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

Optimal. Leaf size=142 \[ \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {357 (3 x+2)^3}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {5281 \sqrt {1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{8784600 \sqrt {5 x+3}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 150, 143, 54, 216} \begin {gather*} \frac {7 (3 x+2)^4}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {357 (3 x+2)^3}{242 \sqrt {1-2 x} (5 x+3)^{3/2}}+\frac {5281 \sqrt {1-2 x} (3 x+2)^2}{39930 (5 x+3)^{3/2}}-\frac {\sqrt {1-2 x} (55300905 x+33035947)}{8784600 \sqrt {5 x+3}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{200 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5281*Sqrt[1 - 2*x]*(2 + 3*x)^2)/(39930*(3 + 5*x)^(3/2)) - (357*(2 + 3*x)^3)/(242*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2
)) + (7*(2 + 3*x)^4)/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (Sqrt[1 - 2*x]*(33035947 + 55300905*x))/(8784600*S
qrt[3 + 5*x]) + (2997*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 143

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> Simp[((b^2*d*e*g - a^2*d*f*h*m - a*b*(d*(f*g + e*h) - c*f*h*(m + 1)) + b*f*h*(b*c - a*d)*(m + 1)*x)*(a + b*x
)^(m + 1)*(c + d*x)^(n + 1))/(b^2*d*(b*c - a*d)*(m + 1)), x] + Dist[(a*d*f*h*m + b*(d*(f*g + e*h) - c*f*h*(m +
2)))/(b^2*d), Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && EqQ[m
+ n + 2, 0] && NeQ[m, -1] &&  !(SumSimplerQ[n, 1] &&  !SumSimplerQ[m, 1])

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 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)^5}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{33} \int \frac {(2+3 x)^3 \left (141+\frac {507 x}{2}\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {1}{363} \int \frac {\left (-6537-\frac {48861 x}{4}\right ) (2+3 x)^2}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {2 \int \frac {\left (-\frac {1453983}{4}-\frac {5027355 x}{8}\right ) (2+3 x)}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997}{400} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{200 \sqrt {5}}\\ &=\frac {5281 \sqrt {1-2 x} (2+3 x)^2}{39930 (3+5 x)^{3/2}}-\frac {357 (2+3 x)^3}{242 \sqrt {1-2 x} (3+5 x)^{3/2}}+\frac {7 (2+3 x)^4}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac {\sqrt {1-2 x} (33035947+55300905 x)}{8784600 \sqrt {3+5 x}}+\frac {2997 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{200 \sqrt {10}}\\ \end {align*}

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Mathematica [C]  time = 1.37, size = 257, normalized size = 1.81 \begin {gather*} -\frac {37 \left (-1320000 (3 x+2)^3 (1-2 x)^{7/2} \, _4F_3\left (\frac {1}{2},2,2,\frac {7}{2};1,1,\frac {9}{2};-\frac {5}{11} (2 x-1)\right )-1050000 (x+3) \left (6 x^2+x-2\right )^2 (1-2 x)^{5/2} \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};-\frac {5}{11} (2 x-1)\right )+77 \sqrt {55} \left (\sqrt {10-20 x} \sqrt {5 x+3} \left (43200 x^5+28080 x^4-400032 x^3+1229303 x^2+2053496 x+1669914\right )-27951 \left (108 x^3+513 x^2+1296 x+374\right ) \sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\right )\right )}{614922000 \sqrt {22} (1-2 x)^3}+\frac {1183 \left (19573 x^3+62232 x^2+52044 x+13040\right )}{878460 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac {3 (3 x+2)^4}{10 (1-2 x)^{3/2} (5 x+3)^{3/2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-3*(2 + 3*x)^4)/(10*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (1183*(13040 + 52044*x + 62232*x^2 + 19573*x^3))/(8784
60*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) - (37*(77*Sqrt[55]*(Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*(1669914 + 2053496*x + 1
229303*x^2 - 400032*x^3 + 28080*x^4 + 43200*x^5) - 27951*(374 + 1296*x + 513*x^2 + 108*x^3)*ArcSin[Sqrt[5/11]*
Sqrt[1 - 2*x]]) - 1050000*(1 - 2*x)^(5/2)*(3 + x)*(-2 + x + 6*x^2)^2*Hypergeometric2F1[3/2, 9/2, 11/2, (-5*(-1
+ 2*x))/11] - 1320000*(1 - 2*x)^(7/2)*(2 + 3*x)^3*HypergeometricPFQ[{1/2, 2, 2, 7/2}, {1, 1, 9/2}, (-5*(-1 +
2*x))/11]))/(614922000*Sqrt[22]*(1 - 2*x)^3)

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IntegrateAlgebraic [A]  time = 0.17, size = 141, normalized size = 0.99 \begin {gather*} \frac {(5 x+3)^{3/2} \left (-\frac {80 (1-2 x)^4}{(5 x+3)^4}-\frac {8240 (1-2 x)^3}{(5 x+3)^3}-\frac {131529519 (1-2 x)^2}{(5 x+3)^2}-\frac {34814500 (1-2 x)}{5 x+3}+3361400\right )}{8784600 (1-2 x)^{3/2} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )}-\frac {2997 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{200 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 5*x)^(3/2)*(3361400 - (80*(1 - 2*x)^4)/(3 + 5*x)^4 - (8240*(1 - 2*x)^3)/(3 + 5*x)^3 - (131529519*(1 - 2*
x)^2)/(3 + 5*x)^2 - (34814500*(1 - 2*x))/(3 + 5*x)))/(8784600*(1 - 2*x)^(3/2)*(2 + (5*(1 - 2*x))/(3 + 5*x))) -
(2997*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(200*Sqrt[10])

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fricas [A]  time = 1.44, size = 121, normalized size = 0.85 \begin {gather*} -\frac {131637231 \, \sqrt {10} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, 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 (213465780 \, x^{4} - 1247811640 \, x^{3} - 1260430251 \, x^{2} + 19593966 \, x + 168318961\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{175692000 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/175692000*(131637231*sqrt(10)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*
x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 20*(213465780*x^4 - 1247811640*x^3 - 1260430251*x^2 + 19593966*x + 1
68318961)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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giac [A]  time = 1.51, size = 191, normalized size = 1.35 \begin {gather*} -\frac {1}{439230000} \, \sqrt {10} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} + \frac {4092 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {2997}{2000} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {{\left (4 \, {\left (10673289 \, \sqrt {5} {\left (5 \, x + 3\right )} - 440040554 \, \sqrt {5}\right )} {\left (5 \, x + 3\right )} + 7233942969 \, \sqrt {5}\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{5490375000 \, {\left (2 \, x - 1\right )}^{2}} + \frac {\sqrt {10} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {1023 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} + 4\right )}}{27451875 \, {\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((2+3*x)^5/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-1/439230000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) + 4092*(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3)) + 2997/2000*sqrt(10)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) - 1/5490375000*(4*(1067328
9*sqrt(5)*(5*x + 3) - 440040554*sqrt(5))*(5*x + 3) + 7233942969*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x -
1)^2 + 1/27451875*sqrt(10)*(5*x + 3)^(3/2)*(1023*(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 = 0.02, size = 182, normalized size = 1.28 \begin {gather*} \frac {\sqrt {-2 x +1}\, \left (13163723100 \sqrt {10}\, x^{4} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4269315600 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {10}\, x^{3} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}-7766596629 \sqrt {10}\, x^{2} \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-789823386 \sqrt {10}\, x \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-391879320 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-3366379220 \sqrt {-10 x^{2}-x +3}\right )}{175692000 \left (2 x -1\right )^{2} \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((3*x+2)^5/(-2*x+1)^(5/2)/(5*x+3)^(5/2),x)

[Out]

1/175692000*(-2*x+1)^(1/2)*(13163723100*10^(1/2)*x^4*arcsin(20/11*x+1/11)[phone]*10^(1/2)*x^3*arcsin(20/11
*x+1/11)-4269315600*(-10*x^2-x+3)^(1/2)*x^4-7766596629*10^(1/2)*x^2*arcsin(20/11*x+1/11)[phone]*(-10*x^2-
x+3)^(1/2)*x^3-789823386*10^(1/2)*x*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x^2+1184735079*10^(1/
2)*arcsin(20/11*x+1/11)-391879320*(-10*x^2-x+3)^(1/2)*x-3366379220*(-10*x^2-x+3)^(1/2))/(2*x-1)^2/(-10*x^2-x+3
)^(1/2)/(5*x+3)^(3/2)

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maxima [A]  time = 1.30, size = 197, normalized size = 1.39 \begin {gather*} -\frac {243 \, x^{4}}{10 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {999}{5856400} \, x {\left (\frac {7220 \, x}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {439230 \, x^{2}}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {361}{\sqrt {-10 \, x^{2} - x + 3}} + \frac {21901 \, x}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {87483}{{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}\right )} - \frac {2997}{4000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {360639}{2928200} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {5842159 \, x}{878460 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3429 \, x^{2}}{25 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} + \frac {947293}{21961500 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {3016649 \, x}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} - \frac {1851167}{90750 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-243/10*x^4/(-10*x^2 - x + 3)^(3/2) + 999/5856400*x*(7220*x/sqrt(-10*x^2 - x + 3) + 439230*x^2/(-10*x^2 - x +
3)^(3/2) + 361/sqrt(-10*x^2 - x + 3) + 21901*x/(-10*x^2 - x + 3)^(3/2) - 87483/(-10*x^2 - x + 3)^(3/2)) - 2997
/4000*sqrt(10)*arcsin(-20/11*x - 1/11) + 360639/2928200*sqrt(-10*x^2 - x + 3) - 5842159/878460*x/sqrt(-10*x^2
- x + 3) + 3429/25*x^2/(-10*x^2 - x + 3)^(3/2) + 947293/21961500/sqrt(-10*x^2 - x + 3) + 3016649/90750*x/(-10*
x^2 - x + 3)^(3/2) - 1851167/90750/(-10*x^2 - x + 3)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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