3.1799 \(\int \frac{\sqrt{1-2 x} (3+5 x)}{(2+3 x)^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{(1-2 x)^{3/2}}{42 (3 x+2)^2}-\frac{23 \sqrt{1-2 x}}{42 (3 x+2)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

[Out]

(1 - 2*x)^(3/2)/(42*(2 + 3*x)^2) - (23*Sqrt[1 - 2*x])/(42*(2 + 3*x)) + (23*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
21*Sqrt[21])

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Rubi [A]  time = 0.0142218, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {78, 47, 63, 206} \[ \frac{(1-2 x)^{3/2}}{42 (3 x+2)^2}-\frac{23 \sqrt{1-2 x}}{42 (3 x+2)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1 - 2*x)^(3/2)/(42*(2 + 3*x)^2) - (23*Sqrt[1 - 2*x])/(42*(2 + 3*x)) + (23*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(
21*Sqrt[21])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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 63

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 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)}{(2+3 x)^3} \, dx &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}+\frac{23}{14} \int \frac{\sqrt{1-2 x}}{(2+3 x)^2} \, dx\\ &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac{23 \sqrt{1-2 x}}{42 (2+3 x)}-\frac{23}{42} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac{23 \sqrt{1-2 x}}{42 (2+3 x)}+\frac{23}{42} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{(1-2 x)^{3/2}}{42 (2+3 x)^2}-\frac{23 \sqrt{1-2 x}}{42 (2+3 x)}+\frac{23 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0388847, size = 69, normalized size = 1.01 \[ \frac{21 \left (142 x^2+19 x-45\right )+46 \sqrt{21-42 x} (3 x+2)^2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{882 \sqrt{1-2 x} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(21*(-45 + 19*x + 142*x^2) + 46*Sqrt[21 - 42*x]*(2 + 3*x)^2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(882*Sqrt[1 - 2*
x]*(2 + 3*x)^2)

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Maple [A]  time = 0.01, size = 48, normalized size = 0.7 \begin{align*} -36\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ( -{\frac{71\, \left ( 1-2\,x \right ) ^{3/2}}{756}}+{\frac{23\,\sqrt{1-2\,x}}{108}} \right ) }+{\frac{23\,\sqrt{21}}{441}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-36*(-71/756*(1-2*x)^(3/2)+23/108*(1-2*x)^(1/2))/(-6*x-4)^2+23/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2
)

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Maxima [A]  time = 1.5965, size = 100, normalized size = 1.47 \begin{align*} -\frac{23}{882} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{71 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 161 \, \sqrt{-2 \, x + 1}}{21 \,{\left (9 \,{\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23/882*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21*(71*(-2*x + 1)^(3/2)
- 161*sqrt(-2*x + 1))/(9*(2*x - 1)^2 + 84*x + 7)

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Fricas [A]  time = 1.36016, size = 192, normalized size = 2.82 \begin{align*} \frac{23 \, \sqrt{21}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (71 \, x + 45\right )} \sqrt{-2 \, x + 1}}{882 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/882*(23*sqrt(21)*(9*x^2 + 12*x + 4)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(71*x + 45)*sqrt
(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 118.096, size = 316, normalized size = 4.65 \begin{align*} - \frac{148 \left (\begin{cases} \frac{\sqrt{21} \left (- \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{4} + \frac{\log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{4} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} - \frac{1}{4 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )}\right )}{147} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{56 \left (\begin{cases} \frac{\sqrt{21} \left (\frac{3 \log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1 \right )}}{16} - \frac{3 \log{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1 \right )}}{16} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )} + \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} + 1\right )^{2}} + \frac{3}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )} - \frac{1}{16 \left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text{for}\: x \leq \frac{1}{2} \wedge x > - \frac{2}{3} \end{cases}\right )}{9} - \frac{20 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-148*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(s
qrt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (x <= 1/2) & (x > -2/3)))/9 - 56*Pi
ecewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sq
rt(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)
) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3)))/9 - 20*Piecewise((-sqrt(21)*acot
h(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3
))/9

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Giac [A]  time = 2.05992, size = 92, normalized size = 1.35 \begin{align*} -\frac{23}{882} \, \sqrt{21} \log \left (\frac{{\left | -2 \, \sqrt{21} + 6 \, \sqrt{-2 \, x + 1} \right |}}{2 \,{\left (\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}\right )}}\right ) + \frac{71 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 161 \, \sqrt{-2 \, x + 1}}{84 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-23/882*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/84*(71*(-2*x +
1)^(3/2) - 161*sqrt(-2*x + 1))/(3*x + 2)^2