3.2599 \(\int \frac{(2+3 x)^3 (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx\)

Optimal. Leaf size=157 \[ \frac{(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{9444023 \sqrt{1-2 x} (5 x+3)^{3/2}}{33792}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac{9444023 \sqrt{1-2 x} \sqrt{5 x+3}}{4096}+\frac{103884253 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096 \sqrt{10}} \]

[Out]

(-9444023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096 - (9444023*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/33792 - (373*(2 + 3*x)^2
*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 +
5*x)^(5/2)*(81191 + 40164*x))/1408 + (103884253*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0481711, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{(5 x+3)^{5/2} (3 x+2)^3}{3 (1-2 x)^{3/2}}-\frac{373 (5 x+3)^{5/2} (3 x+2)^2}{66 \sqrt{1-2 x}}-\frac{9444023 \sqrt{1-2 x} (5 x+3)^{3/2}}{33792}-\frac{\sqrt{1-2 x} (5 x+3)^{5/2} (40164 x+81191)}{1408}-\frac{9444023 \sqrt{1-2 x} \sqrt{5 x+3}}{4096}+\frac{103884253 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{4096 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9444023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096 - (9444023*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/33792 - (373*(2 + 3*x)^2
*(3 + 5*x)^(5/2))/(66*Sqrt[1 - 2*x]) + ((2 + 3*x)^3*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) - (Sqrt[1 - 2*x]*(3 +
5*x)^(5/2)*(81191 + 40164*x))/1408 + (103884253*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096*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)^{5/2}}{(1-2 x)^{5/2}} \, dx &=\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{3} \int \frac{(2+3 x)^2 (3+5 x)^{3/2} \left (52+\frac{165 x}{2}\right )}{(1-2 x)^{3/2}} \, dx\\ &=-\frac{373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{1}{33} \int \frac{\left (-\frac{15989}{2}-\frac{50205 x}{4}\right ) (2+3 x) (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac{9444023 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx}{8448}\\ &=-\frac{9444023 \sqrt{1-2 x} (3+5 x)^{3/2}}{33792}-\frac{373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac{9444023 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx}{2048}\\ &=-\frac{9444023 \sqrt{1-2 x} \sqrt{3+5 x}}{4096}-\frac{9444023 \sqrt{1-2 x} (3+5 x)^{3/2}}{33792}-\frac{373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac{103884253 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{8192}\\ &=-\frac{9444023 \sqrt{1-2 x} \sqrt{3+5 x}}{4096}-\frac{9444023 \sqrt{1-2 x} (3+5 x)^{3/2}}{33792}-\frac{373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac{103884253 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{4096 \sqrt{5}}\\ &=-\frac{9444023 \sqrt{1-2 x} \sqrt{3+5 x}}{4096}-\frac{9444023 \sqrt{1-2 x} (3+5 x)^{3/2}}{33792}-\frac{373 (2+3 x)^2 (3+5 x)^{5/2}}{66 \sqrt{1-2 x}}+\frac{(2+3 x)^3 (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac{\sqrt{1-2 x} (3+5 x)^{5/2} (81191+40164 x)}{1408}+\frac{103884253 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{4096 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0752542, size = 84, normalized size = 0.54 \[ \frac{311652759 \sqrt{10-20 x} (2 x-1) \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-10 \sqrt{5 x+3} \left (1036800 x^5+5477760 x^4+15301008 x^3+40614996 x^2-129940960 x+47216961\right )}{122880 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(47216961 - 129940960*x + 40614996*x^2 + 15301008*x^3 + 5477760*x^4 + 1036800*x^5) + 311652
759*Sqrt[10 - 20*x]*(-1 + 2*x)*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(122880*(1 - 2*x)^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 171, normalized size = 1.1 \begin{align*}{\frac{1}{245760\, \left ( 2\,x-1 \right ) ^{2}} \left ( -20736000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-109555200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-306020160\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-1246611036\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-812299920\,{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}-944339220\,\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)^(5/2)/(1-2*x)^(5/2),x)

[Out]

1/245760*(-20736000*x^5*(-10*x^2-x+3)^(1/2)-109555200*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11
*x+1/11)*x^2-306020160*x^3*(-10*x^2-x+3)^(1/2)-1246611036*10^(1/2)*arcsin(20/11*x+1/11)*x-812299920*x^2*(-10*x
^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-944339220*(-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)

________________________________________________________________________________________

Maxima [C]  time = 3.0553, size = 439, normalized size = 2.8 \begin{align*} \frac{2606989}{2048} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{395307}{81920} i \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x - \frac{21}{11}\right ) + \frac{495}{256} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{343 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (16 \, x^{4} - 32 \, x^{3} + 24 \, x^{2} - 8 \, x + 1\right )}} - \frac{441 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} - \frac{63 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{16 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{64 \,{\left (2 \, x - 1\right )}} - \frac{16335}{1024} \, \sqrt{10 \, x^{2} - 21 \, x + 8} x + \frac{68607}{4096} \, \sqrt{10 \, x^{2} - 21 \, x + 8} - \frac{114345}{512} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{18865 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{192 \,{\left (8 \, x^{3} - 12 \, x^{2} + 6 \, x - 1\right )}} + \frac{24255 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{128 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3465 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{128 \,{\left (2 \, x - 1\right )}} + \frac{207515 \, \sqrt{-10 \, x^{2} - x + 3}}{384 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{3721795 \, \sqrt{-10 \, x^{2} - x + 3}}{768 \,{\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)^(5/2)/(1-2*x)^(5/2),x, algorithm="maxima")

[Out]

2606989/2048*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 395307/81920*I*sqrt(5)*sqrt(2)*arcsin(20/11*x - 21/11) +
495/256*(-10*x^2 - x + 3)^(3/2) - 343/16*(-10*x^2 - x + 3)^(5/2)/(16*x^4 - 32*x^3 + 24*x^2 - 8*x + 1) - 441/3
2*(-10*x^2 - x + 3)^(5/2)/(8*x^3 - 12*x^2 + 6*x - 1) - 63/16*(-10*x^2 - x + 3)^(5/2)/(4*x^2 - 4*x + 1) - 27/64
*(-10*x^2 - x + 3)^(5/2)/(2*x - 1) - 16335/1024*sqrt(10*x^2 - 21*x + 8)*x + 68607/4096*sqrt(10*x^2 - 21*x + 8)
- 114345/512*sqrt(-10*x^2 - x + 3) - 18865/192*(-10*x^2 - x + 3)^(3/2)/(8*x^3 - 12*x^2 + 6*x - 1) + 24255/128
*(-10*x^2 - x + 3)^(3/2)/(4*x^2 - 4*x + 1) + 3465/128*(-10*x^2 - x + 3)^(3/2)/(2*x - 1) + 207515/384*sqrt(-10*
x^2 - x + 3)/(4*x^2 - 4*x + 1) + 3721795/768*sqrt(-10*x^2 - x + 3)/(2*x - 1)

________________________________________________________________________________________

Fricas [A]  time = 1.50699, size = 356, normalized size = 2.27 \begin{align*} -\frac{311652759 \, \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 (1036800 \, x^{5} + 5477760 \, x^{4} + 15301008 \, x^{3} + 40614996 \, x^{2} - 129940960 \, x + 47216961\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{245760 \,{\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)^(5/2)/(1-2*x)^(5/2),x, algorithm="fricas")

[Out]

-1/245760*(311652759*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*(1036800*x^5 + 5477760*x^4 + 15301008*x^3 + 40614996*x^2 - 129940960*x + 47216961)*sqrt(
5*x + 3)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.77776, size = 149, normalized size = 0.95 \begin{align*} \frac{103884253}{40960} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{{\left (4 \,{\left (3 \,{\left (36 \,{\left (8 \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} + 137 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 13627 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9444023 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 1038842530 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 17140901745 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{7680000 \,{\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)^(5/2)/(1-2*x)^(5/2),x, algorithm="giac")

[Out]

103884253/40960*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/7680000*(4*(3*(36*(8*(12*sqrt(5)*(5*x + 3) +
137*sqrt(5))*(5*x + 3) + 13627*sqrt(5))*(5*x + 3) + 9444023*sqrt(5))*(5*x + 3) - 1038842530*sqrt(5))*(5*x + 3)
+ 17140901745*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2