3.2349 \(\int (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=194 \[ -\frac {3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac {9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac {306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac {3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac {37029509 (1-2 x)^{5/2} \sqrt {5 x+3}}{3276800}+\frac {407324599 (1-2 x)^{3/2} \sqrt {5 x+3}}{65536000}+\frac {13441711767 \sqrt {1-2 x} \sqrt {5 x+3}}{655360000}+\frac {147858829437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{655360000 \sqrt {10}} \]

[Out]

-3366319/819200*(1-2*x)^(5/2)*(3+5*x)^(3/2)-306029/256000*(1-2*x)^(5/2)*(3+5*x)^(5/2)-3/80*(1-2*x)^(5/2)*(2+3*
x)^2*(3+5*x)^(7/2)-9/448000*(1-2*x)^(5/2)*(3+5*x)^(7/2)*(25043+16120*x)[phone]/6553600000*arcsin(1/11*22
^(1/2)*(3+5*x)^(1/2))*10^(1/2)[phone]/65536000*(1-2*x)^(3/2)*(3+5*x)^(1/2)-37029509/3276800*(1-2*x)^(5/2)*(
3+5*x)^(1/2)[phone]/655360000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.07, 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} \[ -\frac {3}{80} (1-2 x)^{5/2} (3 x+2)^2 (5 x+3)^{7/2}-\frac {9 (1-2 x)^{5/2} (16120 x+25043) (5 x+3)^{7/2}}{448000}-\frac {306029 (1-2 x)^{5/2} (5 x+3)^{5/2}}{256000}-\frac {3366319 (1-2 x)^{5/2} (5 x+3)^{3/2}}{819200}-\frac {37029509 (1-2 x)^{5/2} \sqrt {5 x+3}}{3276800}+\frac {407324599 (1-2 x)^{3/2} \sqrt {5 x+3}}{65536000}+\frac {13441711767 \sqrt {1-2 x} \sqrt {5 x+3}}{655360000}+\frac {147858829437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{655360000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(13441711767*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/655360000 + (407324599*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/65536000 - (37
029509*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/3276800 - (3366319*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/819200 - (306029*(1
- 2*x)^(5/2)*(3 + 5*x)^(5/2))/256000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/80 - (9*(1 - 2*x)^(5/2)
*(3 + 5*x)^(7/2)*(25043 + 16120*x))/448000 + (147858829437*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(655360000*Sqrt[1
0])

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)^{3/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{80} \int \left (-389-\frac {1209 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {306029 \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx}{25600}\\ &=-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {3366319 \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx}{102400}\\ &=-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {111088527 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{1638400}\\ &=-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {407324599 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{6553600}\\ &=\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {13441711767 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{131072000}\\ &=\frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1310720000}\\ &=\frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{655360000 \sqrt {5}}\\ &=\frac {13441711767 \sqrt {1-2 x} \sqrt {3+5 x}}{655360000}+\frac {407324599 (1-2 x)^{3/2} \sqrt {3+5 x}}{65536000}-\frac {37029509 (1-2 x)^{5/2} \sqrt {3+5 x}}{3276800}-\frac {3366319 (1-2 x)^{5/2} (3+5 x)^{3/2}}{819200}-\frac {306029 (1-2 x)^{5/2} (3+5 x)^{5/2}}{256000}-\frac {3}{80} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {9 (1-2 x)^{5/2} (3+5 x)^{7/2} (25043+16120 x)}{448000}+\frac {147858829437 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{655360000 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.20, size = 94, normalized size = 0.48 \[ \frac {10 \sqrt {5 x+3} \left (1548288000000 x^8+4014489600000 x^7+2714081280000 x^6-1370011136000 x^5-2412933395200 x^4-588662541760 x^3+472622713160 x^2+370542366022 x-116041578381\right )[phone] \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{45875200000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-116041578381 + 370542366022*x + 472622713160*x^2 - 588662541760*x^3 - 2412933395200*x^4 -
1370011136000*x^5 + 2714081280000*x^6 + 4014489600000*x^7 + 1548288000000*x^8) + 1035011806059*Sqrt[-10 + 20*x
]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(45875200000*Sqrt[1 - 2*x])

________________________________________________________________________________________

fricas [A]  time = 1.07, size = 92, normalized size = 0.47 \[ -\frac {1}{4587520000} \, {\left (774144000000 \, x^{7} + 2394316800000 \, x^{6} + 2554199040000 \, x^{5} + 592093952000 \, x^{4} - 910419721600 \, x^{3} - 749541131680 \, x^{2} - 138459209260 \, x + 116041578381\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {147858829437}{13107200000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4587520000*(774144000000*x^7 + 2394316800000*x^6 + 2554199040000*x^5 + 592093952000*x^4 - 910419721600*x^3
- 749541131680*x^2 - 138459209260*x + 116041578381)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 147858829437/13107200000*sq
rt(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 = 1.41, size = 545, normalized size = 2.81 \[ -\frac {9}{229376000000} \, \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 {297}{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 {1851}{2560000000} \, \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 {13943}{960000000} \, \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 {883}{9600000} \, \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 {669}{20000} \, \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 {81}{100} \, \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 {108}{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 )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/229376000000*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(-
10*x + 5) - 309625826895*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 297/71680000000*sqrt(5)*(2*(4*(8*(4*(1
6*(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))) -
1851/2560000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 1538
5695)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3))) - 13943/960000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 40
31895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 883/9600000*sqr
t(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))) + 669/20000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sq
rt(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 81/100*sqrt(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))) + 108/25*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 = 172, normalized size = 0.89 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-15482880000000 \sqrt {-10 x^{2}-x +3}\, x^{7}-47886336000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-51083980800000 \sqrt {-10 x^{2}-x +3}\, x^{5}-11841879040000 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-2320831567620 \sqrt {-10 x^{2}-x +3}\right )}{91750400000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/91750400000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-15482880000000*(-10*x^2-x+3)^(1/2)*x^7-47886336000000*(-10*x^2-x+
3)^(1/2)*x^6-51083980800000*(-10*x^2-x+3)^(1/2)*x^5-11841879040000*(-10*x^2-x+3)^(1/2)*x^4+18208394432000*(-10
*x^2-x+3)^(1/2)*x^3+14990822633600*(-10*x^2-x+3)^(1/2)*x^2+1035011806059*10^(1/2)*arcsin(20/11*x+1/11)+2769184
185200*(-10*x^2-x+3)^(1/2)*x-2320831567620*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.15, size = 133, normalized size = 0.69 \[ -\frac {27}{16} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{3} - \frac {2187}{448} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {100119}{17920} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {5653247}{1792000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {3366319}{409600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {3366319}{8192000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1221973797}{32768000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {147858829437}{13107200000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1221973797}{655360000} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/16*(-10*x^2 - x + 3)^(5/2)*x^3 - 2187/448*(-10*x^2 - x + 3)^(5/2)*x^2 - 100119/17920*(-10*x^2 - x + 3)^(5/
2)*x - 5653247/1792000*(-10*x^2 - x + 3)^(5/2) + 3366319/409600*(-10*x^2 - x + 3)^(3/2)*x + 3366319/8192000*(-
10*x^2 - x + 3)^(3/2) + 1221973797/32768000*sqrt(-10*x^2 - x + 3)*x - 147858829437/13107200000*sqrt(10)*arcsin
(-20/11*x - 1/11) + 1221973797/655360000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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)**(3/2)*(2+3*x)**3*(3+5*x)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________