3.2436 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=186 \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt{5 x+3}}+\frac{13}{50} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^3+\frac{111 (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2}{5000}-\frac{(1-2 x)^{5/2} \sqrt{5 x+3} (1990620 x+2725981)}{8000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{5 x+3}}{32000000}+\frac{118054167 \sqrt{1-2 x} \sqrt{5 x+3}}{320000000}+\frac{1298595837 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000000 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(5*Sqrt[3 + 5*x]) + (118054167*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320000000 + (3577
399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/32000000 + (111*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/5000 + (13*(1 -
2*x)^(5/2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/50 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(2725981 + 1990620*x))/8000000 + (12
98595837*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320000000*Sqrt[10])

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Rubi [A]  time = 0.0635421, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ -\frac{2 (1-2 x)^{5/2} (3 x+2)^4}{5 \sqrt{5 x+3}}+\frac{13}{50} (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^3+\frac{111 (1-2 x)^{5/2} \sqrt{5 x+3} (3 x+2)^2}{5000}-\frac{(1-2 x)^{5/2} \sqrt{5 x+3} (1990620 x+2725981)}{8000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{5 x+3}}{32000000}+\frac{118054167 \sqrt{1-2 x} \sqrt{5 x+3}}{320000000}+\frac{1298595837 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320000000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(5*Sqrt[3 + 5*x]) + (118054167*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320000000 + (3577
399*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/32000000 + (111*(1 - 2*x)^(5/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/5000 + (13*(1 -
2*x)^(5/2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/50 - ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]*(2725981 + 1990620*x))/8000000 + (12
98595837*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320000000*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{(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{2}{5} \int \frac{(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{1}{150} \int \frac{(1-2 x)^{3/2} (2+3 x)^2 \left (-162+\frac{333 x}{2}\right )}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}}{5000}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}+\frac{\int \frac{(1-2 x)^{3/2} (2+3 x) \left (\frac{34731}{2}+\frac{99531 x}{4}\right )}{\sqrt{3+5 x}} \, dx}{7500}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}}{5000}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x} (2725981+1990620 x)}{8000000}+\frac{3577399 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{3200000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{3577399 (1-2 x)^{3/2} \sqrt{3+5 x}}{32000000}+\frac{111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}}{5000}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x} (2725981+1990620 x)}{8000000}+\frac{118054167 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{64000000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{118054167 \sqrt{1-2 x} \sqrt{3+5 x}}{320000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{3+5 x}}{32000000}+\frac{111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}}{5000}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x} (2725981+1990620 x)}{8000000}+\frac{1298595837 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{640000000}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{118054167 \sqrt{1-2 x} \sqrt{3+5 x}}{320000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{3+5 x}}{32000000}+\frac{111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}}{5000}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x} (2725981+1990620 x)}{8000000}+\frac{1298595837 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{320000000 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^4}{5 \sqrt{3+5 x}}+\frac{118054167 \sqrt{1-2 x} \sqrt{3+5 x}}{320000000}+\frac{3577399 (1-2 x)^{3/2} \sqrt{3+5 x}}{32000000}+\frac{111 (1-2 x)^{5/2} (2+3 x)^2 \sqrt{3+5 x}}{5000}+\frac{13}{50} (1-2 x)^{5/2} (2+3 x)^3 \sqrt{3+5 x}-\frac{(1-2 x)^{5/2} \sqrt{3+5 x} (2725981+1990620 x)}{8000000}+\frac{1298595837 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{320000000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0509287, size = 98, normalized size = 0.53 \[ \frac{-10 \left (6912000000 x^7+4631040000 x^6-9103968000 x^5-4815780800 x^4+5550785640 x^3+1793366630 x^2-1029299623 x-168414751\right )-1298595837 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3200000000 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-168414751 - 1029299623*x + 1793366630*x^2 + 5550785640*x^3 - 4815780800*x^4 - 9103968000*x^5 + 46310400
00*x^6 + 6912000000*x^7) - 1298595837*Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3200000
000*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.012, size = 167, normalized size = 0.9 \begin{align*}{\frac{1}{6400000000} \left ( 69120000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}[phone]\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-50604480000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-73460048000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+18777832400\,{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}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6400000000*(69120000000*(-10*x^2-x+3)^(1/2)*x^6+80870400000*x^5*(-10*x^2-x+3)^(1/2)-50604480000*x^4*(-10*x^2
-x+3)^(1/2)-73460048000*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x+18777832400*x^2*(-1
0*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)[phone]*(-10*x^
2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 1.9661, size = 193, normalized size = 1.04 \begin{align*} -\frac{108 \, x^{7}}{5 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1809 \, x^{6}}{125 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{284499 \, x^{5}}{10000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{3009863 \, x^{4}}{200000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{138769641 \, x^{3}}{8000000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{179336663 \, x^{2}}{32000000 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{1298595837}{6400000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{1029299623 \, x}{320000000 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{168414751}{320000000 \, \sqrt{-10 \, x^{2} - x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-108/5*x^7/sqrt(-10*x^2 - x + 3) - 1809/125*x^6/sqrt(-10*x^2 - x + 3) + 284499/10000*x^5/sqrt(-10*x^2 - x + 3)
+ 3009863/200000*x^4/sqrt(-10*x^2 - x + 3) - 138769641/8000000*x^3/sqrt(-10*x^2 - x + 3) - 179336663/32000000
*x^2/sqrt(-10*x^2 - x + 3) - 1298595837/6400000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1029299623/320000000*x/s
qrt(-10*x^2 - x + 3) + 168414751/320000000/sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.88331, size = 379, normalized size = 2.04 \begin{align*} -\frac{1298595837 \, \sqrt{10}{\left (5 \, x + 3\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 (3456000000 \, x^{6} + 4043520000 \, x^{5} - 2530224000 \, x^{4} - 3673002400 \, x^{3} + 938891620 \, x^{2} + 1366129125 \, x + 168414751\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{6400000000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6400000000*(1298595837*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*
x^2 + x - 3)) - 20*(3456000000*x^6 + 4043520000*x^5 - 2530224000*x^4 - 3673002400*x^3 + 938891620*x^2 + 136612
9125*x + 168414751)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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

[Out]

Timed out

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Giac [A]  time = 1.8815, size = 220, normalized size = 1.18 \begin{align*} \frac{1}{8000000000} \,{\left (4 \,{\left (8 \,{\left (108 \,{\left (16 \,{\left (20 \, \sqrt{5}{\left (5 \, x + 3\right )} - 243 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 9263 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 2532859 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} + 3473645 \, \sqrt{5}\right )}{\left (5 \, x + 3\right )} - 533500275 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{1298595837}{3200000000} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{121 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{781250 \, \sqrt{5 \, x + 3}} + \frac{242 \, \sqrt{10} \sqrt{5 \, x + 3}}{390625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8000000000*(4*(8*(108*(16*(20*sqrt(5)*(5*x + 3) - 243*sqrt(5))*(5*x + 3) + 9263*sqrt(5))*(5*x + 3) + 2532859
*sqrt(5))*(5*x + 3) + 3473645*sqrt(5))*(5*x + 3) - 533500275*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 12985958
37/3200000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 121/781250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) + 242/390625*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))