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

Optimal. Leaf size=164 \[ -\frac {376 (1-2 x)^{3/2} (3 x+2)^3}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac {741}{250} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2+\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (4392 x+3185)}{40000}+\frac {69713 \sqrt {1-2 x} \sqrt {5 x+3}}{400000}+\frac {766843 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400000 \sqrt {10}} \]

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Rubi [A]  time = 0.05, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \begin {gather*} -\frac {376 (1-2 x)^{3/2} (3 x+2)^3}{75 \sqrt {5 x+3}}-\frac {2 (1-2 x)^{5/2} (3 x+2)^3}{15 (5 x+3)^{3/2}}+\frac {741}{250} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2+\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (4392 x+3185)}{40000}+\frac {69713 \sqrt {1-2 x} \sqrt {5 x+3}}{400000}+\frac {766843 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{400000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^3)/(15*(3 + 5*x)^(3/2)) - (376*(1 - 2*x)^(3/2)*(2 + 3*x)^3)/(75*Sqrt[3 + 5*x]) +
(69713*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/400000 + (741*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/250 + (21*(1 - 2
*x)^(3/2)*Sqrt[3 + 5*x]*(3185 + 4392*x))/40000 + (766843*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(400000*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)^3}{(3+5 x)^{5/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}+\frac {2}{15} \int \frac {(-1-33 x) (1-2 x)^{3/2} (2+3 x)^2}{(3+5 x)^{3/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {4}{75} \int \frac {\left (\frac {249}{2}-2223 x\right ) \sqrt {1-2 x} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {741}{250} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{750} \int \frac {\sqrt {1-2 x} (2+3 x) \left (1155+\frac {34587 x}{2}\right )}{\sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {741}{250} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (3185+4392 x)}{40000}+\frac {69713 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{80000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {69713 \sqrt {1-2 x} \sqrt {3+5 x}}{400000}+\frac {741}{250} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (3185+4392 x)}{40000}+\frac {766843 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{800000}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {69713 \sqrt {1-2 x} \sqrt {3+5 x}}{400000}+\frac {741}{250} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (3185+4392 x)}{40000}+\frac {766843 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{400000 \sqrt {5}}\\ &=-\frac {2 (1-2 x)^{5/2} (2+3 x)^3}{15 (3+5 x)^{3/2}}-\frac {376 (1-2 x)^{3/2} (2+3 x)^3}{75 \sqrt {3+5 x}}+\frac {69713 \sqrt {1-2 x} \sqrt {3+5 x}}{400000}+\frac {741}{250} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}+\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (3185+4392 x)}{40000}+\frac {766843 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{400000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 93, normalized size = 0.57 \begin {gather*} \frac {2300529 (5 x+3)^{3/2} \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \left (12960000 x^6-4536000 x^5-16421400 x^4+13874190 x^3+12677675 x^2-3232208 x-2322001\right )}{12000000 \sqrt {1-2 x} (5 x+3)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*(-2322001 - 3232208*x + 12677675*x^2 + 13874190*x^3 - 16421400*x^4 - 4536000*x^5 + 12960000*x^6) + 230052
9*(3 + 5*x)^(3/2)*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(12000000*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2)
)

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IntegrateAlgebraic [A]  time = 0.72, size = 151, normalized size = 0.92 \begin {gather*} \frac {\sqrt {11-2 (5 x+3)} \left (10368 \sqrt {5} (5 x+3)^5-147744 \sqrt {5} (5 x+3)^4+530820 \sqrt {5} (5 x+3)^3+1016385 \sqrt {5} (5 x+3)^2-796928 \sqrt {5} (5 x+3)-30976 \sqrt {5}\right )}{30000000 (5 x+3)^{3/2}}-\frac {766843 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{200000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[11 - 2*(3 + 5*x)]*(-30976*Sqrt[5] - 796928*Sqrt[5]*(3 + 5*x) + 1016385*Sqrt[5]*(3 + 5*x)^2 + 530820*Sqrt
[5]*(3 + 5*x)^3 - 147744*Sqrt[5]*(3 + 5*x)^4 + 10368*Sqrt[5]*(3 + 5*x)^5))/(30000000*(3 + 5*x)^(3/2)) - (76684
3*ArcTan[(Sqrt[2]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*x)])])/(200000*Sqrt[10])

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fricas [A]  time = 1.36, size = 106, normalized size = 0.65 \begin {gather*} -\frac {2300529 \, \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 (6480000 \, x^{5} + 972000 \, x^{4} - 7724700 \, x^{3} + 3074745 \, x^{2} + 7876210 \, x + 2322001\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{24000000 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24000000*(2300529*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*(6480000*x^5 + 972000*x^4 - 7724700*x^3 + 3074745*x^2 + 7876210*x + 2322001)*sqrt(5*x
+ 3)*sqrt(-2*x + 1))/(25*x^2 + 30*x + 9)

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giac [A]  time = 2.25, size = 197, normalized size = 1.20 \begin {gather*} \frac {1}{10000000} \, {\left (36 \, {\left (24 \, {\left (4 \, \sqrt {5} {\left (5 \, x + 3\right )} - 57 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 4915 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 338795 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - \frac {11}{3750000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {2268 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {766843}{4000000} \, \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 {567 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{234375 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10000000*(36*(24*(4*sqrt(5)*(5*x + 3) - 57*sqrt(5))*(5*x + 3) + 4915*sqrt(5))*(5*x + 3) + 338795*sqrt(5))*sq
rt(5*x + 3)*sqrt(-10*x + 5) - 11/3750000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 22
68*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3)) + 766843/4000000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x
+ 3)) + 11/234375*sqrt(10)*(5*x + 3)^(3/2)*(567*(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 = 164, normalized size = 1.00 \begin {gather*} \frac {\left (129600000 \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}-154494000 \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}}{24000000 \sqrt {-10 x^{2}-x +3}\, \left (5 x +3\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24000000*(129600000*(-10*x^2-x+3)^(1/2)*x^5+19440000*(-10*x^2-x+3)^(1/2)*x^4+57513225*10^(1/2)*x^2*arcsin(20
/11*x+1/11)-154494000*(-10*x^2-x+3)^(1/2)*x^3+69015870*10^(1/2)*x*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^
(1/2)*x^2+20704761*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x+46440020*(-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 = 1.20, size = 325, normalized size = 1.98 \begin {gather*} -\frac {395307}{8000000} i \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {23}{11}\right ) + \frac {23221}{500000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {99}{5000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{625 \, {\left (625 \, x^{4} + 1500 \, x^{3} + 1350 \, x^{2} + 540 \, x + 81\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1250 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {9 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{625 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {27 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{2500 \, {\left (5 \, x + 3\right )}} + \frac {3267}{20000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} x + \frac {75141}{400000} \, \sqrt {10 \, x^{2} + 23 \, x + \frac {51}{5}} + \frac {3267}{25000} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {11 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{3750 \, {\left (125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27\right )}} + \frac {99 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac {99 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{2500 \, {\left (5 \, x + 3\right )}} - \frac {121 \, \sqrt {-10 \, x^{2} - x + 3}}{18750 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac {9493 \, \sqrt {-10 \, x^{2} - x + 3}}{37500 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-395307/8000000*I*sqrt(5)*sqrt(2)*arcsin(20/11*x + 23/11) + 23221/500000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11
) + 99/5000*(-10*x^2 - x + 3)^(3/2) + 1/625*(-10*x^2 - x + 3)^(5/2)/(625*x^4 + 1500*x^3 + 1350*x^2 + 540*x + 8
1) + 9/1250*(-10*x^2 - x + 3)^(5/2)/(125*x^3 + 225*x^2 + 135*x + 27) + 9/625*(-10*x^2 - x + 3)^(5/2)/(25*x^2 +
30*x + 9) + 27/2500*(-10*x^2 - x + 3)^(5/2)/(5*x + 3) + 3267/20000*sqrt(10*x^2 + 23*x + 51/5)*x + 75141/40000
0*sqrt(10*x^2 + 23*x + 51/5) + 3267/25000*sqrt(-10*x^2 - x + 3) - 11/3750*(-10*x^2 - x + 3)^(3/2)/(125*x^3 + 2
25*x^2 + 135*x + 27) + 99/2500*(-10*x^2 - x + 3)^(3/2)/(25*x^2 + 30*x + 9) + 99/2500*(-10*x^2 - x + 3)^(3/2)/(
5*x + 3) - 121/18750*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 9493/37500*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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