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

Optimal. Leaf size=116 \[ -\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {5 x+3}+\frac {1331}{512} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512 \sqrt {10}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {50, 54, 216} \begin {gather*} -\frac {1}{8} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {55}{96} (1-2 x)^{3/2} (5 x+3)^{3/2}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {5 x+3}+\frac {1331}{512} \sqrt {1-2 x} \sqrt {5 x+3}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{512 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(3 + 5*x)^(5/2),x]

[Out]

(1331*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/512 - (605*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/256 - (55*(1 - 2*x)^(3/2)*(3 + 5*
x)^(3/2))/96 - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/8 + (14641*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(512*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 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 \sqrt {1-2 x} (3+5 x)^{5/2} \, dx &=-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {55}{16} \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {605}{64} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {6655}{512} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1024}\\ &=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{512 \sqrt {5}}\\ &=\frac {1331}{512} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {605}{256} (1-2 x)^{3/2} \sqrt {3+5 x}-\frac {55}{96} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {1}{8} (1-2 x)^{3/2} (3+5 x)^{5/2}+\frac {14641 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{512 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.07, size = 74, normalized size = 0.64 \begin {gather*} \frac {43923 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (19200 x^4+21440 x^3-3848 x^2-13846 x+4005\right )}{15360 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(4005 - 13846*x - 3848*x^2 + 21440*x^3 + 19200*x^4) + 43923*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5
/11]*Sqrt[-1 + 2*x]])/(15360*Sqrt[1 - 2*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.14, size = 125, normalized size = 1.08 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {375 (1-2 x)^3}{(5 x+3)^3}+\frac {550 (1-2 x)^2}{(5 x+3)^2}+\frac {292 (1-2 x)}{5 x+3}-24\right )}{1536 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {14641 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{512 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-24 + (375*(1 - 2*x)^3)/(3 + 5*x)^3 + (550*(1 - 2*x)^2)/(3 + 5*x)^2 + (292*(1 - 2*x))/(
3 + 5*x)))/(1536*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^4) - (14641*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt
[3 + 5*x]])/(512*Sqrt[10])

________________________________________________________________________________________

fricas [A]  time = 1.41, size = 72, normalized size = 0.62 \begin {gather*} \frac {1}{1536} \, {\left (9600 \, x^{3} + 15520 \, x^{2} + 5836 \, x - 4005\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {14641}{10240} \, \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((3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/1536*(9600*x^3 + 15520*x^2 + 5836*x - 4005)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 14641/10240*sqrt(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 = 1.13, size = 203, normalized size = 1.75 \begin {gather*} \frac {1}{76800} \, \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 {3}{1600} \, \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}{400} \, \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 {27}{50} \, \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((3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="giac")

[Out]

1/76800*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184
305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 3/1600*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))) + 27/400*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))) + 27/50*sqrt(5)*(11*sqrt(2)*arcsi
n(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 104, normalized size = 0.90 \begin {gather*} \frac {14641 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{10240 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (5 x +3\right )^{\frac {7}{2}} \sqrt {-2 x +1}}{20}-\frac {11 \left (5 x +3\right )^{\frac {5}{2}} \sqrt {-2 x +1}}{240}-\frac {121 \left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{384}-\frac {1331 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{512} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*(5*x+3)^(7/2)*(-2*x+1)^(1/2)-11/240*(5*x+3)^(5/2)*(-2*x+1)^(1/2)-121/384*(5*x+3)^(3/2)*(-2*x+1)^(1/2)-133
1/512*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+14641/10240*((-2*x+1)*(5*x+3))^(1/2)/(5*x+3)^(1/2)/(-2*x+1)^(1/2)*10^(1/2)*
arcsin(20/11*x+1/11)

________________________________________________________________________________________

maxima [A]  time = 1.15, size = 70, normalized size = 0.60 \begin {gather*} -\frac {5}{8} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {91}{96} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {605}{128} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {14641}{10240} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {121}{512} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/8*(-10*x^2 - x + 3)^(3/2)*x - 91/96*(-10*x^2 - x + 3)^(3/2) + 605/128*sqrt(-10*x^2 - x + 3)*x - 14641/10240
*sqrt(10)*arcsin(-20/11*x - 1/11) + 121/512*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 9.71, size = 272, normalized size = 2.34 \begin {gather*} \begin {cases} \frac {125 i \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {10 x - 5}} - \frac {1925 i \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{24 \sqrt {10 x - 5}} - \frac {605 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{192 \sqrt {10 x - 5}} - \frac {6655 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{768 \sqrt {10 x - 5}} + \frac {14641 i \sqrt {x + \frac {3}{5}}}{512 \sqrt {10 x - 5}} - \frac {14641 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5120} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {14641 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5120} - \frac {125 \left (x + \frac {3}{5}\right )^{\frac {9}{2}}}{2 \sqrt {5 - 10 x}} + \frac {1925 \left (x + \frac {3}{5}\right )^{\frac {7}{2}}}{24 \sqrt {5 - 10 x}} + \frac {605 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{192 \sqrt {5 - 10 x}} + \frac {6655 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{768 \sqrt {5 - 10 x}} - \frac {14641 \sqrt {x + \frac {3}{5}}}{512 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((125*I*(x + 3/5)**(9/2)/(2*sqrt(10*x - 5)) - 1925*I*(x + 3/5)**(7/2)/(24*sqrt(10*x - 5)) - 605*I*(x
+ 3/5)**(5/2)/(192*sqrt(10*x - 5)) - 6655*I*(x + 3/5)**(3/2)/(768*sqrt(10*x - 5)) + 14641*I*sqrt(x + 3/5)/(512
*sqrt(10*x - 5)) - 14641*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/5120, 10*Abs(x + 3/5)/11 > 1), (14641*sq
rt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/5120 - 125*(x + 3/5)**(9/2)/(2*sqrt(5 - 10*x)) + 1925*(x + 3/5)**(7/2)
/(24*sqrt(5 - 10*x)) + 605*(x + 3/5)**(5/2)/(192*sqrt(5 - 10*x)) + 6655*(x + 3/5)**(3/2)/(768*sqrt(5 - 10*x))
- 14641*sqrt(x + 3/5)/(512*sqrt(5 - 10*x)), True))

________________________________________________________________________________________