3.21.54 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\) [2054]

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

[Out]

-6/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+64/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-5/11*(1
-2*x)^(1/2)/(3+5*x)

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Rubi [A]
time = 0.02, 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 = {105, 162, 65, 212} \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 65

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 105

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] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 162

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 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 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 \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {160}{11} \text {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 = 0.18, size = 75, normalized size = 0.97 \begin {gather*} -\frac {5 \sqrt {1-2 x}}{33+55 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 {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])/(33 + 55*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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Maple [A]
time = 0.16, size = 54, normalized size = 0.70

method	result	size
derivativedivides	\(\frac {2 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {64 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(54\)
default	\(\frac {2 \sqrt {1-2 x}}{11 \left (-\frac {6}{5}-2 x \right )}+\frac {64 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(54\)
risch	\(\frac {-\frac {5}{11}+\frac {10 x}{11}}{\left (3+5 x \right ) \sqrt {1-2 x}}+\frac {64 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{121}-\frac {6 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{7}\)	\(59\)
trager	\(-\frac {5 \sqrt {1-2 x}}{11 \left (3+5 x \right )}+\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{7}-\frac {32 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x +55 \sqrt {1-2 x}-8 \RootOf \left (\textit {\_Z}^{2}-55\right )}{3+5 x}\right )}{121}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.48, 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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Fricas [A]
time = 1.37, 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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Sympy [C] Result contains complex when optimal does not.
time = 5.61, size = 515, normalized size = 6.69 \begin {gather*} - \frac {140 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {4340 \sqrt {55} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {7260 \sqrt {21} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {2170 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {3630 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {154 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110}}{10 \sqrt {x - \frac {1}{2}}} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {4774 \sqrt {55} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {7986 \sqrt {21} i \sqrt {x - \frac {1}{2}} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {2387 \sqrt {55} i \pi \sqrt {x - \frac {1}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} + \frac {3993 \sqrt {21} i \pi \sqrt {x - \frac {1}{2}}}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \sqrt {x - \frac {1}{2}}} - \frac {770 \sqrt {2} i \left (x - \frac {1}{2}\right )}{8470 \left (x - \frac {1}{2}\right )^{\frac {3}{2}} + 9317 \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)*I*(x - 1/2)**(3/2)*atan(sqrt(110)/(10*sqrt(x - 1/2)))/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2
)) + 4340*sqrt(55)*I*(x - 1/2)**(3/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x -
1/2)) - 7260*sqrt(21)*I*(x - 1/2)**(3/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x -
1/2)) - 2170*sqrt(55)*I*pi*(x - 1/2)**(3/2)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) + 3630*sqrt(21)*I*pi
*(x - 1/2)**(3/2)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) - 154*sqrt(55)*I*sqrt(x - 1/2)*atan(sqrt(110)/(
10*sqrt(x - 1/2)))/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) + 4774*sqrt(55)*I*sqrt(x - 1/2)*atan(sqrt(110)
*sqrt(x - 1/2)/11)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) - 7986*sqrt(21)*I*sqrt(x - 1/2)*atan(sqrt(42)*
sqrt(x - 1/2)/7)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)) - 2387*sqrt(55)*I*pi*sqrt(x - 1/2)/(8470*(x - 1/
2)**(3/2) + 9317*sqrt(x - 1/2)) + 3993*sqrt(21)*I*pi*sqrt(x - 1/2)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2)
) - 770*sqrt(2)*I*(x - 1/2)/(8470*(x - 1/2)**(3/2) + 9317*sqrt(x - 1/2))

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Giac [A]
time = 1.58, 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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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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