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

Optimal. Leaf size=52 \[ -\frac {2 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}-\frac {2}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {47, 54, 216} \begin {gather*} -\frac {2 \sqrt {1-2 x}}{5 \sqrt {5 x+3}}-\frac {2}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/(3 + 5*x)^(3/2),x]

[Out]

(-2*Sqrt[1 - 2*x])/(5*Sqrt[3 + 5*x]) - (2*Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/5

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 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}}{(3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}-\frac {2}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}-\frac {4 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5 \sqrt {5}}\\ &=-\frac {2 \sqrt {1-2 x}}{5 \sqrt {3+5 x}}-\frac {2}{5} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 58, normalized size = 1.12 \begin {gather*} \frac {2}{25} \sqrt {1-2 x} \left (\frac {\sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{\sqrt {2 x-1}}-\frac {5}{\sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[1 - 2*x]*(-5/Sqrt[3 + 5*x] + (Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/Sqrt[-1 + 2*x]))/25

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IntegrateAlgebraic [A]  time = 0.06, size = 61, normalized size = 1.17 \begin {gather*} \frac {2}{5} \sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x}}{5 \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.46, size = 76, normalized size = 1.46 \begin {gather*} \frac {\sqrt {5} \sqrt {2} {\left (5 \, x + 3\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 10 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{25 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25*(sqrt(5)*sqrt(2)*(5*x + 3)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 10*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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giac [B]  time = 1.32, size = 83, normalized size = 1.60 \begin {gather*} -\frac {1}{50} \, \sqrt {5} {\left (4 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {2} \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/50*sqrt(5)*(4*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)) + sqrt(2)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(2)*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))

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maple [F]  time = 0.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-2 x +1}}{\left (5 x +3\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.36, size = 36, normalized size = 0.69 \begin {gather*} -\frac {1}{25} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {2 \, \sqrt {-10 \, x^{2} - x + 3}}{5 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/25*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 2/5*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {1-2\,x}}{{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 1.55, size = 151, normalized size = 2.90 \begin {gather*} \begin {cases} - \frac {2 \sqrt {10} \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{25} - \frac {\sqrt {10} i \log {\left (\frac {1}{x + \frac {3}{5}} \right )}}{25} - \frac {\sqrt {10} i \log {\left (x + \frac {3}{5} \right )}}{25} - \frac {2 \sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{25} & \text {for}\: \frac {11}{10 \left |{x + \frac {3}{5}}\right |} > 1 \\- \frac {2 \sqrt {10} i \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}}}{25} - \frac {\sqrt {10} i \log {\left (\frac {1}{x + \frac {3}{5}} \right )}}{25} + \frac {2 \sqrt {10} i \log {\left (\sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} + 1 \right )}}{25} & \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)**(3/2),x)

[Out]

Piecewise((-2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/25 - sqrt(10)*I*log(1/(x + 3/5))/25 - sqrt(10)*I*log(x + 3
/5)/25 - 2*sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/25, 11/(10*Abs(x + 3/5)) > 1), (-2*sqrt(10)*I*sqrt(1 - 11
/(10*(x + 3/5)))/25 - sqrt(10)*I*log(1/(x + 3/5))/25 + 2*sqrt(10)*I*log(sqrt(1 - 11/(10*(x + 3/5))) + 1)/25, T
rue))

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