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

Optimal. Leaf size=72 \[ -\frac {1}{4} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {11}{40} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {121 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40 \sqrt {10}} \]

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Rubi [A]  time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, 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}{4} \sqrt {5 x+3} (1-2 x)^{3/2}+\frac {11}{40} \sqrt {5 x+3} \sqrt {1-2 x}+\frac {121 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{40 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*Sqrt[3 + 5*x],x]

[Out]

(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/40 - ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4 + (121*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
])/(40*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} \sqrt {3+5 x} \, dx &=-\frac {1}{4} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {11}{8} \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx\\ &=\frac {11}{40} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {121}{80} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {11}{40} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {121 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{40 \sqrt {5}}\\ &=\frac {11}{40} \sqrt {1-2 x} \sqrt {3+5 x}-\frac {1}{4} (1-2 x)^{3/2} \sqrt {3+5 x}+\frac {121 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{40 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 64, normalized size = 0.89 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (-40 x^2+18 x+1\right )+121 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{400 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(1 + 18*x - 40*x^2) + 121*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(400*Sqrt[1 -
2*x])

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IntegrateAlgebraic [A]  time = 0.14, size = 93, normalized size = 1.29 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {5 (1-2 x)}{5 x+3}-2\right )}{40 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^2}-\frac {121 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{40 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - 2*x]*Sqrt[3 + 5*x],x]

[Out]

(-121*Sqrt[1 - 2*x]*(-2 + (5*(1 - 2*x))/(3 + 5*x)))/(40*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^2) - (121*
ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(40*Sqrt[10])

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fricas [A]  time = 0.82, size = 62, normalized size = 0.86 \begin {gather*} \frac {1}{40} \, {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {121}{800} \, \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((1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/40*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 121/800*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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giac [A]  time = 1.05, size = 86, normalized size = 1.19 \begin {gather*} \frac {1}{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 {3}{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((1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

1/400*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)))
+ 3/50*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.01, size = 72, normalized size = 1.00 \begin {gather*} \frac {121 \sqrt {\left (-2 x +1\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )}{800 \sqrt {5 x +3}\, \sqrt {-2 x +1}}+\frac {\left (5 x +3\right )^{\frac {3}{2}} \sqrt {-2 x +1}}{10}-\frac {11 \sqrt {-2 x +1}\, \sqrt {5 x +3}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(5*x+3)^(3/2)*(-2*x+1)^(1/2)-11/40*(-2*x+1)^(1/2)*(5*x+3)^(1/2)+121/800*((-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)

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maxima [A]  time = 1.21, size = 41, normalized size = 0.57 \begin {gather*} \frac {1}{2} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {121}{800} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1}{40} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(-10*x^2 - x + 3)*x - 121/800*sqrt(10)*arcsin(-20/11*x - 1/11) + 1/40*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 1.71, size = 54, normalized size = 0.75 \begin {gather*} \sqrt {1-2\,x}\,\sqrt {5\,x+3}\,\left (\frac {x}{2}+\frac {1}{40}\right )-\frac {\sqrt {2}\,\sqrt {5}\,\ln \left (x+\frac {1}{20}-\frac {\sqrt {10}\,\sqrt {1-2\,x}\,\sqrt {5\,x+3}\,1{}\mathrm {i}}{10}\right )\,121{}\mathrm {i}}{800} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1 - 2*x)^(1/2)*(5*x + 3)^(1/2)*(x/2 + 1/40) - (2^(1/2)*5^(1/2)*log(x - (10^(1/2)*(1 - 2*x)^(1/2)*(5*x + 3)^(1
/2)*1i)/10 + 1/20)*121i)/800

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sympy [A]  time = 2.62, size = 184, normalized size = 2.56 \begin {gather*} \begin {cases} \frac {5 i \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{\sqrt {10 x - 5}} - \frac {33 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{4 \sqrt {10 x - 5}} + \frac {121 i \sqrt {x + \frac {3}{5}}}{40 \sqrt {10 x - 5}} - \frac {121 \sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{400} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\\frac {121 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{400} - \frac {5 \left (x + \frac {3}{5}\right )^{\frac {5}{2}}}{\sqrt {5 - 10 x}} + \frac {33 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{4 \sqrt {5 - 10 x}} - \frac {121 \sqrt {x + \frac {3}{5}}}{40 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((5*I*(x + 3/5)**(5/2)/sqrt(10*x - 5) - 33*I*(x + 3/5)**(3/2)/(4*sqrt(10*x - 5)) + 121*I*sqrt(x + 3/5
)/(40*sqrt(10*x - 5)) - 121*sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/400, 10*Abs(x + 3/5)/11 > 1), (121*sq
rt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/400 - 5*(x + 3/5)**(5/2)/sqrt(5 - 10*x) + 33*(x + 3/5)**(3/2)/(4*sqrt(
5 - 10*x)) - 121*sqrt(x + 3/5)/(40*sqrt(5 - 10*x)), True))

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