3.2008 \(\int \frac{3+5 x}{\sqrt{1-2 x} (2+3 x)^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{\sqrt{1-2 x}}{21 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

[Out]

Sqrt[1 - 2*x]/(21*(2 + 3*x)) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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Rubi [A]  time = 0.009429, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {78, 63, 206} \[ \frac{\sqrt{1-2 x}}{21 (3 x+2)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(21*(2 + 3*x)) - (68*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 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{3+5 x}{\sqrt{1-2 x} (2+3 x)^2} \, dx &=\frac{\sqrt{1-2 x}}{21 (2+3 x)}+\frac{34}{21} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{21 (2+3 x)}-\frac{34}{21} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{\sqrt{1-2 x}}{21 (2+3 x)}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}}\\ \end{align*}

Mathematica [A]  time = 0.0220071, size = 45, normalized size = 0.94 \[ \frac{\sqrt{1-2 x}}{63 x+42}-\frac{68 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{21 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(42 + 63*x) - (68*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])

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Maple [A]  time = 0.008, size = 36, normalized size = 0.8 \begin{align*} -{\frac{2}{63}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{68\,\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)/(2+3*x)^2/(1-2*x)^(1/2),x)

[Out]

-2/63*(1-2*x)^(1/2)/(-2*x-4/3)-68/441*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)

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Maxima [A]  time = 4.67756, size = 72, normalized size = 1.5 \begin{align*} \frac{34}{441} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{\sqrt{-2 \, x + 1}}{21 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/441*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21*sqrt(-2*x + 1)/(3*x +
2)

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Fricas [A]  time = 1.3851, size = 151, normalized size = 3.15 \begin{align*} \frac{34 \, \sqrt{21}{\left (3 \, x + 2\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, \sqrt{-2 \, x + 1}}{441 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/441*(34*sqrt(21)*(3*x + 2)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*sqrt(-2*x + 1))/(3*x + 2)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.5481, size = 76, normalized size = 1.58 \begin{align*} \frac{34}{441} \, \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{\sqrt{-2 \, x + 1}}{21 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

34/441*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21*sqrt(-2*x +
1)/(3*x + 2)