3.2282 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\)

Optimal. Leaf size=122 \[ \frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}-\frac{11 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)^2}-\frac{121 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

[Out]

(-121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) - (11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(84*(2 + 3*x)^2) + (Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)^3) - (1331*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[
7])

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Rubi [A]  time = 0.029413, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{\sqrt{1-2 x} (5 x+3)^{5/2}}{3 (3 x+2)^3}-\frac{11 \sqrt{1-2 x} (5 x+3)^{3/2}}{84 (3 x+2)^2}-\frac{121 \sqrt{1-2 x} \sqrt{5 x+3}}{392 (3 x+2)}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{392 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

(-121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)) - (11*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(84*(2 + 3*x)^2) + (Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(3*(2 + 3*x)^3) - (1331*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(392*Sqrt[
7])

Rule 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx &=\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{11}{6} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{121}{56} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{121 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)}-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{1331}{784} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{121 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)}-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}+\frac{1331}{392} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{121 \sqrt{1-2 x} \sqrt{3+5 x}}{392 (2+3 x)}-\frac{11 \sqrt{1-2 x} (3+5 x)^{3/2}}{84 (2+3 x)^2}+\frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{3 (2+3 x)^3}-\frac{1331 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{392 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0447017, size = 74, normalized size = 0.61 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (4223 x^2+4478 x+1152\right )}{(3 x+2)^3}-3993 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{8232} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^4,x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1152 + 4478*x + 4223*x^2))/(2 + 3*x)^3 - 3993*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(S
qrt[7]*Sqrt[3 + 5*x])])/8232

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Maple [B]  time = 0.009, size = 202, normalized size = 1.7 \begin{align*}{\frac{1}{16464\, \left ( 2+3\,x \right ) ^{3}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 107811\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+215622\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+143748\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+59122\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+31944\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +62692\,x\sqrt{-10\,{x}^{2}-x+3}+16128\,\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((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^4,x)

[Out]

1/16464*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(107811*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+215
622*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+143748*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2
)/(-10*x^2-x+3)^(1/2))*x+59122*x^2*(-10*x^2-x+3)^(1/2)+31944*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+
3)^(1/2))+62692*x*(-10*x^2-x+3)^(1/2)+16128*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^3

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Maxima [A]  time = 1.75043, size = 163, normalized size = 1.34 \begin{align*} \frac{1331}{5488} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{55}{294} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{33 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{196 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{407 \, \sqrt{-10 \, x^{2} - x + 3}}{1176 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="maxima")

[Out]

1331/5488*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 55/294*sqrt(-10*x^2 - x + 3) - 1/21*(-10
*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 33/196*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 407/117
6*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.80625, size = 301, normalized size = 2.47 \begin{align*} -\frac{3993 \, \sqrt{7}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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 \,{\left (4223 \, x^{2} + 4478 \, x + 1152\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{16464 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="fricas")

[Out]

-1/16464*(3993*sqrt(7)*(27*x^3 + 54*x^2 + 36*x + 8)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3)) - 14*(4223*x^2 + 4478*x + 1152)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8
)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**4,x)

[Out]

Timed out

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Giac [B]  time = 2.5226, size = 429, normalized size = 3.52 \begin{align*} \frac{1331}{54880} \, \sqrt{70} \sqrt{10}{\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 )} - \frac{1331 \,{\left (3 \, \sqrt{10}{\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 )}^{5} + 2240 \, \sqrt{10}{\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 )}^{3} - 235200 \, \sqrt{10}{\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 )}\right )}}{588 \,{\left ({\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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^4,x, algorithm="giac")

[Out]

1331/54880*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*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)))) - 1331/588*(3*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 2240*sqrt(10)*((sqrt(2)*sqr
t(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 235200*sqrt
(10)*((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)^3