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

Optimal. Leaf size=187 \[ -\frac {47}{400} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{7/2}-\frac {783 (1-2 x)^{5/2} (5 x+3)^{5/2}}{1600}-\frac {8613 (1-2 x)^{5/2} (5 x+3)^{3/2}}{5120}-\frac {94743 (1-2 x)^{5/2} \sqrt {5 x+3}}{20480}+\frac {1042173 (1-2 x)^{3/2} \sqrt {5 x+3}}{409600}+\frac {34391709 \sqrt {1-2 x} \sqrt {5 x+3}}{4096000}+\frac {378308799 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4096000 \sqrt {10}} \]

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Rubi [A]  time = 0.06, antiderivative size = 187, 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 = {90, 80, 50, 54, 216} \begin {gather*} -\frac {47}{400} (1-2 x)^{5/2} (5 x+3)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (3 x+2) (5 x+3)^{7/2}-\frac {783 (1-2 x)^{5/2} (5 x+3)^{5/2}}{1600}-\frac {8613 (1-2 x)^{5/2} (5 x+3)^{3/2}}{5120}-\frac {94743 (1-2 x)^{5/2} \sqrt {5 x+3}}{20480}+\frac {1042173 (1-2 x)^{3/2} \sqrt {5 x+3}}{409600}+\frac {34391709 \sqrt {1-2 x} \sqrt {5 x+3}}{4096000}+\frac {378308799 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{4096000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(34391709*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/4096000 + (1042173*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/409600 - (94743*(1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/20480 - (8613*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/5120 - (783*(1 - 2*x)^(5/2)*(3 + 5*x)
^(5/2))/1600 - (47*(1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/400 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/70 + (
378308799*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(4096000*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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 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)^2 (3+5 x)^{5/2} \, dx &=-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}-\frac {1}{70} \int \left (-322-\frac {987 x}{2}\right ) (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {783}{160} \int (1-2 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {8613}{640} \int (1-2 x)^{3/2} (3+5 x)^{3/2} \, dx\\ &=-\frac {8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {284229 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{10240}\\ &=-\frac {94743 (1-2 x)^{5/2} \sqrt {3+5 x}}{20480}-\frac {8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {1042173 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{40960}\\ &=\frac {1042173 (1-2 x)^{3/2} \sqrt {3+5 x}}{409600}-\frac {94743 (1-2 x)^{5/2} \sqrt {3+5 x}}{20480}-\frac {8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {34391709 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{819200}\\ &=\frac {34391709 \sqrt {1-2 x} \sqrt {3+5 x}}{4096000}+\frac {1042173 (1-2 x)^{3/2} \sqrt {3+5 x}}{409600}-\frac {94743 (1-2 x)^{5/2} \sqrt {3+5 x}}{20480}-\frac {8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {378308799 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{8192000}\\ &=\frac {34391709 \sqrt {1-2 x} \sqrt {3+5 x}}{4096000}+\frac {1042173 (1-2 x)^{3/2} \sqrt {3+5 x}}{409600}-\frac {94743 (1-2 x)^{5/2} \sqrt {3+5 x}}{20480}-\frac {8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {378308799 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{4096000 \sqrt {5}}\\ &=\frac {34391709 \sqrt {1-2 x} \sqrt {3+5 x}}{4096000}+\frac {1042173 (1-2 x)^{3/2} \sqrt {3+5 x}}{409600}-\frac {94743 (1-2 x)^{5/2} \sqrt {3+5 x}}{20480}-\frac {8613 (1-2 x)^{5/2} (3+5 x)^{3/2}}{5120}-\frac {783 (1-2 x)^{5/2} (3+5 x)^{5/2}}{1600}-\frac {47}{400} (1-2 x)^{5/2} (3+5 x)^{7/2}-\frac {3}{70} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{7/2}+\frac {378308799 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{4096000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 89, normalized size = 0.48 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (3686400000 x^7+6932480000 x^6+1347072000 x^5-4641542400 x^4-2692390720 x^3+690468120 x^2+1044112194 x-247243887\right )[phone] \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{286720000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-247243887 + 1044112194*x + 690468120*x^2 - 2692390720*x^3 - 4641542400*x^4 + 1347072000*x^
5 + 6932480000*x^6 + 3686400000*x^7) + 2648161593*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(286720
000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.30, size = 173, normalized size = 0.93 \begin {gather*} -\frac {161051 \sqrt {1-2 x} \left (\frac {256921875 (1-2 x)^6}{(5 x+3)^6}+\frac {685125000 (1-2 x)^5}{(5 x+3)^5}+\frac {775561500 (1-2 x)^4}{(5 x+3)^4}+\frac {479436800 (1-2 x)^3}{(5 x+3)^3}+\frac {156895760 (1-2 x)^2}{(5 x+3)^2}-\frac {17539200 (1-2 x)}{5 x+3}-1052352\right )}{28672000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^7}-\frac {378308799 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{4096000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-161051*Sqrt[1 - 2*x]*(-1052352 + (256921875*(1 - 2*x)^6)/(3 + 5*x)^6 + (685125000*(1 - 2*x)^5)/(3 + 5*x)^5 +
(775561500*(1 - 2*x)^4)/(3 + 5*x)^4 + (479436800*(1 - 2*x)^3)/(3 + 5*x)^3 + (156895760*(1 - 2*x)^2)/(3 + 5*x)
^2 - (17539200*(1 - 2*x))/(3 + 5*x)))/(28672000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^7) - (378308799*Ar
cTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(4096000*Sqrt[10])

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fricas [A]  time = 1.33, size = 87, normalized size = 0.47 \begin {gather*} -\frac {1}{28672000} \, {\left (1843200000 \, x^{6} + 4387840000 \, x^{5} + 2867456000 \, x^{4} - 887043200 \, x^{3} - 1789716960 \, x^{2} - 549624420 \, x + 247243887\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {378308799}{81920000} \, \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)^(3/2)*(2+3*x)^2*(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/28672000*(1843200000*x^6 + 4387840000*x^5 + 2867456000*x^4 - 887043200*x^3 - 1789716960*x^2 - 549624420*x +
247243887)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 378308799/81920000*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 = 1.42, size = 446, normalized size = 2.39 \begin {gather*} -\frac {3}{7168000000} \, \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 {79}{512000000} \, \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 {1061}{192000000} \, \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 {1111}{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 {69}{8000} \, \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}{250} \, \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 {54}{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)^(3/2)*(2+3*x)^2*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-3/7168000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(120*x - 443)*(5*x + 3) + 94933)*(5*x + 3) - 7838433)*(5*x + 3) + 98
794353)*(5*x + 3) - 1568443065)*(5*x + 3) + 8438816295)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 17534989395*sqrt(2)*ar
csin(1/11*sqrt(22)*sqrt(5*x + 3))) - 79/512000000*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))) - 1061/192000000*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)*s
qrt(5*x + 3))) - 1111/9600000*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))) + 69/8000*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))) + 81/250*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))) + 54/25*sq
rt(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 = 155, normalized size = 0.83 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-36864000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-87756800000 \sqrt {-10 x^{2}-x +3}\, x^{5}-57349120000 \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 )-4944877740 \sqrt {-10 x^{2}-x +3}\right )}{573440000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/573440000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-36864000000*(-10*x^2-x+3)^(1/2)*x^6-87756800000*(-10*x^2-x+3)^(1/2)
*x^5-57349120000*(-10*x^2-x+3)^(1/2)*x^4+17740864000*(-10*x^2-x+3)^(1/2)*x^3+35794339200*(-10*x^2-x+3)^(1/2)*x
^2+2648161593*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-4944877740*(-10*x^2-x+3)^(1/2))/
(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.48, size = 116, normalized size = 0.62 \begin {gather*} -\frac {9}{14} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x^{2} - \frac {157}{112} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {12309}{11200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {8613}{2560} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {8613}{51200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3126519}{204800} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {378308799}{81920000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3126519}{4096000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/14*(-10*x^2 - x + 3)^(5/2)*x^2 - 157/112*(-10*x^2 - x + 3)^(5/2)*x - 12309/11200*(-10*x^2 - x + 3)^(5/2) +
8613/2560*(-10*x^2 - x + 3)^(3/2)*x + 8613/51200*(-10*x^2 - x + 3)^(3/2) + 3126519/204800*sqrt(-10*x^2 - x + 3
)*x - 378308799/81920000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3126519/4096000*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 )}^{3/2}\,{\left (3\,x+2\right )}^2\,{\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)^(3/2)*(3*x + 2)^2*(5*x + 3)^(5/2),x)

[Out]

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

[Out]

Timed out

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