3.2520 \(\int \frac{(2+3 x)^3 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=110 \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{7}{4} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{\sqrt{1-2 x} \sqrt{5 x+3} (73380 x+176833)}{3200}-\frac{1463447 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

[Out]

(7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4 + ((2 + 3*x)^3*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (Sqrt[1 - 2*x]*Sqr
t[3 + 5*x]*(176833 + 73380*x))/3200 - (1463447*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*Sqrt[10])

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Rubi [A]  time = 0.0281858, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {97, 153, 147, 54, 216} \[ \frac{\sqrt{5 x+3} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{7}{4} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^2+\frac{\sqrt{1-2 x} \sqrt{5 x+3} (73380 x+176833)}{3200}-\frac{1463447 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{3200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/4 + ((2 + 3*x)^3*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + (Sqrt[1 - 2*x]*Sqr
t[3 + 5*x]*(176833 + 73380*x))/3200 - (1463447*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(3200*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 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{(2+3 x)^3 \sqrt{3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^2 \left (32+\frac{105 x}{2}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7}{4} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{1}{30} \int \frac{\left (-\frac{5625}{2}-\frac{18345 x}{4}\right ) (2+3 x)}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{7}{4} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} \sqrt{3+5 x} (176833+73380 x)}{3200}-\frac{1463447 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{6400}\\ &=\frac{7}{4} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} \sqrt{3+5 x} (176833+73380 x)}{3200}-\frac{1463447 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{3200 \sqrt{5}}\\ &=\frac{7}{4} \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}+\frac{(2+3 x)^3 \sqrt{3+5 x}}{\sqrt{1-2 x}}+\frac{\sqrt{1-2 x} \sqrt{3+5 x} (176833+73380 x)}{3200}-\frac{1463447 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{3200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0336668, size = 69, normalized size = 0.63 \[ \frac{1463447 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (14400 x^3+57960 x^2+142686 x-224833\right )}{32000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-224833 + 142686*x + 57960*x^2 + 14400*x^3) + 1463447*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sq
rt[1 - 2*x]])/(32000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.011, size = 123, normalized size = 1.1 \begin{align*} -{\frac{1}{128000\,x-64000} \left ( -288000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2926894\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-1159200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-1463447\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -2853720\,x\sqrt{-10\,{x}^{2}-x+3}+4496660\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64000*(-288000*x^3*(-10*x^2-x+3)^(1/2)+2926894*10^(1/2)*arcsin(20/11*x+1/11)*x-1159200*x^2*(-10*x^2-x+3)^(1
/2)-1463447*10^(1/2)*arcsin(20/11*x+1/11)-2853720*x*(-10*x^2-x+3)^(1/2)+4496660*(-10*x^2-x+3)^(1/2))*(1-2*x)^(
1/2)*(3+5*x)^(1/2)/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.04009, size = 107, normalized size = 0.97 \begin{align*} -\frac{1463447}{64000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{9}{40} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{1593}{160} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{89793}{3200} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \, \sqrt{-10 \, x^{2} - x + 3}}{8 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1463447/64000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 9/40*(-10*x^2 - x + 3)^(3/2) + 1593/160*sqrt(-10*x^2 -
x + 3)*x + 89793/3200*sqrt(-10*x^2 - x + 3) - 343/8*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.78956, size = 277, normalized size = 2.52 \begin{align*} \frac{1463447 \, \sqrt{10}{\left (2 \, x - 1\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 (14400 \, x^{3} + 57960 \, x^{2} + 142686 \, x - 224833\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{64000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64000*(1463447*sqrt(10)*(2*x - 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x -
3)) + 20*(14400*x^3 + 57960*x^2 + 142686*x - 224833)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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

[Out]

Timed out

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Giac [A]  time = 2.15169, size = 113, normalized size = 1.03 \begin{align*} -\frac{1463447}{32000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (18 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 89 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4927 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1463447 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{80000 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1463447/32000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/80000*(18*(4*(8*sqrt(5)*(5*x + 3) + 89*sqrt(5)
)*(5*x + 3) + 4927*sqrt(5))*(5*x + 3) - 1463447*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)