1/4*sqrt(5*x +
3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 9240*sqrt(2
)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/
(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2
+ 280))