3.2371 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=135 \[ -\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{27}{100} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{5 x+3}}{16000}+\frac{35511 \sqrt{1-2 x} \sqrt{5 x+3}}{160000}+\frac{390621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (35511*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/160000 - (63*(35 - 8*
x)*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16000 + (27*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/100 + (390621*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(160000*Sqrt[10])

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Rubi [A]  time = 0.0368265, antiderivative size = 135, 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, 153, 147, 50, 54, 216} \[ -\frac{2 (1-2 x)^{3/2} (3 x+2)^3}{5 \sqrt{5 x+3}}+\frac{27}{100} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{5 x+3}}{16000}+\frac{35511 \sqrt{1-2 x} \sqrt{5 x+3}}{160000}+\frac{390621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{160000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(5*Sqrt[3 + 5*x]) + (35511*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/160000 - (63*(35 - 8*
x)*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/16000 + (27*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/100 + (390621*ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]])/(160000*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 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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, 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]

Rubi steps

\begin{align*} \int \frac{(1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{(3-27 x) \sqrt{1-2 x} (2+3 x)^2}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{100} \int \frac{\sqrt{1-2 x} (2+3 x) \left (-105+\frac{63 x}{2}\right )}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt{3+5 x}}-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{3+5 x}}{16000}+\frac{27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{35511 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{32000}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{35511 \sqrt{1-2 x} \sqrt{3+5 x}}{160000}-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{3+5 x}}{16000}+\frac{27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{390621 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{320000}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{35511 \sqrt{1-2 x} \sqrt{3+5 x}}{160000}-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{3+5 x}}{16000}+\frac{27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{390621 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{160000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^3}{5 \sqrt{3+5 x}}+\frac{35511 \sqrt{1-2 x} \sqrt{3+5 x}}{160000}-\frac{63 (35-8 x) (1-2 x)^{3/2} \sqrt{3+5 x}}{16000}+\frac{27}{100} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}+\frac{390621 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{160000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0354649, size = 88, normalized size = 0.65 \[ \frac{10 \left (864000 x^5+446400 x^4-1014120 x^3-346790 x^2+223559 x+46783\right )-390621 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1600000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(46783 + 223559*x - 346790*x^2 - 1014120*x^3 + 446400*x^4 + 864000*x^5) - 390621*Sqrt[10 - 20*x]*Sqrt[3 +
5*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1600000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.011, size = 133, normalized size = 1. \begin{align*}{\frac{1}{3200000} \left ( -8640000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-8784000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+1953105\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+5749200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+1171863\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +6342500\,x\sqrt{-10\,{x}^{2}-x+3}+935660\,\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((1-2*x)^(3/2)*(2+3*x)^3/(3+5*x)^(3/2),x)

[Out]

1/3200000*(-8640000*x^4*(-10*x^2-x+3)^(1/2)-8784000*x^3*(-10*x^2-x+3)^(1/2)+1953105*10^(1/2)*arcsin(20/11*x+1/
11)*x+5749200*x^2*(-10*x^2-x+3)^(1/2)+1171863*10^(1/2)*arcsin(20/11*x+1/11)+6342500*x*(-10*x^2-x+3)^(1/2)+9356
60*(-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 [C]  time = 2.86404, size = 248, normalized size = 1.84 \begin{align*} \frac{27}{500} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{35937}{1000000} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{23}{11}\right ) + \frac{1378113}{16000000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{171}{10000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{297}{2500} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} x + \frac{9801}{40000} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{6831}{50000} \, \sqrt{10 \, x^{2} + 23 \, x + \frac{51}{5}} + \frac{28809}{800000} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{625 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1250 \,{\left (5 \, x + 3\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{3125 \,{\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)^(3/2),x, algorithm="maxima")

[Out]

27/500*(-10*x^2 - x + 3)^(3/2)*x - 35937/1000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 1378113/16000000*
sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 171/10000*(-10*x^2 - x + 3)^(3/2) + 297/2500*sqrt(10*x^2 + 23*x + 51/
5)*x + 9801/40000*sqrt(-10*x^2 - x + 3)*x + 6831/50000*sqrt(10*x^2 + 23*x + 51/5) + 28809/800000*sqrt(-10*x^2
- x + 3) + 1/625*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 9/1250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 33/3
125*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.52848, size = 298, normalized size = 2.21 \begin{align*} -\frac{390621 \, \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 (432000 \, x^{4} + 439200 \, x^{3} - 287460 \, x^{2} - 317125 \, x - 46783\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3200000 \,{\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)^(3/2),x, algorithm="fricas")

[Out]

-1/3200000*(390621*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*(432000*x^4 + 439200*x^3 - 287460*x^2 - 317125*x - 46783)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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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)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.62263, size = 185, normalized size = 1.37 \begin{align*} -\frac{1}{4000000} \,{\left (36 \,{\left (8 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 83 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 805 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 128915 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{390621}{1600000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{31250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{15625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\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)^(3/2),x, algorithm="giac")

[Out]

-1/4000000*(36*(8*(12*sqrt(5)*(5*x + 3) - 83*sqrt(5))*(5*x + 3) - 805*sqrt(5))*(5*x + 3) + 128915*sqrt(5))*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 390621/1600000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 11/31250*sqrt(10)*(
sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/15625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))