3.2590 \(\int \frac{(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=135 \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt{1-2 x}}-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac{4246733 \sqrt{1-2 x} \sqrt{5 x+3}}{14080}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280 \sqrt{10}} \]

[Out]

(-4246733*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/14080 - (101*(2 + 3*x)^2*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x]) + ((2 + 3*
x)^3*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(59719 + 28200*x))/3520 + (424673
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280*Sqrt[10])

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Rubi [A]  time = 0.0372092, 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, 150, 147, 50, 54, 216} \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{101 (5 x+3)^{3/2} (3 x+2)^2}{22 \sqrt{1-2 x}}-\frac{3 \sqrt{1-2 x} (5 x+3)^{3/2} (28200 x+59719)}{3520}-\frac{4246733 \sqrt{1-2 x} \sqrt{5 x+3}}{14080}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1280 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4246733*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/14080 - (101*(2 + 3*x)^2*(3 + 5*x)^(3/2))/(22*Sqrt[1 - 2*x]) + ((2 + 3*
x)^3*(3 + 5*x)^(3/2))/(3*(1 - 2*x)^(3/2)) - (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(59719 + 28200*x))/3520 + (424673
3*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280*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 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{(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^2 \sqrt{3+5 x} \left (42+\frac{135 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-\frac{9969}{2}-\frac{31725 x}{4}\right ) (2+3 x) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{7040}\\ &=-\frac{4246733 \sqrt{1-2 x} \sqrt{3+5 x}}{14080}-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2560}\\ &=-\frac{4246733 \sqrt{1-2 x} \sqrt{3+5 x}}{14080}-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1280 \sqrt{5}}\\ &=-\frac{4246733 \sqrt{1-2 x} \sqrt{3+5 x}}{14080}-\frac{101 (2+3 x)^2 (3+5 x)^{3/2}}{22 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{3 (1-2 x)^{3/2}}-\frac{3 \sqrt{1-2 x} (3+5 x)^{3/2} (59719+28200 x)}{3520}+\frac{4246733 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1280 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0697551, size = 79, normalized size = 0.59 \[ \frac{12740199 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (86400 x^4+447120 x^3+1544724 x^2-5349344 x+1925361\right )}{38400 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(1925361 - 5349344*x + 1544724*x^2 + 447120*x^3 + 86400*x^4) + 12740199*Sqrt[10 - 20*x]*(-1
+ 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(38400*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.013, size = 154, normalized size = 1.1 \begin{align*}{\frac{1}{76800\, \left ( 2\,x-1 \right ) ^{2}} \left ( -1728000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-8942400\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-50960796\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-30894480\,{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}-38507220\,\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)^(3/2)/(1-2*x)^(5/2),x)

[Out]

1/76800*(-1728000*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-8942400*x^3*(-10*x^2-x+3)
^(1/2)-50960796*10^(1/2)*arcsin(20/11*x+1/11)*x-30894480*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/1
1*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-38507220*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2*x-1)^2/
(-10*x^2-x+3)^(1/2)

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Maxima [C]  time = 3.65015, size = 285, normalized size = 2.11 \begin{align*} \frac{428267}{2560} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{35937}{25600} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{9}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{297}{64} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{6237}{1280} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{6237}{128} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{48 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{189 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x - 1\right )}} + \frac{3773 \, \sqrt{-10 \, x^{2} - x + 3}}{96 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3479 \, \sqrt{-10 \, x^{2} - x + 3}}{6 \,{\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)^(3/2)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

428267/2560*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 35937/25600*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) + 9
/16*(-10*x^2 - x + 3)^(3/2) - 297/64*sqrt(10*x^2 - 21*x + 8)*x + 6237/1280*sqrt(10*x^2 - 21*x + 8) - 6237/128*
sqrt(-10*x^2 - x + 3) - 343/48*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 441/16*(-10*x^2 - x + 3)^(
3/2)/(4*x^2 - 4*x + 1) + 189/32*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 3773/96*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x
+ 1) + 3479/6*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.51703, size = 324, normalized size = 2.4 \begin{align*} -\frac{12740199 \, \sqrt{10}{\left (4 \, x^{2} - 4 \, 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 (86400 \, x^{4} + 447120 \, x^{3} + 1544724 \, x^{2} - 5349344 \, x + 1925361\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{76800 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/76800*(12740199*sqrt(10)*(4*x^2 - 4*x + 1)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10
*x^2 + x - 3)) + 20*(86400*x^4 + 447120*x^3 + 1544724*x^2 - 5349344*x + 1925361)*sqrt(5*x + 3)*sqrt(-2*x + 1))
/(4*x^2 - 4*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)**(3/2)/(1-2*x)**(5/2),x)

[Out]

Timed out

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Giac [A]  time = 1.48061, size = 131, normalized size = 0.97 \begin{align*} \frac{4246733}{12800} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (27 \,{\left (4 \,{\left (8 \, \sqrt{5}{\left (5 \, x + 3\right )} + 111 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 8579 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 8493466 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 140142189 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{480000 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4246733/12800*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/480000*(4*(27*(4*(8*sqrt(5)*(5*x + 3) + 111*sqr
t(5))*(5*x + 3) + 8579*sqrt(5))*(5*x + 3) - 8493466*sqrt(5))*(5*x + 3) + 140142189*sqrt(5))*sqrt(5*x + 3)*sqrt
(-10*x + 5)/(2*x - 1)^2