3.2380 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{128 \sqrt{1-2 x} (3 x+2)^3}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac{378}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (1140 x+853)}{10000}+\frac{13153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (128*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(25*Sqrt[3 + 5*x]) + (
378*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/125 + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(853 + 1140*x))/10000 + (13
153*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10000*Sqrt[10])

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Rubi [A]  time = 0.0427989, antiderivative size = 142, 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, 150, 153, 147, 54, 216} \[ -\frac{128 \sqrt{1-2 x} (3 x+2)^3}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac{378}{125} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{21 \sqrt{1-2 x} \sqrt{5 x+3} (1140 x+853)}{10000}+\frac{13153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{10000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (128*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(25*Sqrt[3 + 5*x]) + (
378*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/125 + (21*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(853 + 1140*x))/10000 + (13
153*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10000*Sqrt[10])

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 150

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] && IntegersQ[2*m, 2*n, 2*p]

Rule 153

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] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 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{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{(3-27 x) \sqrt{1-2 x} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{128 \sqrt{1-2 x} (2+3 x)^3}{25 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{\left (\frac{1029}{2}-1701 x\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{128 \sqrt{1-2 x} (2+3 x)^3}{25 \sqrt{3+5 x}}+\frac{378}{125} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}-\frac{2 \int \frac{(2+3 x) \left (-1953+\frac{17955 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1125}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{128 \sqrt{1-2 x} (2+3 x)^3}{25 \sqrt{3+5 x}}+\frac{378}{125} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (853+1140 x)}{10000}+\frac{13153 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{20000}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{128 \sqrt{1-2 x} (2+3 x)^3}{25 \sqrt{3+5 x}}+\frac{378}{125} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (853+1140 x)}{10000}+\frac{13153 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{10000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac{128 \sqrt{1-2 x} (2+3 x)^3}{25 \sqrt{3+5 x}}+\frac{378}{125} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{21 \sqrt{1-2 x} \sqrt{3+5 x} (853+1140 x)}{10000}+\frac{13153 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{10000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0521605, size = 88, normalized size = 0.62 \[ \frac{10 \left (216000 x^5+59400 x^4-320490 x^3-141425 x^2+67568 x+31171\right )-39459 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{300000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(31171 + 67568*x - 141425*x^2 - 320490*x^3 + 59400*x^4 + 216000*x^5) - 39459*Sqrt[10 - 20*x]*(3 + 5*x)^(3/
2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(300000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.011, size = 147, normalized size = 1. \begin{align*}{\frac{1}{600000} \left ( -2160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}+986475\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-1674000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1183770\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+2367900\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+355131\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +2598200\,x\sqrt{-10\,{x}^{2}-x+3}+623420\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/600000*(-2160000*x^4*(-10*x^2-x+3)^(1/2)+986475*10^(1/2)*arcsin(20/11*x+1/11)*x^2-1674000*x^3*(-10*x^2-x+3)^
(1/2)+1183770*10^(1/2)*arcsin(20/11*x+1/11)*x+2367900*x^2*(-10*x^2-x+3)^(1/2)+355131*10^(1/2)*arcsin(20/11*x+1
/11)+2598200*x*(-10*x^2-x+3)^(1/2)+623420*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(3/2)

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Maxima [C]  time = 3.03578, size = 285, normalized size = 2.01 \begin{align*} -\frac{35937}{1000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{7457}{250000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{9}{625} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{297}{2500} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{6831}{50000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{891}{12500} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1875 \,{\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1250 \,{\left (5 \, x + 3\right )}} - \frac{11 \, \sqrt{-10 \, x^{2} - x + 3}}{9375 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{877 \, \sqrt{-10 \, x^{2} - x + 3}}{9375 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-35937/1000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 7457/250000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11)
+ 9/625*(-10*x^2 - x + 3)^(3/2) + 297/2500*sqrt(10*x^2 + 23*x + 51/5)*x + 6831/50000*sqrt(10*x^2 + 23*x + 51/5
) + 891/12500*sqrt(-10*x^2 - x + 3) - 1/1875*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 9/625*
(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 27/1250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 11/9375*sqrt(-10*x^2
- x + 3)/(25*x^2 + 30*x + 9) - 877/9375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.58395, size = 321, normalized size = 2.26 \begin{align*} -\frac{39459 \, \sqrt{10}{\left (25 \, x^{2} + 30 \, x + 9\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 (108000 \, x^{4} + 83700 \, x^{3} - 118395 \, x^{2} - 129910 \, x - 31171\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{600000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/600000*(39459*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10
*x^2 + x - 3)) + 20*(108000*x^4 + 83700*x^3 - 118395*x^2 - 129910*x - 31171)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25
*x^2 + 30*x + 9)

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

[Out]

Timed out

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Giac [A]  time = 2.84274, size = 255, normalized size = 1.8 \begin{align*} -\frac{9}{250000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} - 65 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 265 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - \frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{750000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{13153}{100000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{193 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{62500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{579 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{46875 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/250000*(4*(8*sqrt(5)*(5*x + 3) - 65*sqrt(5))*(5*x + 3) - 265*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/750
000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 13153/100000*sqrt(10)*arcsin(1/11*sqrt(2
2)*sqrt(5*x + 3)) - 193/62500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/46875*(579*sqrt(
10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))^3