3.19.52 \(\int \frac {\sqrt {1-2 x}}{(3+5 x)^3} \, dx\) [1852]

Optimal. Leaf size=68 \[ -\frac {\sqrt {1-2 x}}{10 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{110 (3+5 x)}+\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \]

[Out]

1/3025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-1/10*(1-2*x)^(1/2)/(3+5*x)^2+1/110*(1-2*x)^(1/2)/(3+5*x)

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Rubi [A]
time = 0.01, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 212} \begin {gather*} \frac {\sqrt {1-2 x}}{110 (5 x+3)}-\frac {\sqrt {1-2 x}}{10 (5 x+3)^2}+\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/(3 + 5*x)^3,x]

[Out]

-1/10*Sqrt[1 - 2*x]/(3 + 5*x)^2 + Sqrt[1 - 2*x]/(110*(3 + 5*x)) + ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]/(55*Sqrt[5
5])

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin {align*} \int \frac {\sqrt {1-2 x}}{(3+5 x)^3} \, dx &=-\frac {\sqrt {1-2 x}}{10 (3+5 x)^2}-\frac {1}{10} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx\\ &=-\frac {\sqrt {1-2 x}}{10 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{110 (3+5 x)}-\frac {1}{110} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {\sqrt {1-2 x}}{10 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{110 (3+5 x)}+\frac {1}{110} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {\sqrt {1-2 x}}{10 (3+5 x)^2}+\frac {\sqrt {1-2 x}}{110 (3+5 x)}+\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 53, normalized size = 0.78 \begin {gather*} \frac {\sqrt {1-2 x} (-8+5 x)}{110 (3+5 x)^2}+\frac {\tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{55 \sqrt {55}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(-8 + 5*x))/(110*(3 + 5*x)^2) + ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]/(55*Sqrt[55])

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Maple [A]
time = 0.10, size = 48, normalized size = 0.71

method	result	size
risch	\(-\frac {10 x^{2}-21 x +8}{110 \left (3+5 x \right )^{2} \sqrt {1-2 x}}+\frac {\arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\)	\(46\)
derivativedivides	\(\frac {-\frac {\left (1-2 x \right )^{\frac {3}{2}}}{11}-\frac {\sqrt {1-2 x}}{5}}{\left (-6-10 x \right )^{2}}+\frac {\arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\)	\(48\)
default	\(\frac {-\frac {\left (1-2 x \right )^{\frac {3}{2}}}{11}-\frac {\sqrt {1-2 x}}{5}}{\left (-6-10 x \right )^{2}}+\frac {\arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{3025}\)	\(48\)
trager	\(\frac {\left (-8+5 x \right ) \sqrt {1-2 x}}{110 \left (3+5 x \right )^{2}}-\frac {\RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{6050}\)	\(67\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

200*(-1/2200*(1-2*x)^(3/2)-1/1000*(1-2*x)^(1/2))/(-6-10*x)^2+1/3025*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1
/2)

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Maxima [A]
time = 0.58, size = 74, normalized size = 1.09 \begin {gather*} -\frac {1}{6050} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 11 \, \sqrt {-2 \, x + 1}}{55 \, {\left (25 \, {\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/55*(5*(-2*x + 1)^(3/2)
+ 11*sqrt(-2*x + 1))/(25*(2*x - 1)^2 + 220*x + 11)

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Fricas [A]
time = 0.92, size = 69, normalized size = 1.01 \begin {gather*} \frac {\sqrt {55} {\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \, {\left (5 \, x - 8\right )} \sqrt {-2 \, x + 1}}{6050 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6050*(sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(5*x - 8)*sqrt(-2
*x + 1))/(25*x^2 + 30*x + 9)

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Sympy [C] Result contains complex when optimal does not.
time = 2.73, size = 231, normalized size = 3.40 \begin {gather*} \begin {cases} \frac {\sqrt {55} \operatorname {acosh}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{3025} - \frac {\sqrt {2}}{550 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} + \frac {3 \sqrt {2}}{500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} - \frac {11 \sqrt {2}}{2500 \sqrt {-1 + \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \text {for}\: \frac {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\- \frac {\sqrt {55} i \operatorname {asin}{\left (\frac {\sqrt {110}}{10 \sqrt {x + \frac {3}{5}}} \right )}}{3025} + \frac {\sqrt {2} i}{550 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \sqrt {x + \frac {3}{5}}} - \frac {3 \sqrt {2} i}{500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {3}{2}}} + \frac {11 \sqrt {2} i}{2500 \sqrt {1 - \frac {11}{10 \left (x + \frac {3}{5}\right )}} \left (x + \frac {3}{5}\right )^{\frac {5}{2}}} & \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,x)

[Out]

Piecewise((sqrt(55)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/3025 - sqrt(2)/(550*sqrt(-1 + 11/(10*(x + 3/5)))*sqrt(
x + 3/5)) + 3*sqrt(2)/(500*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(3/2)) - 11*sqrt(2)/(2500*sqrt(-1 + 11/(10*
(x + 3/5)))*(x + 3/5)**(5/2)), 1/Abs(x + 3/5) > 10/11), (-sqrt(55)*I*asin(sqrt(110)/(10*sqrt(x + 3/5)))/3025 +
sqrt(2)*I/(550*sqrt(1 - 11/(10*(x + 3/5)))*sqrt(x + 3/5)) - 3*sqrt(2)*I/(500*sqrt(1 - 11/(10*(x + 3/5)))*(x +
3/5)**(3/2)) + 11*sqrt(2)*I/(2500*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(5/2)), True))

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Giac [A]
time = 1.24, size = 68, normalized size = 1.00 \begin {gather*} -\frac {1}{6050} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 11 \, \sqrt {-2 \, x + 1}}{220 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/220*(5*(-2*x
+ 1)^(3/2) + 11*sqrt(-2*x + 1))/(5*x + 3)^2

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Mupad [B]
time = 1.19, size = 54, normalized size = 0.79 \begin {gather*} \frac {\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{3025}-\frac {\frac {\sqrt {1-2\,x}}{125}+\frac {{\left (1-2\,x\right )}^{3/2}}{275}}{\frac {44\,x}{5}+{\left (2\,x-1\right )}^2+\frac {11}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)/(5*x + 3)^3,x)

[Out]

(55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/3025 - ((1 - 2*x)^(1/2)/125 + (1 - 2*x)^(3/2)/275)/((44*x)/5 +
(2*x - 1)^2 + 11/25)

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