3.1835 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=95 \[ \frac{69 \sqrt{1-2 x}}{14 (3 x+2)}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2}+\frac{793}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

Sqrt[1 - 2*x]/(2*(2 + 3*x)^2) + (69*Sqrt[1 - 2*x])/(14*(2 + 3*x)) + (793*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/7 - 10*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0352092, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {99, 151, 156, 63, 206} \[ \frac{69 \sqrt{1-2 x}}{14 (3 x+2)}+\frac{\sqrt{1-2 x}}{2 (3 x+2)^2}+\frac{793}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(2*(2 + 3*x)^2) + (69*Sqrt[1 - 2*x])/(14*(2 + 3*x)) + (793*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/7 - 10*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

Rule 99

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

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, 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}}{(2+3 x)^3 (3+5 x)} \, dx &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2}-\frac{1}{2} \int \frac{-13+15 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2}+\frac{69 \sqrt{1-2 x}}{14 (2+3 x)}-\frac{1}{14} \int \frac{-563+345 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2}+\frac{69 \sqrt{1-2 x}}{14 (2+3 x)}-\frac{2379}{14} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+275 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{2 (2+3 x)^2}+\frac{69 \sqrt{1-2 x}}{14 (2+3 x)}+\frac{2379}{14} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-275 \operatorname{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}}{2 (2+3 x)^2}+\frac{69 \sqrt{1-2 x}}{14 (2+3 x)}+\frac{793}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0686515, size = 80, normalized size = 0.84 \[ \frac{\sqrt{1-2 x} (207 x+145)}{14 (3 x+2)^2}+\frac{793}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-10 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(145 + 207*x))/(14*(2 + 3*x)^2) + (793*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - 10*Sqrt[
55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.01, size = 66, normalized size = 0.7 \begin{align*} -18\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{23\, \left ( 1-2\,x \right ) ^{3/2}}{14}}-{\frac{71\,\sqrt{1-2\,x}}{18}} \right ) }+{\frac{793\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-10\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-18*(23/14*(1-2*x)^(3/2)-71/18*(1-2*x)^(1/2))/(-6*x-4)^2+793/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1
0*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.06695, size = 149, normalized size = 1.57 \begin{align*} 5 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{793}{98} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{207 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \sqrt{-2 \, x + 1}}{7 \,{\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((1-2*x)^(1/2)/(2+3*x)^3/(3+5*x),x, algorithm="maxima")

[Out]

5*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 793/98*sqrt(21)*log(-(sqrt(21)
- 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/7*(207*(-2*x + 1)^(3/2) - 497*sqrt(-2*x + 1))/(9*(2*x -
1)^2 + 84*x + 7)

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Fricas [A]  time = 1.59079, size = 328, normalized size = 3.45 \begin{align*} \frac{793 \, \sqrt{7} \sqrt{3}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 490 \, \sqrt{55}{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 7 \,{\left (207 \, x + 145\right )} \sqrt{-2 \, x + 1}}{98 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/98*(793*sqrt(7)*sqrt(3)*(9*x^2 + 12*x + 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 490*
sqrt(55)*(9*x^2 + 12*x + 4)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 7*(207*x + 145)*sqrt(-2*x + 1
))/(9*x^2 + 12*x + 4)

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Sympy [A]  time = 62.3178, size = 376, normalized size = 3.96 \begin{align*} 132 \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 ) - 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 ) - 330 \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 ) + 550 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

132*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*(sq
rt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) - 56*Piece
wise((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*(sqrt(
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))) - 330*Piecewise((-sqrt(21)*acoth(sq
rt(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)) +
550*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(55)*sqr
t(1 - 2*x)/11)/55, 2*x - 1 > -11/5))

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Giac [A]  time = 1.99294, size = 144, normalized size = 1.52 \begin{align*} 5 \, \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{793}{98} \, \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{207 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 497 \, \sqrt{-2 \, x + 1}}{28 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 793/98*sqrt(21)*log(1
/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/28*(207*(-2*x + 1)^(3/2) - 497*sqrt(
-2*x + 1))/(3*x + 2)^2