3.2394 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac{19}{18} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

[Out]

(19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/18 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (5*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(4*(2 + 3*x)) + (118*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (155*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/108

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Rubi [A]  time = 0.0538979, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ -\frac{\sqrt{5 x+3} (1-2 x)^{5/2}}{6 (3 x+2)^2}+\frac{5 \sqrt{5 x+3} (1-2 x)^{3/2}}{4 (3 x+2)}+\frac{19}{18} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(19*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/18 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(6*(2 + 3*x)^2) + (5*(1 - 2*x)^(3/2)*Sqr
t[3 + 5*x])/(4*(2 + 3*x)) + (118*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/27 - (155*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/108

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 149

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

Rule 154

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

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{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^3} \, dx &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{1}{6} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^2 \sqrt{3+5 x}} \, dx\\ &=-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}-\frac{1}{18} \int \frac{\left (-\frac{915}{4}-285 x\right ) \sqrt{1-2 x}}{(2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}-\frac{1}{270} \int \frac{-\frac{14865}{4}-3540 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{118}{27} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{1085}{216} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{1085}{108} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )+\frac{236 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{27 \sqrt{5}}\\ &=\frac{19}{18} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{6 (2+3 x)^2}+\frac{5 (1-2 x)^{3/2} \sqrt{3+5 x}}{4 (2+3 x)}+\frac{118}{27} \sqrt{\frac{2}{5}} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{155}{108} \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.145071, size = 126, normalized size = 0.88 \[ \frac{-15 \sqrt{5 x+3} \left (96 x^3+822 x^2+37 x-236\right )-472 \sqrt{10-20 x} (3 x+2)^2 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-775 \sqrt{7-14 x} (3 x+2)^2 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{540 \sqrt{1-2 x} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[3 + 5*x]*(-236 + 37*x + 822*x^2 + 96*x^3) - 472*Sqrt[10 - 20*x]*(2 + 3*x)^2*ArcSin[Sqrt[5/11]*Sqrt[1
- 2*x]] - 775*Sqrt[7 - 14*x]*(2 + 3*x)^2*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(540*Sqrt[1 - 2*x]*(2
+ 3*x)^2)

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Maple [A]  time = 0.01, size = 208, normalized size = 1.4 \begin{align*}{\frac{1}{1080\, \left ( 2+3\,x \right ) ^{2}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6975\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+4248\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}+9300\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+5664\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+1440\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3100\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) +1888\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +13050\,x\sqrt{-10\,{x}^{2}-x+3}+7080\,\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)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^3,x)

[Out]

1/1080*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(6975*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+4248*1
0^(1/2)*arcsin(20/11*x+1/11)*x^2+9300*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+5664*10^(1/
2)*arcsin(20/11*x+1/11)*x+1440*x^2*(-10*x^2-x+3)^(1/2)+3100*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3
)^(1/2))+1888*10^(1/2)*arcsin(20/11*x+1/11)+13050*x*(-10*x^2-x+3)^(1/2)+7080*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3
)^(1/2)/(2+3*x)^2

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Maxima [A]  time = 5.78158, size = 136, normalized size = 0.94 \begin{align*} \frac{59}{135} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{155}{216} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{13}{9} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{7 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{6 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{49 \, \sqrt{-10 \, x^{2} - x + 3}}{36 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

59/135*sqrt(10)*arcsin(20/11*x + 1/11) + 155/216*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1
3/9*sqrt(-10*x^2 - x + 3) + 7/6*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 49/36*sqrt(-10*x^2 - x + 3)/(3*x
+ 2)

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Fricas [A]  time = 1.59275, size = 437, normalized size = 3.03 \begin{align*} -\frac{472 \, \sqrt{5} \sqrt{2}{\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac{\sqrt{5} \sqrt{2}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) + 775 \, \sqrt{7}{\left (9 \, x^{2} + 12 \, x + 4\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 ) - 30 \,{\left (48 \, x^{2} + 435 \, x + 236\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1080 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1080*(472*sqrt(5)*sqrt(2)*(9*x^2 + 12*x + 4)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) + 775*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2
*x + 1)/(10*x^2 + x - 3)) - 30*(48*x^2 + 435*x + 236)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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

[Out]

Timed out

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Giac [B]  time = 2.31783, size = 463, normalized size = 3.22 \begin{align*} \frac{31}{432} \, \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{59}{135} \, \sqrt{10}{\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{4}{135} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{77 \,{\left (17 \, \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} - 13720 \, \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 )}}{54 \,{\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 )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

31/432*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)))) + 59/135*sqrt(10)*(pi + 2*arctan(-1/4*sqrt(5*x + 3)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 4/135*sqrt(5)*sqrt
(5*x + 3)*sqrt(-10*x + 5) - 77/54*(17*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)))^3 - 13720*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)^2