3.25.11 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx\) [2411]

Optimal. Leaf size=216 \[ \frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}} \]

[Out]

-5787133/3072000*(1-2*x)^(7/2)*(3+5*x)^(3/2)-526103/768000*(1-2*x)^(7/2)*(3+5*x)^(5/2)-1/30*(1-2*x)^(7/2)*(2+3
*x)^2*(3+5*x)^(7/2)-1/672000*(1-2*x)^(7/2)*(3+5*x)^(7/2)*(245011+170940*x)[phone]/65536000000*arcsin(1/1
1*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)[phone]/1966080000*(1-2*x)^(3/2)*(3+5*x)^(1/2)[phone]/491520000*(1-2*
x)^(5/2)*(3+5*x)^(1/2)-63658463/16384000*(1-2*x)^(7/2)*(3+5*x)^(1/2)[phone]/6553600000*(1-2*x)^(1/2)*(3+5
*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {102, 152, 52, 56, 222} \begin {gather*} \frac {932023556783 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{6553600000 \sqrt {10}}-\frac {1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac {5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac {(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac {63658463 \sqrt {5 x+3} (1-2 x)^{7/2}}{16384000}+\frac {700243093 \sqrt {5 x+3} (1-2 x)^{5/2}}{491520000}+\frac {7702674023 \sqrt {5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac {84729414253 \sqrt {5 x+3} \sqrt {1-2 x}}{6553600000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(84729414253*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6553600000 + (7702674023*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1966080000 +
(700243093*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/491520000 - (63658463*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/16384000 - (57
87133*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/3072000 - (526103*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/768000 - ((1 - 2*x)^
(7/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/30 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(7/2)*(245011 + 170940*x))/672000 + (932023
556783*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6553600000*Sqrt[10])

Rule 52

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 56

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 102

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 152

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 222

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)^{5/2} \, dx &=-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{90} \int \left (-393-\frac {1221 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {526103 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{64000}\\ &=-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {5787133 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{307200}\\ &=-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {63658463 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{2048000}\\ &=-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {700243093 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{32768000}\\ &=\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {7702674023 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{196608000}\\ &=\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {84729414253 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{1310720000}\\ &=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{13107200000}\\ &=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{6553600000 \sqrt {5}}\\ &=\frac {84729414253 \sqrt {1-2 x} \sqrt {3+5 x}}{6553600000}+\frac {7702674023 (1-2 x)^{3/2} \sqrt {3+5 x}}{1966080000}+\frac {700243093 (1-2 x)^{5/2} \sqrt {3+5 x}}{491520000}-\frac {63658463 (1-2 x)^{7/2} \sqrt {3+5 x}}{16384000}-\frac {5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac {526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac {1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac {(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac {932023556783 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{6553600000 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.34, size = 103, normalized size = 0.48 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-4490138164311+11708962285875 x+81324587821180 x^2+65091129546400 x^3-209312726736000 x^4-440233726720000 x^5-52760857600000 x^6+605463552000000 x^7+636088320000000 x^8+206438400000000 x^9\right )-19572494692443 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{1376256000000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(-4490138164311 + 11708962285875*x + 81324587821180*x^2 + 65091129546400*x^3 - 2093127267360
00*x^4 - 440233726720000*x^5 - 52760857600000*x^6 + 605463552000000*x^7 + 636088320000000*x^8 + 20643840000000
0*x^9) - 19572494692443*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(1376256000000*Sqrt[3 + 5*x])

________________________________________________________________________________________

Maple [A]
time = 0.10, size = 189, normalized size = 0.88

method	result	size
risch	\(-\frac {\left (41287680000000 x^{8}[phone] x^{7}[phone] x^{6}-46327577600000 x^{5}-60250198784000 x^{4}-5712426076800 x^{3}[phone] x^{2}[phone] x -1496712721437\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{137625600000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {932023556783 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{131072000000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(128\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (825753600000000 \sqrt {-10 x^{2}-x +3}\, x^{8}+2048901120000000 \sqrt {-10 x^{2}-x +3}\, x^{7}+1192513536000000 \sqrt {-10 x^{2}-x +3}\, x^{6}-926551552000000 x^{5} \sqrt {-10 x^{2}-x +3}-1205003975680000 x^{4} \sqrt {-10 x^{2}-x +3}-114248521536000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-29934254428740 \sqrt {-10 x^{2}-x +3}\right )}{2752512000000 \sqrt {-10 x^{2}-x +3}}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2752512000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(825753600000000*(-10*x^2-x+3)^(1/2)*x^8+2048901120000000*(-10*x^2
-x+3)^(1/2)*x^7+1192513536000000*(-10*x^2-x+3)^(1/2)*x^6-926551552000000*x^5*(-10*x^2-x+3)^(1/2)-1205003975680
000*x^4*(-10*x^2-x+3)^(1/2)-114248521536000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)+19
572494692443*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-29934254428740*(-10*x^2-x+3)^
(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.51, size = 145, normalized size = 0.67 \begin {gather*} -\frac {3}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x^{2} - \frac {1047}{1600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {111537}{224000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {526103}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {526103}{7680000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {63658463}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {63658463}{245760000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {7702674023}{327680000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {932023556783}{131072000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {7702674023}{6553600000} \, \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)^(5/2),x, algorithm="maxima")

[Out]

-3/10*(-10*x^2 - x + 3)^(7/2)*x^2 - 1047/1600*(-10*x^2 - x + 3)^(7/2)*x - 111537/224000*(-10*x^2 - x + 3)^(7/2
) + 526103/384000*(-10*x^2 - x + 3)^(5/2)*x + 526103/7680000*(-10*x^2 - x + 3)^(5/2) + 63658463/12288000*(-10*
x^2 - x + 3)^(3/2)*x + 63658463/245760000*(-10*x^2 - x + 3)^(3/2) + 7702674023/327680000*sqrt(-10*x^2 - x + 3)
*x - 932023556783/131072000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 7702674023/6553600000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]
time = 0.81, size = 97, normalized size = 0.45 \begin {gather*} \frac {1}{137625600000} \, {\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {932023556783}{131072000000} \, \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)^(5/2),x, algorithm="fricas")

[Out]

1/137625600000*(41287680000000*x^8 + 102445056000000*x^7 + 59625676800000*x^6 - 46327577600000*x^5 - 602501987
84000*x^4 - 5712426076800*x^3 + 16445681555360*x^2 + 6397508631020*x - 1496712721437)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 932023556783/131072000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2
+ x - 3))

________________________________________________________________________________________

Sympy [F(-1)] Timed out
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)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 653 vs. \(2 (159) = 318\).
time = 0.70, size = 653, normalized size = 3.02 \begin {gather*} \frac {3}{2293760000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (4 \, {\left (8 \, {\left (28 \, {\left (160 \, x - 779\right )} {\left (5 \, x + 3\right )} + 297993\right )} {\left (5 \, x + 3\right )} - 16954963\right )} {\left (5 \, x + 3\right )} + 311501761\right )} {\left (5 \, x + 3\right )} - 15396301917\right )} {\left (5 \, x + 3\right )} + 129214816941\right )} {\left (5 \, x + 3\right )} - 1465300159701\right )} {\left (5 \, x + 3\right )} + 5967262275723\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10642307691015 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{40960000000} \, \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 {2217}{358400000000} \, \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 {121}{38400000000} \, \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 {15709}{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 {1429}{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 {129}{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 {297}{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 {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 )} \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)^(5/2),x, algorithm="giac")

[Out]

3/2293760000000*sqrt(5)*(2*(4*(8*(4*(16*(4*(8*(28*(160*x - 779)*(5*x + 3) + 297993)*(5*x + 3) - 16954963)*(5*x
+ 3) + 311501761)*(5*x + 3) - 15396301917)*(5*x + 3) + 129214816941)*(5*x + 3) - 1465300159701)*(5*x + 3) + 5
967262275723)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10642307691015*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/
40960000000*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))) + 2217/358400000000*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
21/38400000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 153856
95)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)
)) - 15709/960000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031
895)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1429/1920000*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)*
arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 129/20000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqr
t(-10*x + 5) + 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 297/500*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))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________