3.2530 \(\int \frac{(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=132 \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{27}{16} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2+\frac{9 \sqrt{1-2 x} (5 x+3)^{3/2} (29320 x+62091)}{12800}+\frac{13246251 \sqrt{1-2 x} \sqrt{5 x+3}}{51200}-\frac{145708761 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200 \sqrt{10}} \]

[Out]

(13246251*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200 + (27*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/16 + ((2 + 3*x)^
3*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(62091 + 29320*x))/12800 - (145708761*ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200*Sqrt[10])

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Rubi [A]  time = 0.0325839, antiderivative size = 132, 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} \[ \frac{(5 x+3)^{3/2} (3 x+2)^3}{\sqrt{1-2 x}}+\frac{27}{16} \sqrt{1-2 x} (5 x+3)^{3/2} (3 x+2)^2+\frac{9 \sqrt{1-2 x} (5 x+3)^{3/2} (29320 x+62091)}{12800}+\frac{13246251 \sqrt{1-2 x} \sqrt{5 x+3}}{51200}-\frac{145708761 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{51200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13246251*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200 + (27*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/16 + ((2 + 3*x)^
3*(3 + 5*x)^(3/2))/Sqrt[1 - 2*x] + (9*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)*(62091 + 29320*x))/12800 - (145708761*ArcS
in[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200*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{(2+3 x)^3 (3+5 x)^{3/2}}{(1-2 x)^{3/2}} \, dx &=\frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}}-\int \frac{(2+3 x)^2 \sqrt{3+5 x} \left (42+\frac{135 x}{2}\right )}{\sqrt{1-2 x}} \, dx\\ &=\frac{27}{16} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{1}{40} \int \frac{\left (-\frac{10365}{2}-\frac{32985 x}{4}\right ) (2+3 x) \sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=\frac{27}{16} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac{13246251 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{25600}\\ &=\frac{13246251 \sqrt{1-2 x} \sqrt{3+5 x}}{51200}+\frac{27}{16} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac{145708761 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{102400}\\ &=\frac{13246251 \sqrt{1-2 x} \sqrt{3+5 x}}{51200}+\frac{27}{16} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac{145708761 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{51200 \sqrt{5}}\\ &=\frac{13246251 \sqrt{1-2 x} \sqrt{3+5 x}}{51200}+\frac{27}{16} \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}+\frac{(2+3 x)^3 (3+5 x)^{3/2}}{\sqrt{1-2 x}}+\frac{9 \sqrt{1-2 x} (3+5 x)^{3/2} (62091+29320 x)}{12800}-\frac{145708761 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{51200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0406129, size = 74, normalized size = 0.56 \[ \frac{145708761 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (864000 x^4+3729600 x^3+8057880 x^2+15218818 x-22217679\right )}{512000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-22217679 + 15218818*x + 8057880*x^2 + 3729600*x^3 + 864000*x^4) + 145708761*Sqrt[10 - 20*
x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(512000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.013, size = 140, normalized size = 1.1 \begin{align*} -{\frac{1}{2048000\,x-1024000} \left ( -17280000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-74592000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-161157600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-145708761\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -304376360\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\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)^(3/2),x)

[Out]

-1/1024000*(-17280000*x^4*(-10*x^2-x+3)^(1/2)-74592000*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11
*x+1/11)*x-161157600*x^2*(-10*x^2-x+3)^(1/2)-145708761*10^(1/2)*arcsin(20/11*x+1/11)-304376360*x*(-10*x^2-x+3)
^(1/2)[phone]*(-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 [C]  time = 3.16083, size = 248, normalized size = 1.88 \begin{align*} -\frac{27}{32} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{155771121}{1024000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{251559}{25600} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) - \frac{2547}{640} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{2079}{64} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x - \frac{9801}{2560} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{43659}{1280} \, \sqrt{10 \, x^{2} - 21 \, x + 8} + \frac{5811399}{51200} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{343 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{32 \,{\left (2 \, x - 1\right )}} - \frac{11319 \, \sqrt{-10 \, x^{2} - x + 3}}{32 \,{\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)^(3/2),x, algorithm="maxima")

[Out]

-27/32*(-10*x^2 - x + 3)^(3/2)*x - 155771121/1024000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 251559/25600*I*s
qrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) - 2547/640*(-10*x^2 - x + 3)^(3/2) + 2079/64*sqrt(10*x^2 - 21*x + 8)*x
- 9801/2560*sqrt(-10*x^2 - x + 3)*x - 43659/1280*sqrt(10*x^2 - 21*x + 8) + 5811399/51200*sqrt(-10*x^2 - x + 3)
- 343/16*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) - 441/32*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) - 11319/32*sqrt
(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.73601, size = 311, normalized size = 2.36 \begin{align*} \frac{145708761 \, \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 (864000 \, x^{4} + 3729600 \, x^{3} + 8057880 \, x^{2} + 15218818 \, x - 22217679\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{1024000 \,{\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)^(3/2),x, algorithm="fricas")

[Out]

1/1024000*(145708761*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*(864000*x^4 + 3729600*x^3 + 8057880*x^2 + 15218818*x - 22217679)*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)**(3/2)/(1-2*x)**(3/2),x)

[Out]

Timed out

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Giac [A]  time = 1.35699, size = 131, normalized size = 0.99 \begin{align*} -\frac{145708761}{512000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) + \frac{{\left (2 \,{\left (36 \,{\left (8 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 115 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 8919 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 4415417 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 145708761 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1280000 \,{\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)^(3/2),x, algorithm="giac")

[Out]

-145708761/512000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 1/1280000*(2*(36*(8*(12*sqrt(5)*(5*x + 3) + 1
15*sqrt(5))*(5*x + 3) + 8919*sqrt(5))*(5*x + 3) + 4415417*sqrt(5))*(5*x + 3) - 145708761*sqrt(5))*sqrt(5*x + 3
)*sqrt(-10*x + 5)/(2*x - 1)