3.2318 \(\int \frac{\sqrt{1-2 x} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=142 \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{15 (5 x+3)^{3/2}}-\frac{524 \sqrt{1-2 x} (3 x+2)^3}{825 \sqrt{5 x+3}}+\frac{623 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (8940 x+2563)}{220000}+\frac{35511 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20000 \sqrt{10}} \]

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(15*(3 + 5*x)^(3/2)) - (524*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(825*Sqrt[3 + 5*x]) + (6
23*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/1375 + (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2563 + 8940*x))/220000 + (3
5511*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20000*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0429995, 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} \[ -\frac{2 \sqrt{1-2 x} (3 x+2)^4}{15 (5 x+3)^{3/2}}-\frac{524 \sqrt{1-2 x} (3 x+2)^3}{825 \sqrt{5 x+3}}+\frac{623 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} (8940 x+2563)}{220000}+\frac{35511 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(2 + 3*x)^4)/(15*(3 + 5*x)^(3/2)) - (524*Sqrt[1 - 2*x]*(2 + 3*x)^3)/(825*Sqrt[3 + 5*x]) + (6
23*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/1375 + (7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2563 + 8940*x))/220000 + (3
5511*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20000*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{\sqrt{1-2 x} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{(10-27 x) (2+3 x)^3}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{524 \sqrt{1-2 x} (2+3 x)^3}{825 \sqrt{3+5 x}}+\frac{4}{825} \int \frac{\left (882-\frac{5607 x}{2}\right ) (2+3 x)^2}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{524 \sqrt{1-2 x} (2+3 x)^3}{825 \sqrt{3+5 x}}+\frac{623 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{1375}-\frac{2 \int \frac{(2+3 x) \left (-\frac{10521}{2}+\frac{46935 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{12375}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{524 \sqrt{1-2 x} (2+3 x)^3}{825 \sqrt{3+5 x}}+\frac{623 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (2563+8940 x)}{220000}+\frac{35511 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{40000}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{524 \sqrt{1-2 x} (2+3 x)^3}{825 \sqrt{3+5 x}}+\frac{623 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (2563+8940 x)}{220000}+\frac{35511 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{20000 \sqrt{5}}\\ &=-\frac{2 \sqrt{1-2 x} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac{524 \sqrt{1-2 x} (2+3 x)^3}{825 \sqrt{3+5 x}}+\frac{623 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}{1375}+\frac{7 \sqrt{1-2 x} \sqrt{3+5 x} (2563+8940 x)}{220000}+\frac{35511 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{20000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0585051, size = 88, normalized size = 0.62 \[ \frac{-10 \left (7128000 x^5+14434200 x^4+3768930 x^3-4392275 x^2-1433776 x+218953\right )-1171863 \sqrt{10-20 x} (5 x+3)^{3/2} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6600000 \sqrt{1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(218953 - 1433776*x - 4392275*x^2 + 3768930*x^3 + 14434200*x^4 + 7128000*x^5) - 1171863*Sqrt[10 - 20*x]*(
3 + 5*x)^(3/2)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(6600000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 147, normalized size = 1. \begin{align*}{\frac{1}{13200000} \left ( 71280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+127680300\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-4379060\,\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((2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)^(5/2),x)

[Out]

1/13200000*(71280000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2+179982000*x^3*(-10*x^2
-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x+127680300*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsi
n(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-4379060*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/
(3+5*x)^(3/2)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 1.60607, size = 333, normalized size = 2.35 \begin{align*} -\frac{1171863 \, \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 (3564000 \, x^{4} + 8999100 \, x^{3} + 6384015 \, x^{2} + 995870 \, x - 218953\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{13200000 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13200000*(1171863*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*(3564000*x^4 + 8999100*x^3 + 6384015*x^2 + 995870*x - 218953)*sqrt(5*x + 3)*sqrt(-2*x
+ 1))/(25*x^2 + 30*x + 9)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 2.73914, size = 255, normalized size = 1.8 \begin{align*} \frac{27}{500000} \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 5 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 475 \, \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}}{8250000 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} + \frac{35511}{200000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{263 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{687500 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{789 \, \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}}}{515625 \,{\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((2+3*x)^4*(1-2*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

27/500000*(4*(8*sqrt(5)*(5*x + 3) + 5*sqrt(5))*(5*x + 3) - 475*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1/8250
000*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 35511/200000*sqrt(10)*arcsin(1/11*sqrt(2
2)*sqrt(5*x + 3)) - 263/687500*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1/515625*(789*sqr
t(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