3.25.12 \(\int (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{5/2} \, dx\) [2412]

Optimal. Leaf size=209 \[ \frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{327680000 \sqrt {10}} \]

[Out]

-132451/153600*(1-2*x)^(7/2)*(3+5*x)^(3/2)-12041/38400*(1-2*x)^(7/2)*(3+5*x)^(5/2)-999/11200*(1-2*x)^(7/2)*(3+
5*x)^(7/2)-3/80*(1-2*x)^(7/2)*(2+3*x)*(3+5*x)^(7/2)[phone]/3276800000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))
*10^(1/2)[phone]/98304000*(1-2*x)^(3/2)*(3+5*x)^(1/2)[phone]/24576000*(1-2*x)^(5/2)*(3+5*x)^(1/2)-1456961
/819200*(1-2*x)^(7/2)*(3+5*x)^(1/2)[phone]/327680000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 209, 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 = {92, 81, 52, 56, 222} \begin {gather*} \frac {21331366001 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{327680000 \sqrt {10}}-\frac {3}{80} (3 x+2) (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac {999 (5 x+3)^{7/2} (1-2 x)^{7/2}}{11200}-\frac {12041 (5 x+3)^{5/2} (1-2 x)^{7/2}}{38400}-\frac {132451 (5 x+3)^{3/2} (1-2 x)^{7/2}}{153600}-\frac {1456961 \sqrt {5 x+3} (1-2 x)^{7/2}}{819200}+\frac {16026571 \sqrt {5 x+3} (1-2 x)^{5/2}}{24576000}+\frac {176292281 \sqrt {5 x+3} (1-2 x)^{3/2}}{98304000}+\frac {1939215091 \sqrt {5 x+3} \sqrt {1-2 x}}{327680000} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1939215091*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/327680000 + (176292281*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/98304000 + (160
26571*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/24576000 - (1456961*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/819200 - (132451*(1 -
2*x)^(7/2)*(3 + 5*x)^(3/2))/153600 - (12041*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/38400 - (999*(1 - 2*x)^(7/2)*(3 +
5*x)^(7/2))/11200 - (3*(1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(7/2))/80 + (21331366001*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(327680000*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 81

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 92

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 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)^2 (3+5 x)^{5/2} \, dx &=-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}-\frac {1}{80} \int \left (-326-\frac {999 x}{2}\right ) (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx\\ &=-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {12041 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{3200}\\ &=-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {132451 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{15360}\\ &=-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {1456961 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{102400}\\ &=-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {16026571 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{1638400}\\ &=\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {176292281 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{9830400}\\ &=\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {1939215091 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{65536000}\\ &=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{655360000}\\ &=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{327680000 \sqrt {5}}\\ &=\frac {1939215091 \sqrt {1-2 x} \sqrt {3+5 x}}{327680000}+\frac {176292281 (1-2 x)^{3/2} \sqrt {3+5 x}}{98304000}+\frac {16026571 (1-2 x)^{5/2} \sqrt {3+5 x}}{24576000}-\frac {1456961 (1-2 x)^{7/2} \sqrt {3+5 x}}{819200}-\frac {132451 (1-2 x)^{7/2} (3+5 x)^{3/2}}{153600}-\frac {12041 (1-2 x)^{7/2} (3+5 x)^{5/2}}{38400}-\frac {999 (1-2 x)^{7/2} (3+5 x)^{7/2}}{11200}-\frac {3}{80} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{7/2}+\frac {21331366001 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{327680000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 98, normalized size = 0.47 \begin {gather*} \frac {10 \sqrt {1-2 x} \left (-67244995017+395916406125 x+1604753427460 x^2+22475240800 x^3-5109421872000 x^4-4777381120000 x^5+4571417600000 x^6+9133056000000 x^7+3870720000000 x^8\right )-447958686021 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{68812800000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*(-67244995017 + 395916406125*x + 1604753427460*x^2 + 22475240800*x^3 - 5109421872000*x^4 - 4
777381120000*x^5 + 4571417600000*x^6 + 9133056000000*x^7 + 3870720000000*x^8) - 447958686021*Sqrt[30 + 50*x]*A
rcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(68812800000*Sqrt[3 + 5*x])

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Maple [A]
time = 0.10, size = 172, normalized size = 0.82

method	result	size
risch	\(-\frac {\left (774144000000 x^{7}[phone] x^{6}[phone] x^{5}-1013681408000 x^{4}-413675529600 x^{3}[phone] x^{2}[phone] x -22414998339\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6881280000 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {21331366001 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{6553600000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(123\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (15482880000000 \sqrt {-10 x^{2}-x +3}\, x^{7}[phone] \sqrt {-10 x^{2}-x +3}\, x^{6}[phone] x^{5} \sqrt {-10 x^{2}-x +3}-20273628160000 x^{4} \sqrt {-10 x^{2}-x +3}-8273510592000 x^{3} \sqrt {-10 x^{2}-x +3}[phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )[phone] x \sqrt {-10 x^{2}-x +3}-448299966780 \sqrt {-10 x^{2}-x +3}\right )}{137625600000 \sqrt {-10 x^{2}-x +3}}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/137625600000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(15482880000000*(-10*x^2-x+3)^(1/2)*x^7+27242496000000*(-10*x^2-x+3
)^(1/2)*x^6+1940172800000*x^5*(-10*x^2-x+3)^(1/2)-20273628160000*x^4*(-10*x^2-x+3)^(1/2)-8273510592000*x^3*(-1
0*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)[phone]
0*x*(-10*x^2-x+3)^(1/2)-448299966780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]
time = 0.54, size = 128, normalized size = 0.61 \begin {gather*} -\frac {9}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} x - \frac {1839}{11200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {7}{2}} + \frac {12041}{19200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x + \frac {12041}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {1456961}{614400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {1456961}{12288000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {176292281}{16384000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {21331366001}{6553600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {176292281}{327680000} \, \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)^2*(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

-9/80*(-10*x^2 - x + 3)^(7/2)*x - 1839/11200*(-10*x^2 - x + 3)^(7/2) + 12041/19200*(-10*x^2 - x + 3)^(5/2)*x +
12041/384000*(-10*x^2 - x + 3)^(5/2) + 1456961/614400*(-10*x^2 - x + 3)^(3/2)*x + 1456961/12288000*(-10*x^2 -
x + 3)^(3/2) + 176292281/16384000*sqrt(-10*x^2 - x + 3)*x - 21331366001/6553600000*sqrt(10)*arcsin(-20/11*x -
1/11) + 176292281/327680000*sqrt(-10*x^2 - x + 3)

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Fricas [A]
time = 0.53, size = 92, normalized size = 0.44 \begin {gather*} \frac {1}{6881280000} \, {\left (774144000000 \, x^{7} + 1362124800000 \, x^{6} + 97008640000 \, x^{5} - 1013681408000 \, x^{4} - 413675529600 \, x^{3} + 252700365920 \, x^{2} + 169330465940 \, x - 22414998339\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {21331366001}{6553600000} \, \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)^2*(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

1/6881280000*(774144000000*x^7 + 1362124800000*x^6 + 97008640000*x^5 - 1013681408000*x^4 - 413675529600*x^3 +
252700365920*x^2 + 169330465940*x - 22414998339)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 21331366001/6553600000*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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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)**2*(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (152) = 304\).
time = 0.76, size = 545, normalized size = 2.61 \begin {gather*} \frac {3}{114688000000} \, \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 {1}{560000000} \, \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 {937}{7680000000} \, \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 {3083}{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 {3181}{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 {87}{40000} \, \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 {27}{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 {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)^(5/2)*(2+3*x)^2*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

3/114688000000*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))) + 1/560000000*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))) + 937/
7680000000*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))) -
3083/960000000*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))) - 3181/9600000*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)*arcsi
n(1/11*sqrt(22)*sqrt(5*x + 3))) - 87/40000*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))) + 27/125*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*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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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 )}^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)^(5/2)*(3*x + 2)^2*(5*x + 3)^(5/2),x)

[Out]

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

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