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

Optimal. Leaf size=179 \[ -\frac {3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac {403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac {52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480000}+\frac {6384004649 \sqrt {1-2 x} \sqrt {5 x+3}}{204800000}+\frac {70224051139 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800000 \sqrt {10}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, 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 = {100, 153, 147, 50, 54, 216} \begin {gather*} -\frac {3}{70} (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^3-\frac {403 (1-2 x)^{3/2} (5 x+3)^{5/2} (3 x+2)^2}{2800}-\frac {52760369 (1-2 x)^{3/2} (5 x+3)^{3/2}}{7680000}-\frac {(1-2 x)^{3/2} (5 x+3)^{5/2} (874608 x+1480103)}{640000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {5 x+3}}{20480000}+\frac {6384004649 \sqrt {1-2 x} \sqrt {5 x+3}}{204800000}+\frac {70224051139 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{204800000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6384004649*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/204800000 - (580364059*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/20480000 - (527
60369*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/7680000 - (403*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/2800 - (3*(
1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(5/2))/70 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2)*(1480103 + 874608*x))/640000
+ (70224051139*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(204800000*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 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 \sqrt {1-2 x} (2+3 x)^4 (3+5 x)^{3/2} \, dx &=-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {1}{70} \int \left (-382-\frac {1209 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx\\ &=-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}+\frac {\int \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2} \left (\frac {121905}{2}+\frac {382641 x}{4}\right ) \, dx}{4200}\\ &=-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {52760369 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{1280000}\\ &=-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {580364059 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{5120000}\\ &=-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {6384004649 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{40960000}\\ &=\frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{409600000}\\ &=\frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{204800000 \sqrt {5}}\\ &=\frac {6384004649 \sqrt {1-2 x} \sqrt {3+5 x}}{204800000}-\frac {580364059 (1-2 x)^{3/2} \sqrt {3+5 x}}{20480000}-\frac {52760369 (1-2 x)^{3/2} (3+5 x)^{3/2}}{7680000}-\frac {403 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}}{2800}-\frac {3}{70} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2}-\frac {(1-2 x)^{3/2} (3+5 x)^{5/2} (1480103+874608 x)}{640000}+\frac {70224051139 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{204800000 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 89, normalized size = 0.50 \begin {gather*} \frac {1474705073919 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (497664000000 x^7+1651968000000 x^6+2010963456000 x^5+842711443200 x^4-356020459840 x^3-548703531560 x^2-330110729902 x+201521732121\right )}{43008000000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(201521732121 - 330110729902*x - 548703531560*x^2 - 356020459840*x^3 + 842711443200*x^4 + 2
010963456000*x^5 + 1651968000000*x^6 + 497664000000*x^7) + 1474705073919*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*S
qrt[-1 + 2*x]])/(43008000000*Sqrt[1 - 2*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.32, size = 173, normalized size = 0.97 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {17311996078125 (1-2 x)^6}{(5 x+3)^6}+\frac {46165322875000 (1-2 x)^5}{(5 x+3)^5}+\frac {52258801430500 (1-2 x)^4}{(5 x+3)^4}+\frac {32408912486400 (1-2 x)^3}{(5 x+3)^3}+\frac {11750081040880 (1-2 x)^2}{(5 x+3)^2}+\frac {2260241334400 (1-2 x)}{5 x+3}-70909935936\right )}{4300800000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^7}-\frac {70224051139 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{204800000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(-70909935936 + (17311996078125*(1 - 2*x)^6)/(3 + 5*x)^6 + (46165322875000*(1 - 2*x)^5)/(
3 + 5*x)^5 + (52258801430500*(1 - 2*x)^4)/(3 + 5*x)^4 + (32408912486400*(1 - 2*x)^3)/(3 + 5*x)^3 + (1175008104
0880*(1 - 2*x)^2)/(3 + 5*x)^2 + (2260241334400*(1 - 2*x))/(3 + 5*x)))/(4300800000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2
*x))/(3 + 5*x))^7) - (70224051139*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(204800000*Sqrt[10])

________________________________________________________________________________________

fricas [A]  time = 1.51, size = 87, normalized size = 0.49 \begin {gather*} \frac {1}{4300800000} \, {\left (248832000000 \, x^{6} + 950400000000 \, x^{5} + 1480681728000 \, x^{4} + 1161696585600 \, x^{3} + 402838062880 \, x^{2} - 72932734340 \, x - 201521732121\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {70224051139}{4096000000} \, \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((2+3*x)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/4300800000*(248832000000*x^6 + 950400000000*x^5 + 1480681728000*x^4 + 1161696585600*x^3 + 402838062880*x^2 -
72932734340*x - 201521732121)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 70224051139/4096000000*sqrt(10)*arctan(1/20*sqrt
(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

giac [B]  time = 1.50, size = 446, normalized size = 2.49 \begin {gather*} \frac {27}{71680000000} \, \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 {261}{1280000000} \, \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 {4203}{320000000} \, \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 {451}{400000} \, \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 {653}{15000} \, \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 {84}{125} \, \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 {72}{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((2+3*x)^4*(3+5*x)^(3/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

27/71680000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933)*(5*x + 3) - 7838433)*(5*x + 3) + 9
8794353)*(5*x + 3) - 1568443065)*(5*x + 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*a
rcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 261/1280000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*
(5*x + 3) - 775911)*(5*x + 3) + 15385695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt
(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 4203/320000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)
*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22
)*sqrt(5*x + 3))) + 451/400000*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)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 653/15000*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))) + 84/125*s
qrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 72/25
*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 155, normalized size = 0.87 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4976640000000 \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}-1458654686800 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-4030434642420 \sqrt {-10 x^{2}-x +3}\right )}{86016000000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/86016000000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(4976640000000*(-10*x^2-x+3)^(1/2)*x^6+19008000000000*(-10*x^2-x+3)
^(1/2)*x^5+29613634560000*(-10*x^2-x+3)^(1/2)*x^4+23233931712000*(-10*x^2-x+3)^(1/2)*x^3+8056761257600*(-10*x^
2-x+3)^(1/2)*x^2+1474705073919*10^(1/2)*arcsin(20/11*x+1/11)-1458654686800*(-10*x^2-x+3)^(1/2)*x-4030434642420
*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.31, size = 121, normalized size = 0.68 \begin {gather*} -\frac {81}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{4} - \frac {12051}{560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {1904661}{56000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {134695173}{4480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {890455739}{53760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {580364059}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {70224051139}{4096000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {580364059}{204800000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/14*(-10*x^2 - x + 3)^(3/2)*x^4 - 12051/560*(-10*x^2 - x + 3)^(3/2)*x^3 - 1904661/56000*(-10*x^2 - x + 3)^(
3/2)*x^2 - 134695173/4480000*(-10*x^2 - x + 3)^(3/2)*x - 890455739/53760000*(-10*x^2 - x + 3)^(3/2) + 58036405
9/10240000*sqrt(-10*x^2 - x + 3)*x - 70224051139/4096000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 580364059/20480
0000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________