3.2428 \(\int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=94 \[ \frac{1}{15} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{60} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/60 + ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]
)/15 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*Sqrt[10])

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Rubi [A]  time = 0.023463, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {50, 54, 216} \[ \frac{1}{15} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{11}{60} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{121}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/Sqrt[3 + 5*x],x]

[Out]

(121*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 + (11*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/60 + ((1 - 2*x)^(5/2)*Sqrt[3 + 5*x]
)/15 + (1331*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(200*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 \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx &=\frac{1}{15} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{11}{6} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{11}{60} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{1}{15} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{121}{40} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{121}{200} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{60} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{1}{15} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{1331}{400} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{121}{200} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{60} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{1}{15} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{1331 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{200 \sqrt{5}}\\ &=\frac{121}{200} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{11}{60} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{1}{15} (1-2 x)^{5/2} \sqrt{3+5 x}+\frac{1331 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0274014, size = 69, normalized size = 0.73 \[ \frac{10 \sqrt{5 x+3} \left (-320 x^3+920 x^2-1406 x+513\right )-3993 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{6000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(513 - 1406*x + 920*x^2 - 320*x^3) - 3993*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/
(6000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.003, size = 88, normalized size = 0.9 \begin{align*}{\frac{1}{15} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}\sqrt{3+5\,x}}+{\frac{11}{60} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}\sqrt{3+5\,x}}+{\frac{121}{200}\sqrt{1-2\,x}\sqrt{3+5\,x}}+{\frac{1331\,\sqrt{10}}{4000}\sqrt{ \left ( 1-2\,x \right ) \left ( 3+5\,x \right ) }\arcsin \left ({\frac{20\,x}{11}}+{\frac{1}{11}} \right ){\frac{1}{\sqrt{1-2\,x}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(1-2*x)^(5/2)*(3+5*x)^(1/2)+11/60*(1-2*x)^(3/2)*(3+5*x)^(1/2)+121/200*(1-2*x)^(1/2)*(3+5*x)^(1/2)+1331/40
00*((1-2*x)*(3+5*x))^(1/2)/(3+5*x)^(1/2)/(1-2*x)^(1/2)*10^(1/2)*arcsin(20/11*x+1/11)

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Maxima [A]  time = 2.58292, size = 78, normalized size = 0.83 \begin{align*} \frac{4}{15} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} - \frac{19}{30} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1331}{4000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{171}{200} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/15*sqrt(-10*x^2 - x + 3)*x^2 - 19/30*sqrt(-10*x^2 - x + 3)*x - 1331/4000*sqrt(10)*arcsin(-20/11*x - 1/11) +
171/200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.97831, size = 216, normalized size = 2.3 \begin{align*} \frac{1}{600} \,{\left (160 \, x^{2} - 380 \, x + 513\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1331}{4000} \, \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((1-2*x)^(5/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/600*(160*x^2 - 380*x + 513)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1331/4000*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 = 10.7723, size = 230, normalized size = 2.45 \begin{align*} \begin{cases} \frac{8 i \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{10 x - 5}} - \frac{187 i \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{15 \sqrt{10 x - 5}} + \frac{7139 i \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{300 \sqrt{10 x - 5}} - \frac{14641 i \sqrt{x + \frac{3}{5}}}{1000 \sqrt{10 x - 5}} - \frac{1331 \sqrt{10} i \operatorname{acosh}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2000} & \text{for}\: \frac{10 \left |{x + \frac{3}{5}}\right |}{11} > 1 \\\frac{1331 \sqrt{10} \operatorname{asin}{\left (\frac{\sqrt{110} \sqrt{x + \frac{3}{5}}}{11} \right )}}{2000} - \frac{8 \left (x + \frac{3}{5}\right )^{\frac{7}{2}}}{3 \sqrt{5 - 10 x}} + \frac{187 \left (x + \frac{3}{5}\right )^{\frac{5}{2}}}{15 \sqrt{5 - 10 x}} - \frac{7139 \left (x + \frac{3}{5}\right )^{\frac{3}{2}}}{300 \sqrt{5 - 10 x}} + \frac{14641 \sqrt{x + \frac{3}{5}}}{1000 \sqrt{5 - 10 x}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((8*I*(x + 3/5)**(7/2)/(3*sqrt(10*x - 5)) - 187*I*(x + 3/5)**(5/2)/(15*sqrt(10*x - 5)) + 7139*I*(x +
3/5)**(3/2)/(300*sqrt(10*x - 5)) - 14641*I*sqrt(x + 3/5)/(1000*sqrt(10*x - 5)) - 1331*sqrt(10)*I*acosh(sqrt(11
0)*sqrt(x + 3/5)/11)/2000, 10*Abs(x + 3/5)/11 > 1), (1331*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/2000 - 8*(
x + 3/5)**(7/2)/(3*sqrt(5 - 10*x)) + 187*(x + 3/5)**(5/2)/(15*sqrt(5 - 10*x)) - 7139*(x + 3/5)**(3/2)/(300*sqr
t(5 - 10*x)) + 14641*sqrt(x + 3/5)/(1000*sqrt(5 - 10*x)), True))

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Giac [B]  time = 2.6998, size = 189, normalized size = 2.01 \begin{align*} \frac{1}{30000} \, \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{1}{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{1}{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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30000*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*s
qrt(22)*sqrt(5*x + 3))) - 1/500*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))) + 1/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqr
t(-10*x + 5))