3.25.78 \(\int \frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\) [2478]

Optimal. Leaf size=150 \[ -\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{819200 \sqrt {10}} \]

[Out]

2678359321/8192000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-22135201/614400*(3+5*x)^(3/2)*(1-2*x)^(1/2)-20
12291/384000*(3+5*x)^(5/2)*(1-2*x)^(1/2)-1/20*(2+3*x)^2*(3+5*x)^(7/2)*(1-2*x)^(1/2)-1/32000*(3+5*x)^(7/2)*(374
39+18960*x)*(1-2*x)^(1/2)-243487211/819200*(1-2*x)^(1/2)*(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 7, 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 {2678359321 \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{819200 \sqrt {10}}-\frac {1}{20} \sqrt {1-2 x} (3 x+2)^2 (5 x+3)^{7/2}-\frac {\sqrt {1-2 x} (18960 x+37439) (5 x+3)^{7/2}}{32000}-\frac {2012291 \sqrt {1-2 x} (5 x+3)^{5/2}}{384000}-\frac {22135201 \sqrt {1-2 x} (5 x+3)^{3/2}}{614400}-\frac {243487211 \sqrt {1-2 x} \sqrt {5 x+3}}{819200} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-243487211*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/819200 - (22135201*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/614400 - (2012291*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/384000 - (Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(7/2))/20 - (Sqrt[1 - 2*x]*(3 + 5*
x)^(7/2)*(37439 + 18960*x))/32000 + (2678359321*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(819200*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 \frac {(2+3 x)^3 (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx &=-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {1}{60} \int \frac {\left (-381-\frac {1185 x}{2}\right ) (2+3 x) (3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2012291 \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x}} \, dx}{64000}\\ &=-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {22135201 \int \frac {(3+5 x)^{3/2}}{\sqrt {1-2 x}} \, dx}{153600}\\ &=-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {243487211 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{409600}\\ &=-\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1638400}\\ &=-\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{819200 \sqrt {5}}\\ &=-\frac {243487211 \sqrt {1-2 x} \sqrt {3+5 x}}{819200}-\frac {22135201 \sqrt {1-2 x} (3+5 x)^{3/2}}{614400}-\frac {2012291 \sqrt {1-2 x} (3+5 x)^{5/2}}{384000}-\frac {1}{20} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{7/2}-\frac {\sqrt {1-2 x} (3+5 x)^{7/2} (37439+18960 x)}{32000}+\frac {2678359321 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{819200 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.22, size = 88, normalized size = 0.59 \begin {gather*} \frac {-10 \sqrt {1-2 x} \left (3608689671+10102628445 x+11328597700 x^2+11213711200 x^3+7993296000 x^4+3490560000 x^5+691200000 x^6\right )-8035077963 \sqrt {30+50 x} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right )}{24576000 \sqrt {3+5 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*(3608689671 + 10102628445*x + 11328597700*x^2 + 11213711200*x^3 + 7993296000*x^4 + 34905600
00*x^5 + 691200000*x^6) - 8035077963*Sqrt[30 + 50*x]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])/(24576000*Sqrt[3 +
5*x])

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 138, normalized size = 0.92

method	result	size
risch	\(\frac {\left (138240000 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2457600 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {2678359321 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{16384000 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(113\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (-2764800000 x^{5} \sqrt {-10 x^{2}-x +3}-12303360000 x^{4} \sqrt {-10 x^{2}-x +3}-24591168000 x^{3} \sqrt {-10 x^{2}-x +3}-30100144000 x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-27254304400 x \sqrt {-10 x^{2}-x +3}-24057931140 \sqrt {-10 x^{2}-x +3}\right )}{49152000 \sqrt {-10 x^{2}-x +3}}\)	\(138\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/49152000*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-2764800000*x^5*(-10*x^2-x+3)^(1/2)-12303360000*x^4*(-10*x^2-x+3)^(1/2
)-24591168000*x^3*(-10*x^2-x+3)^(1/2)-30100144000*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1
/11)-27254304400*x*(-10*x^2-x+3)^(1/2)-24057931140*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]
time = 0.52, size = 109, normalized size = 0.73 \begin {gather*} -\frac {225}{4} \, \sqrt {-10 \, x^{2} - x + 3} x^{5} - \frac {4005}{16} \, \sqrt {-10 \, x^{2} - x + 3} x^{4} - \frac {128079}{256} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {1881259}{3072} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {68135761}{122880} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {2678359321}{16384000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {400965519}{819200} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-225/4*sqrt(-10*x^2 - x + 3)*x^5 - 4005/16*sqrt(-10*x^2 - x + 3)*x^4 - 128079/256*sqrt(-10*x^2 - x + 3)*x^3 -
1881259/3072*sqrt(-10*x^2 - x + 3)*x^2 - 68135761/122880*sqrt(-10*x^2 - x + 3)*x - 2678359321/16384000*sqrt(10
)*arcsin(-20/11*x - 1/11) - 400965519/819200*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]
time = 0.40, size = 82, normalized size = 0.55 \begin {gather*} -\frac {1}{2457600} \, {\left (138240000 \, x^{5} + 615168000 \, x^{4} + 1229558400 \, x^{3} + 1505007200 \, x^{2} + 1362715220 \, x + 1202896557\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {2678359321}{16384000} \, \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)^3*(3+5*x)^(5/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2457600*(138240000*x^5 + 615168000*x^4 + 1229558400*x^3 + 1505007200*x^2 + 1362715220*x + 1202896557)*sqrt(
5*x + 3)*sqrt(-2*x + 1) - 2678359321/16384000*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((2+3*x)**3*(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]
time = 1.92, size = 81, normalized size = 0.54 \begin {gather*} -\frac {1}{122880000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (108 \, {\left (16 \, {\left (20 \, x + 41\right )} {\left (5 \, x + 3\right )} + 2903\right )} {\left (5 \, x + 3\right )} + 2012291\right )} {\left (5 \, x + 3\right )} + 110676005\right )} {\left (5 \, x + 3\right )} + 3652308165\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 40175389815 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/122880000*sqrt(5)*(2*(4*(8*(108*(16*(20*x + 41)*(5*x + 3) + 2903)*(5*x + 3) + 2012291)*(5*x + 3) + 11067600
5)*(5*x + 3) + 3652308165)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 40175389815*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x +
3)))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________