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

Optimal. Leaf size=106 \[ -\frac {340 \sqrt {1-2 x}}{77 (5 x+3)}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 106, 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 = {103, 151, 156, 63, 206} \begin {gather*} -\frac {340 \sqrt {1-2 x}}{77 (5 x+3)}+\frac {3 \sqrt {1-2 x}}{7 (3 x+2) (5 x+3)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-340*Sqrt[1 - 2*x])/(77*(3 + 5*x)) + (3*Sqrt[1 - 2*x])/(7*(2 + 3*x)*(3 + 5*x)) - (426*Sqrt[3/7]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/7 + (650*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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 {1}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx &=\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}+\frac {1}{7} \int \frac {41-45 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {1}{77} \int \frac {1663-1020 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}+\frac {639}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {1625}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {639}{7} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {1625}{11} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {340 \sqrt {1-2 x}}{77 (3+5 x)}+\frac {3 \sqrt {1-2 x}}{7 (2+3 x) (3+5 x)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 100, normalized size = 0.94 \begin {gather*} \frac {4550 \sqrt {55} \left (15 x^2+19 x+6\right ) \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )-11 \sqrt {1-2 x} (1020 x+647)}{847 (3 x+2) (5 x+3)}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-426*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 + (-11*Sqrt[1 - 2*x]*(647 + 1020*x) + 4550*Sqrt[55]*(6 + 1
9*x + 15*x^2)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(847*(2 + 3*x)*(3 + 5*x))

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IntegrateAlgebraic [A]  time = 0.23, size = 101, normalized size = 0.95 \begin {gather*} \frac {4 \sqrt {1-2 x} (510 (1-2 x)-1157)}{77 \left (15 (1-2 x)^2-68 (1-2 x)+77\right )}-\frac {426}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {650}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(-1157 + 510*(1 - 2*x))*Sqrt[1 - 2*x])/(77*(77 - 68*(1 - 2*x) + 15*(1 - 2*x)^2)) - (426*Sqrt[3/7]*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/7 + (650*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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fricas [A]  time = 1.42, size = 122, normalized size = 1.15 \begin {gather*} \frac {15925 \, \sqrt {11} \sqrt {5} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 25773 \, \sqrt {7} \sqrt {3} {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (1020 \, x + 647\right )} \sqrt {-2 \, x + 1}}{5929 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5929*(15925*sqrt(11)*sqrt(5)*(15*x^2 + 19*x + 6)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3))
+ 25773*sqrt(7)*sqrt(3)*(15*x^2 + 19*x + 6)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(1
020*x + 647)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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giac [A]  time = 1.20, size = 116, normalized size = 1.09 \begin {gather*} -\frac {325}{121} \, \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 {213}{49} \, \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 {4 \, {\left (510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1157 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-325/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 213/49*sqrt(21
)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/77*(510*(-2*x + 1)^(3/2) - 11
57*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)

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maple [A]  time = 0.02, size = 70, normalized size = 0.66 \begin {gather*} -\frac {426 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{49}+\frac {650 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {10 \sqrt {-2 x +1}}{11 \left (-2 x -\frac {6}{5}\right )}+\frac {6 \sqrt {-2 x +1}}{7 \left (-2 x -\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

10/11*(-2*x+1)^(1/2)/(-2*x-6/5)+650/121*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+6/7*(-2*x+1)^(1/2)/(-2*
x-4/3)-426/49*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.28, size = 110, normalized size = 1.04 \begin {gather*} -\frac {325}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {213}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (510 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 1157 \, \sqrt {-2 \, x + 1}\right )}}{77 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-325/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 213/49*sqrt(21)*log(-(sq
rt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/77*(510*(-2*x + 1)^(3/2) - 1157*sqrt(-2*x + 1))/
(15*(2*x - 1)^2 + 136*x + 9)

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mupad [B]  time = 0.10, size = 72, normalized size = 0.68 \begin {gather*} \frac {650\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {426\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}-\frac {\frac {4628\,\sqrt {1-2\,x}}{1155}-\frac {136\,{\left (1-2\,x\right )}^{3/2}}{77}}{\frac {136\,x}{15}+{\left (2\,x-1\right )}^2+\frac {3}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(650*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - (426*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/4
9 - ((4628*(1 - 2*x)^(1/2))/1155 - (136*(1 - 2*x)^(3/2))/77)/((136*x)/15 + (2*x - 1)^2 + 3/5)

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sympy [C]  time = 19.90, size = 988, normalized size = 9.32

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

314160*sqrt(2)*I*(x - 1/2)**(5/2)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 356356*
sqrt(2)*I*(x - 1/2)**(3/2)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 29400*sqrt(55)
*I*(x - 1/2)**3*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 45
6533/2) - 1881600*sqrt(55)*I*(x - 1/2)**3*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-456533*x - 355740*(x - 1/2)**3 -
806344*(x - 1/2)**2 + 456533/2) + 43560*sqrt(21)*I*(x - 1/2)**3*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(-456533*x -
355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 3136320*sqrt(21)*I*(x - 1/2)**3*atan(sqrt(42)*sqrt(x -
1/2)/7)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) - 1568160*sqrt(21)*I*pi*(x - 1/2)*
*3/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 940800*sqrt(55)*I*pi*(x - 1/2)**3/(-45
6533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 66640*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)/
(10*sqrt(x - 1/2)))/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) - 4264960*sqrt(55)*I*(x
- 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2)
+ 98736*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)/(6*sqrt(x - 1/2)))/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x
- 1/2)**2 + 456533/2) + 7108992*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-456533*x - 355740*(x
- 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) - 3554496*sqrt(21)*I*pi*(x - 1/2)**2/(-456533*x - 355740*(x - 1/2
)**3 - 806344*(x - 1/2)**2 + 456533/2) + 2132480*sqrt(55)*I*pi*(x - 1/2)**2/(-456533*x - 355740*(x - 1/2)**3 -
806344*(x - 1/2)**2 + 456533/2) + 37730*sqrt(55)*I*(x - 1/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(-456533*x -
355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) - 2414720*sqrt(55)*I*(x - 1/2)*atan(sqrt(110)*sqrt(x - 1
/2)/11)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 55902*sqrt(21)*I*(x - 1/2)*atan(s
qrt(42)/(6*sqrt(x - 1/2)))/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 4024944*sqrt(2
1)*I*(x - 1/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/
2) - 2012472*sqrt(21)*I*pi*(x - 1/2)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2) + 1207
360*sqrt(55)*I*pi*(x - 1/2)/(-456533*x - 355740*(x - 1/2)**3 - 806344*(x - 1/2)**2 + 456533/2)

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