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]

793/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-10*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+1/2*(1-2*
x)^(1/2)/(2+3*x)^2+69/14*(1-2*x)^(1/2)/(2+3*x)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00, 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 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 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 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 = 0.09, 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]]

________________________________________________________________________________________

fricas [A]  time = 0.96, size = 116, normalized size = 1.22 \[ \frac {793 \, \sqrt {7} \sqrt {3} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 490 \, \sqrt {55} {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (\frac {5 \, x + \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 7 \, {\left (207 \, x + 145\right )} \sqrt {-2 \, x + 1}}{98 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \]

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)

________________________________________________________________________________________

giac [A]  time = 1.20, size = 107, normalized size = 1.13 \[ 5 \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {793}{98} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {207 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{28 \, {\left (3 \, x + 2\right )}^{2}} \]

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

________________________________________________________________________________________

maple [A]  time = 0.01, size = 66, normalized size = 0.69 \[ \frac {793 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{49}-10 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )-\frac {18 \left (\frac {23 \left (-2 x +1\right )^{\frac {3}{2}}}{14}-\frac {71 \sqrt {-2 x +1}}{18}\right )}{\left (-6 x -4\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-10*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-18*(23/14*(-2*x+1)^(3/2)-71/18*(-2*x+1)^(1/2))/(-6*x-4)^2+7
93/49*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.33, size = 110, normalized size = 1.16 \[ 5 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {793}{98} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {207 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 497 \, \sqrt {-2 \, x + 1}}{7 \, {\left (9 \, {\left (2 \, x - 1\right )}^{2} + 84 \, x + 7\right )}} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 71, normalized size = 0.75 \[ \frac {793\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-10\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )+\frac {\frac {71\,\sqrt {1-2\,x}}{9}-\frac {23\,{\left (1-2\,x\right )}^{3/2}}{7}}{\frac {28\,x}{3}+{\left (2\,x-1\right )}^2+\frac {7}{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(793*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/49 - 10*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) + ((7
1*(1 - 2*x)^(1/2))/9 - (23*(1 - 2*x)^(3/2))/7)/((28*x)/3 + (2*x - 1)^2 + 7/9)

________________________________________________________________________________________

sympy [A]  time = 142.01, size = 376, normalized size = 3.96 \[ 132 \left (\begin {cases} \frac {\sqrt {21} \left (- \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{4} + \frac {\log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{4} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} - \frac {1}{4 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )}\right )}{147} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {2}{3} \end {cases}\right ) - 56 \left (\begin {cases} \frac {\sqrt {21} \left (\frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1 \right )}}{16} - \frac {3 \log {\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1 \right )}}{16} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )} + \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} + 1\right )^{2}} + \frac {3}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )} - \frac {1}{16 \left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} - 1\right )^{2}}\right )}{1029} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {2}{3} \end {cases}\right ) - 330 \left (\begin {cases} - \frac {\sqrt {21} \operatorname {acoth}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 < - \frac {7}{3} \\- \frac {\sqrt {21} \operatorname {atanh}{\left (\frac {\sqrt {21} \sqrt {1 - 2 x}}{7} \right )}}{21} & \text {for}\: 2 x - 1 > - \frac {7}{3} \end {cases}\right ) + 550 \left (\begin {cases} - \frac {\sqrt {55} \operatorname {acoth}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 < - \frac {11}{5} \\- \frac {\sqrt {55} \operatorname {atanh}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{55} & \text {for}\: 2 x - 1 > - \frac {11}{5} \end {cases}\right ) \]

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))

________________________________________________________________________________________