3.2470 \(\int \frac{(2+3 x) (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{1}{10} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{59}{80} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{1947}{320} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{21417 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]

[Out]

(-1947*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320 - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/80 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/
2))/10 + (21417*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320*Sqrt[10])

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Rubi [A]  time = 0.0220849, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{1}{10} \sqrt{1-2 x} (5 x+3)^{5/2}-\frac{59}{80} \sqrt{1-2 x} (5 x+3)^{3/2}-\frac{1947}{320} \sqrt{1-2 x} \sqrt{5 x+3}+\frac{21417 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{320 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1947*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/320 - (59*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/80 - (Sqrt[1 - 2*x]*(3 + 5*x)^(5/
2))/10 + (21417*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(320*Sqrt[10])

Rule 80

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 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{(2+3 x) (3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx &=-\frac{1}{10} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{59}{20} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{59}{80} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{1947}{160} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x}} \, dx\\ &=-\frac{1947}{320} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{59}{80} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{21417}{640} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{1947}{320} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{59}{80} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{21417 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{320 \sqrt{5}}\\ &=-\frac{1947}{320} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{59}{80} \sqrt{1-2 x} (3+5 x)^{3/2}-\frac{1}{10} \sqrt{1-2 x} (3+5 x)^{5/2}+\frac{21417 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{320 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0312744, size = 60, normalized size = 0.64 \[ \frac{-10 \sqrt{1-2 x} \sqrt{5 x+3} \left (800 x^2+2140 x+2943\right )-21417 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3200} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2943 + 2140*x + 800*x^2) - 21417*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/
3200

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Maple [A]  time = 0.008, size = 87, normalized size = 0.9 \begin{align*}{\frac{1}{6400}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -16000\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+21417\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -42800\,x\sqrt{-10\,{x}^{2}-x+3}-58860\,\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+5*x)^(3/2)/(1-2*x)^(1/2),x)

[Out]

1/6400*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(-16000*x^2*(-10*x^2-x+3)^(1/2)+21417*10^(1/2)*arcsin(20/11*x+1/11)-42800*x
*(-10*x^2-x+3)^(1/2)-58860*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.23491, size = 78, normalized size = 0.83 \begin{align*} -\frac{5}{2} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{107}{16} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{21417}{6400} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) - \frac{2943}{320} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2*sqrt(-10*x^2 - x + 3)*x^2 - 107/16*sqrt(-10*x^2 - x + 3)*x - 21417/6400*sqrt(10)*arcsin(-20/11*x - 1/11)
- 2943/320*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.66855, size = 221, normalized size = 2.35 \begin{align*} -\frac{1}{320} \,{\left (800 \, x^{2} + 2140 \, x + 2943\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{21417}{6400} \, \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+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/320*(800*x^2 + 2140*x + 2943)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 21417/6400*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 [A]  time = 58.0122, size = 224, normalized size = 2.38 \begin{align*} \frac{2 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (\frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{968} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{3 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{8}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} + \frac{6 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (\frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} + \frac{3 \sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3}}{22} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x
)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5) & (x < 1/2)))/25 + 6*sqrt(5)*Piecewis
e((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5
*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5)
& (x < 1/2)))/25

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Giac [A]  time = 2.04201, size = 73, normalized size = 0.78 \begin{align*} -\frac{1}{3200} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x + 83\right )}{\left (5 \, x + 3\right )} + 1947\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 21417 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3200*sqrt(5)*(2*(4*(40*x + 83)*(5*x + 3) + 1947)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 21417*sqrt(2)*arcsin(1/11*
sqrt(22)*sqrt(5*x + 3)))