3.2301 \(\int \frac{\sqrt{1-2 x} (2+3 x)^3}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac{3}{40} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (216 x+335)}{6400}+\frac{47761 \sqrt{1-2 x} \sqrt{5 x+3}}{64000}+\frac{525371 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]

[Out]

(47761*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/40 - (21*(1 - 2*x)^(
3/2)*Sqrt[3 + 5*x]*(335 + 216*x))/6400 + (525371*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0259666, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac{3}{40} (1-2 x)^{3/2} \sqrt{5 x+3} (3 x+2)^2-\frac{21 (1-2 x)^{3/2} \sqrt{5 x+3} (216 x+335)}{6400}+\frac{47761 \sqrt{1-2 x} \sqrt{5 x+3}}{64000}+\frac{525371 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{64000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(47761*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/40 - (21*(1 - 2*x)^(
3/2)*Sqrt[3 + 5*x]*(335 + 216*x))/6400 + (525371*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*Sqrt[10])

Rule 100

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 147

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 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 \frac{\sqrt{1-2 x} (2+3 x)^3}{\sqrt{3+5 x}} \, dx &=-\frac{3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{1}{40} \int \frac{\left (-175-\frac{567 x}{2}\right ) \sqrt{1-2 x} (2+3 x)}{\sqrt{3+5 x}} \, dx\\ &=-\frac{3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (335+216 x)}{6400}+\frac{47761 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{12800}\\ &=\frac{47761 \sqrt{1-2 x} \sqrt{3+5 x}}{64000}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (335+216 x)}{6400}+\frac{525371 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{128000}\\ &=\frac{47761 \sqrt{1-2 x} \sqrt{3+5 x}}{64000}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (335+216 x)}{6400}+\frac{525371 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{64000 \sqrt{5}}\\ &=\frac{47761 \sqrt{1-2 x} \sqrt{3+5 x}}{64000}-\frac{3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}-\frac{21 (1-2 x)^{3/2} \sqrt{3+5 x} (335+216 x)}{6400}+\frac{525371 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{64000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.115353, size = 74, normalized size = 0.7 \[ \frac{-10 \sqrt{5 x+3} \left (172800 x^4+239040 x^3-10440 x^2-159718 x+41789\right )-525371 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{640000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(41789 - 159718*x - 10440*x^2 + 239040*x^3 + 172800*x^4) - 525371*Sqrt[10 - 20*x]*ArcSin[Sq
rt[5/11]*Sqrt[1 - 2*x]])/(640000*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 104, normalized size = 1. \begin{align*}{\frac{1}{1280000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1728000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+3254400\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+525371\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1522800\,x\sqrt{-10\,{x}^{2}-x+3}-835780\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1280000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(1728000*x^3*(-10*x^2-x+3)^(1/2)+3254400*x^2*(-10*x^2-x+3)^(1/2)+525371*
10^(1/2)*arcsin(20/11*x+1/11)+1522800*x*(-10*x^2-x+3)^(1/2)-835780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.74334, size = 99, normalized size = 0.93 \begin{align*} -\frac{27}{200} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{525371}{1280000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{963}{4000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{21663}{16000} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{887}{12800} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/200*(-10*x^2 - x + 3)^(3/2)*x + 525371/1280000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 963/4000*(-10*x^2
- x + 3)^(3/2) + 21663/16000*sqrt(-10*x^2 - x + 3)*x + 887/12800*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.34594, size = 251, normalized size = 2.37 \begin{align*} \frac{1}{64000} \,{\left (86400 \, x^{3} + 162720 \, x^{2} + 76140 \, x - 41789\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{525371}{1280000} \, \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64000*(86400*x^3 + 162720*x^2 + 76140*x - 41789)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 525371/1280000*sqrt(10)*arct
an(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

________________________________________________________________________________________

Sympy [A]  time = 25.3667, size = 462, normalized size = 4.36 \begin{align*} - \frac{343 \sqrt{2} \left (\begin{cases} \frac{11 \sqrt{5} \left (- \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6}}{22} + \frac{\operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{2}\right )}{25} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} + \frac{441 \sqrt{2} \left (\begin{cases} \frac{121 \sqrt{5} \left (\frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (20 x + 1\right )}{968} - \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} - \frac{189 \sqrt{2} \left (\begin{cases} \frac{1331 \sqrt{5} \left (\frac{5 \sqrt{5} \left (1 - 2 x\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{7986} + \frac{3 \sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (20 x + 1\right )}{1936} - \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} + \frac{27 \sqrt{2} \left (\begin{cases} \frac{14641 \sqrt{5} \left (\frac{5 \sqrt{5} \left (1 - 2 x\right )^{\frac{3}{2}} \left (10 x + 6\right )^{\frac{3}{2}}}{3993} + \frac{7 \sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (20 x + 1\right )}{3872} + \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac{\sqrt{5} \sqrt{1 - 2 x} \sqrt{10 x + 6}}{22} + \frac{35 \operatorname{asin}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{3}{5} \end{cases}\right )}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343*sqrt(2)*Piecewise((11*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + asin(sqrt(55)*sqrt(1 - 2*x)/11)
/2)/25, (x <= 1/2) & (x > -3/5)))/8 + 441*sqrt(2)*Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)
*(20*x + 1)/968 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(1 - 2*x)/11)/8)/125, (x <= 1/
2) & (x > -3/5)))/8 - 189*sqrt(2)*Piecewise((1331*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 +
3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/1936 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 5*asin(sqr
t(55)*sqrt(1 - 2*x)/11)/16)/625, (x <= 1/2) & (x > -3/5)))/8 + 27*sqrt(2)*Piecewise((14641*sqrt(5)*(5*sqrt(5)*
(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/3993 + 7*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 + sqrt(5)*sqr
t(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 - sqrt(5)*sqrt(1 -
2*x)*sqrt(10*x + 6)/22 + 35*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128)/3125, (x <= 1/2) & (x > -3/5)))/8

________________________________________________________________________________________

Giac [B]  time = 2.31224, size = 274, normalized size = 2.58 \begin{align*} \frac{9}{3200000} \, \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{9}{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{9}{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{4}{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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/3200000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1
84305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/20000*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))) + 9/500*sqrt(5)*(2*(20*x - 23)*sq
rt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 4/25*sqrt(5)*(11*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))