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

Optimal. Leaf size=77 \[ -\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}-6 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {103, 156, 63, 206} \begin {gather*} -\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}-6 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{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)*(3 + 5*x)^2),x]

[Out]

(-5*Sqrt[1 - 2*x])/(11*(3 + 5*x)) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (64*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 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) (3+5 x)^2} \, dx &=-\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-\frac {1}{11} \int \frac {23-15 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}+9 \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {160}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-9 \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {160}{11} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {5 \sqrt {1-2 x}}{11 (3+5 x)}-6 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{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 = 77, normalized size = 1.00 \begin {gather*} -\frac {5 \sqrt {1-2 x}}{11 (5 x+3)}-6 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*Sqrt[1 - 2*x])/(11*(3 + 5*x)) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (64*Sqrt[5/11]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]])/11

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IntegrateAlgebraic [A]  time = 0.17, size = 81, normalized size = 1.05 \begin {gather*} \frac {10 \sqrt {1-2 x}}{11 (5 (1-2 x)-11)}-6 \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {64}{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)*(3 + 5*x)^2),x]

[Out]

(10*Sqrt[1 - 2*x])/(11*(-11 + 5*(1 - 2*x))) - 6*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + (64*Sqrt[5/11]*Ar
cTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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fricas [A]  time = 1.27, size = 102, normalized size = 1.32 \begin {gather*} \frac {224 \, \sqrt {11} \sqrt {5} {\left (5 \, x + 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 363 \, \sqrt {7} \sqrt {3} {\left (5 \, x + 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 385 \, \sqrt {-2 \, x + 1}}{847 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/847*(224*sqrt(11)*sqrt(5)*(5*x + 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 363*sqrt(7
)*sqrt(3)*(5*x + 3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 385*sqrt(-2*x + 1))/(5*x + 3)

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giac [A]  time = 1.23, size = 95, normalized size = 1.23 \begin {gather*} -\frac {32}{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 {3}{7} \, \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 {5 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 3/7*sqrt(21)*lo
g(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5/11*sqrt(-2*x + 1)/(5*x + 3)

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maple [A]  time = 0.01, size = 54, normalized size = 0.70 \begin {gather*} -\frac {6 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{7}+\frac {64 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {2 \sqrt {-2 x +1}}{11 \left (-2 x -\frac {6}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/11*(-2*x+1)^(1/2)/(-2*x-6/5)+64/121*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-6/7*arctanh(1/7*21^(1/2)*
(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.20, size = 89, normalized size = 1.16 \begin {gather*} -\frac {32}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {3}{7} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {5 \, \sqrt {-2 \, x + 1}}{11 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-32/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 3/7*sqrt(21)*log(-(sqrt(2
1) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 5/11*sqrt(-2*x + 1)/(5*x + 3)

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mupad [B]  time = 0.10, size = 53, normalized size = 0.69 \begin {gather*} \frac {64\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121}-\frac {6\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{7}-\frac {2\,\sqrt {1-2\,x}}{11\,\left (2\,x+\frac {6}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(64*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121 - (6*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/7 -
(2*(1 - 2*x)^(1/2))/(11*(2*x + 6/5))

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sympy [C]  time = 13.45, size = 552, normalized size = 7.17 \begin {gather*} - \frac {140 \sqrt {55} \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} + \frac {4340 \sqrt {55} \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} - \frac {7260 \sqrt {21} \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} - \frac {2170 \sqrt {55} \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} + \frac {3630 \sqrt {21} \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} - \frac {154 \sqrt {55} \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} + \frac {4774 \sqrt {55} \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} - \frac {7986 \sqrt {21} \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} - \frac {2387 \sqrt {55} \pi \sqrt {x - \frac {1}{2}}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} + \frac {3993 \sqrt {21} \pi \sqrt {x - \frac {1}{2}}}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} - \frac {770 \sqrt {2} \left (x - \frac {1}{2}\right )}{- 8470 i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} - 9317 i \sqrt {x - \frac {1}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-140*sqrt(55)*(x - 1/2)**(3/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x -
1/2)) + 4340*sqrt(55)*(x - 1/2)**(3/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqr
t(x - 1/2)) - 7260*sqrt(21)*(x - 1/2)**(3/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-8470*I*(x - 1/2)**(3/2) - 9317*I
*sqrt(x - 1/2)) - 2170*sqrt(55)*pi*(x - 1/2)**(3/2)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2)) + 3630*s
qrt(21)*pi*(x - 1/2)**(3/2)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2)) - 154*sqrt(55)*sqrt(x - 1/2)*ata
n(sqrt(110)/(10*sqrt(x - 1/2)))/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2)) + 4774*sqrt(55)*sqrt(x - 1/2
)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2)) - 7986*sqrt(21)*sqrt(x -
1/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2)) - 2387*sqrt(55)*pi*sqrt(
x - 1/2)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2)) + 3993*sqrt(21)*pi*sqrt(x - 1/2)/(-8470*I*(x - 1/2)
**(3/2) - 9317*I*sqrt(x - 1/2)) - 770*sqrt(2)*(x - 1/2)/(-8470*I*(x - 1/2)**(3/2) - 9317*I*sqrt(x - 1/2))

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