3.2447 \(\int \frac {(1-2 x)^{5/2} (2+3 x)^4}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=193 \[ -\frac {508 (1-2 x)^{3/2} (3 x+2)^4}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}+\frac {2514}{625} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3+\frac {23991 (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2}{25000}+\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (118392 x+64435)}{4000000}+\frac {8026963 \sqrt {1-2 x} \sqrt {5 x+3}}{40000000}+\frac {88296593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40000000 \sqrt {10}} \]

[Out]

-2/15*(1-2*x)^(5/2)*(2+3*x)^4/(3+5*x)^(3/2)[phone]/400000000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-50
8/75*(1-2*x)^(3/2)*(2+3*x)^4/(3+5*x)^(1/2)+23991/25000*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2)+2514/625*(1-2*x)^
(3/2)*(2+3*x)^3*(3+5*x)^(1/2)+21/4000000*(1-2*x)^(3/2)*(64435+118392*x)*(3+5*x)^(1/2)+8026963/40000000*(1-2*x)
^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.07, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {97, 150, 153, 147, 50, 54, 216} \[ -\frac {508 (1-2 x)^{3/2} (3 x+2)^4}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^4}{15 (5 x+3)^{3/2}}+\frac {2514}{625} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^3+\frac {23991 (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2}{25000}+\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (118392 x+64435)}{4000000}+\frac {8026963 \sqrt {1-2 x} \sqrt {5 x+3}}{40000000}+\frac {88296593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40000000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^4)/(15*(3 + 5*x)^(3/2)) - (508*(1 - 2*x)^(3/2)*(2 + 3*x)^4)/(75*Sqrt[3 + 5*x]) +
(8026963*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40000000 + (23991*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/25000 + (2
514*(1 - 2*x)^(3/2)*(2 + 3*x)^3*Sqrt[3 + 5*x])/625 + (21*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]*(64435 + 118392*x))/400
0000 + (88296593*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(40000000*Sqrt[10])

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 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 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 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 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 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)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(2-39 x) (1-2 x)^{3/2} (2+3 x)^3}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {(552-3771 x) \sqrt {1-2 x} (2+3 x)^3}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}-\frac {2 \int \frac {\sqrt {1-2 x} (2+3 x)^2 \left (-2406+\frac {71973 x}{2}\right )}{\sqrt {3+5 x}} \, dx}{1875}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}}{25000}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {\left (\frac {25095}{2}-\frac {932337 x}{4}\right ) \sqrt {1-2 x} (2+3 x)}{\sqrt {3+5 x}} \, dx}{37500}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}}{25000}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (64435+118392 x)}{4000000}+\frac {8026963 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{8000000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {8026963 \sqrt {1-2 x} \sqrt {3+5 x}}{40000000}+\frac {23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}}{25000}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (64435+118392 x)}{4000000}+\frac {88296593 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{80000000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {8026963 \sqrt {1-2 x} \sqrt {3+5 x}}{40000000}+\frac {23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}}{25000}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (64435+118392 x)}{4000000}+\frac {88296593 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{40000000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^4}{15 (3+5 x)^{3/2}}-\frac {508 (1-2 x)^{3/2} (2+3 x)^4}{75 \sqrt {3+5 x}}+\frac {8026963 \sqrt {1-2 x} \sqrt {3+5 x}}{40000000}+\frac {23991 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}}{25000}+\frac {2514}{625} (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (64435+118392 x)}{4000000}+\frac {88296593 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40000000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 98, normalized size = 0.51 \[ \frac {264889779 (5 x+3)^{3/2} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (3110400000 x^7+1697760000 x^6-4464936000 x^5-1391171400 x^4+3137091690 x^3+1095371425 x^2-558948208 x-210855251\right )}{1200000000 \sqrt {1-2 x} (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-210855251 - 558948208*x + 1095371425*x^2 + 3137091690*x^3 - 1391171400*x^4 - 4464936000*x^5 + 169776000
0*x^6 + 3110400000*x^7) + 264889779*(3 + 5*x)^(3/2)*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(1200
000000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))

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fricas [A]  time = 1.06, size = 111, normalized size = 0.58 \[ -\frac {264889779 \, \sqrt {10} {\left (25 \, x^{2} + 30 \, x + 9\right )} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 20 \, {\left (1555200000 \, x^{6} + 1626480000 \, x^{5} - 1419228000 \, x^{4} - 1405199700 \, x^{3} + 865945995 \, x^{2} + 980658710 \, x + 210855251\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2400000000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2400000000*(264889779*sqrt(10)*(25*x^2 + 30*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x
+ 1)/(10*x^2 + x - 3)) - 20*(1555200000*x^6 + 1626480000*x^5 - 1419228000*x^4 - 1405199700*x^3 + 865945995*x^2
+ 980658710*x + 210855251)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [A]  time = 2.64, size = 210, normalized size = 1.09 \[ \frac {1}{1000000000} \, {\left (12 \, {\left (24 \, {\left (12 \, {\left (48 \, \sqrt {5} {\left (5 \, x + 3\right )} - 613 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 19439 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 1264235 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} - 10674335 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11}{18750000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {3060 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {88296593}{400000000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {11 \, \sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {765 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{1171875 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1000000000*(12*(24*(12*(48*sqrt(5)*(5*x + 3) - 613*sqrt(5))*(5*x + 3) + 19439*sqrt(5))*(5*x + 3) + 1264235*s
qrt(5))*(5*x + 3) - 10674335*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) - 11/18750000*sqrt(10)*((sqrt(2)*sqrt(-10*
x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 3060*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)) + 88296593/400
000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 11/1171875*sqrt(10)*(5*x + 3)^(3/2)*(765*(sqrt(2)*sqrt(-
10*x + 5) - sqrt(22))^2/(5*x + 3) + 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3

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maple [A]  time = 0.01, size = 181, normalized size = 0.94 \[ \frac {\left (31104000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] \sqrt {-10 x^{2}-x +3}\, x^{5}-28384560000 \sqrt {-10 x^{2}-x +3}\, x^{4}-28103994000 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}}{2400000000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2400000000*(31104000000*(-10*x^2-x+3)^(1/2)*x^6+32529600000*(-10*x^2-x+3)^(1/2)*x^5-28384560000*(-10*x^2-x+3
)^(1/2)*x^4+6622244475*10^(1/2)*x^2*arcsin(20/11*x+1/11)-28103994000*(-10*x^2-x+3)^(1/2)*x^3+7946693370*10^(1/
2)*x*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x^2+2384008011*10^(1/2)*arcsin(20/11*x+1/11)+1961317
4200*(-10*x^2-x+3)^(1/2)*x+4217105020*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)/(-10*x^2-x+3)^(1/2)/(5*x+3)^(3/2)

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maxima [C]  time = 1.11, size = 354, normalized size = 1.83 \[ \frac {81}{15625} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {891}{25000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {70759953}{800000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {27401}{1250000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {8811}{500000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {6 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {18 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{3125 \, {\left (5 \, x + 3\right )}} + \frac {584793}{2000000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {13450239}{40000000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {3267}{62500} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{18750 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {33 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3125 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {99 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{6250 \, {\left (5 \, x + 3\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{93750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {638 \, \sqrt {-10 \, x^{2} - x + 3}}{9375 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/15625*(-10*x^2 - x + 3)^(5/2) + 891/25000*(-10*x^2 - x + 3)^(3/2)*x - 70759953/800000000*I*sqrt(5)*sqrt(2)*
arcsin(20/11*x + 23/11) + 27401/1250000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 8811/500000*(-10*x^2 - x + 3)
^(3/2) + 1/3125*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 81) + 6/3125*(-10*x^2 - x + 3
)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 18/3125*(-10*x^2 - x + 3)^(5/2)/(25*x^2 + 30*x + 9) + 27/3125*(-10*
x^2 - x + 3)^(5/2)/(5*x + 3) + 584793/2000000*sqrt(10*x^2 + 23*x + 51/5)*x + 13450239/40000000*sqrt(10*x^2 + 2
3*x + 51/5) + 3267/62500*sqrt(-10*x^2 - x + 3) - 11/18750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 225*x^2 + 135*x +
27) + 33/3125*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 99/6250*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 121/9
3750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 638/9375*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^4}{{\left (5\,x+3\right )}^{5/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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