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

Optimal. Leaf size=194 \[ -\frac {3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac {9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac {24269487 \sqrt {5 x+3} (1-2 x)^{7/2}}{20480000}+\frac {88988119 \sqrt {5 x+3} (1-2 x)^{5/2}}{204800000}+\frac {978869309 \sqrt {5 x+3} (1-2 x)^{3/2}}{819200000}+\frac {32302687197 \sqrt {5 x+3} \sqrt {1-2 x}}{8192000000}+\frac {355329559167 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8192000000 \sqrt {10}} \]

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Rubi [A]  time = 0.06, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \begin {gather*} -\frac {3}{80} (3 x+2)^2 (5 x+3)^{5/2} (1-2 x)^{7/2}-\frac {735439 (5 x+3)^{3/2} (1-2 x)^{7/2}}{1280000}-\frac {9 (5 x+3)^{5/2} (13480 x+18399) (1-2 x)^{7/2}}{448000}-\frac {24269487 \sqrt {5 x+3} (1-2 x)^{7/2}}{20480000}+\frac {88988119 \sqrt {5 x+3} (1-2 x)^{5/2}}{204800000}+\frac {978869309 \sqrt {5 x+3} (1-2 x)^{3/2}}{819200000}+\frac {32302687197 \sqrt {5 x+3} \sqrt {1-2 x}}{8192000000}+\frac {355329559167 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{8192000000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(32302687197*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/8192000000 + (978869309*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/819200000 + (
88988119*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/204800000 - (24269487*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/20480000 - (73543
9*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/1280000 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/80 - (9*(1 - 2*x)
^(7/2)*(3 + 5*x)^(5/2)*(18399 + 13480*x))/448000 + (355329559167*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(8192000000
*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 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 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 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 (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2} \, dx &=-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{80} \int \left (-323-\frac {1011 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2} \, dx\\ &=-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {735439 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{128000}\\ &=-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {24269487 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{2560000}\\ &=-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {266964357 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{40960000}\\ &=\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {978869309 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{81920000}\\ &=\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {32302687197 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1638400000}\\ &=\frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{16384000000}\\ &=\frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{8192000000 \sqrt {5}}\\ &=\frac {32302687197 \sqrt {1-2 x} \sqrt {3+5 x}}{8192000000}+\frac {978869309 (1-2 x)^{3/2} \sqrt {3+5 x}}{819200000}+\frac {88988119 (1-2 x)^{5/2} \sqrt {3+5 x}}{204800000}-\frac {24269487 (1-2 x)^{7/2} \sqrt {3+5 x}}{20480000}-\frac {735439 (1-2 x)^{7/2} (3+5 x)^{3/2}}{1280000}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {9 (1-2 x)^{7/2} (3+5 x)^{5/2} (18399+13480 x)}{448000}+\frac {355329559167 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{8192000000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.22, size = 94, normalized size = 0.48 \begin {gather*} \frac {2487306914169 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (7741440000000 x^8+10340352000000 x^7-5488281600000 x^6-11337362944000 x^5+569714643200 x^4+4956975460160 x^3+580113118440 x^2-1173301694402 x+115416461871\right )}{573440000000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(115416461871 - 1173301694402*x + 580113118440*x^2 + 4956975460160*x^3 + 569714643200*x^4 -
11337362944000*x^5 - 5488281600000*x^6 + 10340352000000*x^7 + 7741440000000*x^8) + 2487306914169*Sqrt[-10 + 2
0*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(573440000000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.38, size = 189, normalized size = 0.97 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {1206579609375 (1-2 x)^7}{(5 x+3)^7}+\frac {3700177468750 (1-2 x)^6}{(5 x+3)^6}+\frac {4924692377500 (1-2 x)^5}{(5 x+3)^5}+\frac {3644476883000 (1-2 x)^4}{(5 x+3)^4}+\frac {1323386693200 (1-2 x)^3}{(5 x+3)^3}-\frac {315473913440 (1-2 x)^2}{(5 x+3)^2}-\frac {37889817280 (1-2 x)}{5 x+3}-1976860032\right )}{57344000000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^8}-\frac {355329559167 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{8192000000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-161051*Sqrt[1 - 2*x]*(-1976860032 + (1206579609375*(1 - 2*x)^7)/(3 + 5*x)^7 + (3700177468750*(1 - 2*x)^6)/(3
+ 5*x)^6 + (4924692377500*(1 - 2*x)^5)/(3 + 5*x)^5 + (3644476883000*(1 - 2*x)^4)/(3 + 5*x)^4 + (1323386693200
*(1 - 2*x)^3)/(3 + 5*x)^3 - (315473913440*(1 - 2*x)^2)/(3 + 5*x)^2 - (37889817280*(1 - 2*x))/(3 + 5*x)))/(5734
4000000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^8) - (355329559167*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3
+ 5*x]])/(8192000000*Sqrt[10])

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fricas [A]  time = 0.82, size = 92, normalized size = 0.47 \begin {gather*} \frac {1}{57344000000} \, {\left (3870720000000 \, x^{7} + 7105536000000 \, x^{6} + 808627200000 \, x^{5} - 5264367872000 \, x^{4} - 2347326614400 \, x^{3} + 1304824422880 \, x^{2} + 942468770660 \, x - 115416461871\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {355329559167}{163840000000} \, \sqrt {10} \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 ) \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)^(3/2),x, algorithm="fricas")

[Out]

1/57344000000*(3870720000000*x^7 + 7105536000000*x^6 + 808627200000*x^5 - 5264367872000*x^4 - 2347326614400*x^
3 + 1304824422880*x^2 + 942468770660*x - 115416461871)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 355329559167/16384000000
0*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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giac [B]  time = 1.36, size = 545, normalized size = 2.81 \begin {gather*} \frac {9}{573440000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (24 \, {\left (140 \, x - 599\right )} {\left (5 \, x + 3\right )} + 175163\right )} {\left (5 \, x + 3\right )} - 4295993\right )} {\left (5 \, x + 3\right )} + 265620213\right )} {\left (5 \, x + 3\right )} - 2676516549\right )} {\left (5 \, x + 3\right )} + 35390483373\right )} {\left (5 \, x + 3\right )} - 164483997363\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 309625826895 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {99}{89600000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (20 \, {\left (120 \, x - 443\right )} {\left (5 \, x + 3\right )} + 94933\right )} {\left (5 \, x + 3\right )} - 7838433\right )} {\left (5 \, x + 3\right )} + 98794353\right )} {\left (5 \, x + 3\right )} - 1568443065\right )} {\left (5 \, x + 3\right )} + 8438816295\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 17534989395 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1029}{12800000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {457}{240000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {409}{1920000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {101}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {69}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {36}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\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)^(3/2),x, algorithm="giac")

[Out]

9/573440000000*sqrt(5)*(2*(4*(8*(4*(16*(4*(24*(140*x - 599)*(5*x + 3) + 175163)*(5*x + 3) - 4295993)*(5*x + 3)
+ 265620213)*(5*x + 3) - 2676516549)*(5*x + 3) + 35390483373)*(5*x + 3) - 164483997363)*sqrt(5*x + 3)*sqrt(-1
0*x + 5) - 309625826895*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 99/89600000000*sqrt(5)*(2*(4*(8*(4*(16*
(20*(120*x - 443)*(5*x + 3) + 94933)*(5*x + 3) - 7838433)*(5*x + 3) + 98794353)*(5*x + 3) - 1568443065)*(5*x +
3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1
029/12800000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385
695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3
))) - 457/240000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 40318
95)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 409/1920000*sqrt(5
)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*ar
csin(1/11*sqrt(22)*sqrt(5*x + 3))) - 101/60000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(
-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 69/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*s
qrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 36/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt
(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.01, size = 172, normalized size = 0.89 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (77414400000000 \sqrt {-10 x^{2}-x +3}\, x^{7}[phone] \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] \sqrt {-10 x^{2}-x +3}\, x^{5}-105287357440000 \sqrt {-10 x^{2}-x +3}\, x^{4}-46946532288000 \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2308329237420 \sqrt {-10 x^{2}-x +3}\right )}{1146880000000 \sqrt {-10 x^{2}-x +3}} \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)^(3/2),x)

[Out]

1/1146880000000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(77414400000000*(-10*x^2-x+3)^(1/2)*x^7+142110720000000*(-10*x^2-
x+3)^(1/2)*x^6+16172544000000*(-10*x^2-x+3)^(1/2)*x^5-105287357440000*(-10*x^2-x+3)^(1/2)*x^4-46946532288000*(
-10*x^2-x+3)^(1/2)*x^3+26096488457600*(-10*x^2-x+3)^(1/2)*x^2+2487306914169*10^(1/2)*arcsin(20/11*x+1/11)+1884
9375413200*(-10*x^2-x+3)^(1/2)*x-2308329237420*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.26, size = 133, normalized size = 0.69 \begin {gather*} \frac {27}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} + \frac {6183}{5600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} + \frac {71331}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {6491477}{22400000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {8089829}{5120000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {8089829}{102400000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {2936607927}{409600000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {355329559167}{163840000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {2936607927}{8192000000} \, \sqrt {-10 \, x^{2} - x + 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)^(3/2),x, algorithm="maxima")

[Out]

27/40*(-10*x^2 - x + 3)^(5/2)*x^3 + 6183/5600*(-10*x^2 - x + 3)^(5/2)*x^2 + 71331/224000*(-10*x^2 - x + 3)^(5/
2)*x - 6491477/22400000*(-10*x^2 - x + 3)^(5/2) + 8089829/5120000*(-10*x^2 - x + 3)^(3/2)*x + 8089829/10240000
0*(-10*x^2 - x + 3)^(3/2) + 2936607927/409600000*sqrt(-10*x^2 - x + 3)*x - 355329559167/163840000000*sqrt(10)*
arcsin(-20/11*x - 1/11) + 2936607927/8192000000*sqrt(-10*x^2 - x + 3)

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

[Out]

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

[Out]

Timed out

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