3.2303 \(\int \frac{\sqrt{1-2 x} (2+3 x)}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=72 \[ -\frac{3}{20} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{41}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

[Out]

(41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 - (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/20 + (451*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(200*Sqrt[10])

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Rubi [A]  time = 0.0154829, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{20} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{41}{200} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{200 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/200 - (3*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/20 + (451*ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]])/(200*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{\sqrt{1-2 x} (2+3 x)}{\sqrt{3+5 x}} \, dx &=-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{41}{40} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=\frac{41}{200} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{451}{400} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{41}{200} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{451 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{200 \sqrt{5}}\\ &=\frac{41}{200} \sqrt{1-2 x} \sqrt{3+5 x}-\frac{3}{20} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{451 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{200 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0280298, size = 64, normalized size = 0.89 \[ \frac{10 \sqrt{5 x+3} \left (-120 x^2+38 x+11\right )-451 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{2000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(11 + 38*x - 120*x^2) - 451*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(2000*Sqrt[1 -
2*x])

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Maple [A]  time = 0.007, size = 70, normalized size = 1. \begin{align*}{\frac{1}{4000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 451\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +1200\,x\sqrt{-10\,{x}^{2}-x+3}+220\,\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)*(1-2*x)^(1/2)/(3+5*x)^(1/2),x)

[Out]

1/4000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(451*10^(1/2)*arcsin(20/11*x+1/11)+1200*x*(-10*x^2-x+3)^(1/2)+220*(-10*x^2-
x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.26988, size = 59, normalized size = 0.82 \begin{align*} \frac{451}{4000} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{3}{10} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{11}{200} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

451/4000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 3/10*sqrt(-10*x^2 - x + 3)*x + 11/200*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.37428, size = 198, normalized size = 2.75 \begin{align*} \frac{1}{200} \,{\left (60 \, x + 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{451}{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((2+3*x)*(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/200*(60*x + 11)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 451/4000*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 = 7.14694, size = 165, normalized size = 2.29 \begin{align*} - \frac{7 \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 )}{2} + \frac{3 \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 )}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*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)))/2 + 3*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)))/2

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Giac [A]  time = 1.97485, size = 116, normalized size = 1.61 \begin{align*} \frac{3}{2000} \, \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}{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)*(1-2*x)^(1/2)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

3/2000*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)))
+ 1/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))