3.22.76 \(\int \frac {1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\) [2176]

Optimal. Leaf size=85 \[ \frac {4}{231 (1-2 x)^{3/2}}+\frac {272}{5929 \sqrt {1-2 x}}+\frac {18}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

[Out]

4/231/(1-2*x)^(3/2)+18/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/1331*arctanh(1/11*55^(1/2)*(1-2*x)^
(1/2))*55^(1/2)+272/5929/(1-2*x)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 85, 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 = {87, 157, 162, 65, 212} \begin {gather*} \frac {272}{5929 \sqrt {1-2 x}}+\frac {4}{231 (1-2 x)^{3/2}}+\frac {18}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{121} \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/((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)),x]

[Out]

4/(231*(1 - 2*x)^(3/2)) + 272/(5929*Sqrt[1 - 2*x]) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*
Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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 87

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

Rule 157

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] && IntegersQ[2*m, 2*n, 2*p]

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}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx &=\frac {4}{231 (1-2 x)^{3/2}}+\frac {1}{77} \int \frac {53+30 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2}}+\frac {272}{5929 \sqrt {1-2 x}}-\frac {2 \int \frac {-\frac {2449}{2}-1020 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{5929}\\ &=\frac {4}{231 (1-2 x)^{3/2}}+\frac {272}{5929 \sqrt {1-2 x}}-\frac {27}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {125}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {4}{231 (1-2 x)^{3/2}}+\frac {272}{5929 \sqrt {1-2 x}}+\frac {27}{49} \text {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {125}{121} \text {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {4}{231 (1-2 x)^{3/2}}+\frac {272}{5929 \sqrt {1-2 x}}+\frac {18}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{121} \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 = 0.13, size = 77, normalized size = 0.91 \begin {gather*} -\frac {4 (-281+408 x)}{17787 (1-2 x)^{3/2}}+\frac {18}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{121} \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/((1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)),x]

[Out]

(-4*(-281 + 408*x))/(17787*(1 - 2*x)^(3/2)) + (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (50*Sqrt[5/
11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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Maple [A]
time = 0.14, size = 56, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {4}{231 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {18 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {272}{5929 \sqrt {1-2 x}}\)	\(56\)
default	\(\frac {4}{231 \left (1-2 x \right )^{\frac {3}{2}}}+\frac {18 \arctanh \left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{343}-\frac {50 \arctanh \left (\frac {\sqrt {55}\, \sqrt {1-2 x}}{11}\right ) \sqrt {55}}{1331}+\frac {272}{5929 \sqrt {1-2 x}}\)	\(56\)
trager	\(-\frac {4 \left (408 x -281\right ) \sqrt {1-2 x}}{17787 \left (-1+2 x \right )^{2}}-\frac {9 \RootOf \left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{2}-21\right ) x -5 \RootOf \left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{343}-\frac {25 \RootOf \left (\textit {\_Z}^{2}-55\right ) \ln \left (-\frac {5 \RootOf \left (\textit {\_Z}^{2}-55\right ) x -8 \RootOf \left (\textit {\_Z}^{2}-55\right )-55 \sqrt {1-2 x}}{3+5 x}\right )}{1331}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/231/(1-2*x)^(3/2)+18/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50/1331*arctanh(1/11*55^(1/2)*(1-2*x)^
(1/2))*55^(1/2)+272/5929/(1-2*x)^(1/2)

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Maxima [A]
time = 0.50, size = 87, normalized size = 1.02 \begin {gather*} \frac {25}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {9}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {4 \, {\left (408 \, x - 281\right )}}{17787 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/343*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 4/17787*(408*x - 281)/(-2*x + 1)^(3/2)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (55) = 110\).
time = 1.10, size = 122, normalized size = 1.44 \begin {gather*} \frac {25725 \, \sqrt {11} \sqrt {5} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 35937 \, \sqrt {7} \sqrt {3} {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) - 308 \, {\left (408 \, x - 281\right )} \sqrt {-2 \, x + 1}}{1369599 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1369599*(25725*sqrt(11)*sqrt(5)*(4*x^2 - 4*x + 1)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3))
+ 35937*sqrt(7)*sqrt(3)*(4*x^2 - 4*x + 1)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) - 308*(4
08*x - 281)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [C] Result contains complex when optimal does not.
time = 5.07, size = 105, normalized size = 1.24 \begin {gather*} - \frac {50 \sqrt {55} i \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{1331} + \frac {18 \sqrt {21} i \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{343} - \frac {136 \sqrt {2} i}{5929 \sqrt {x - \frac {1}{2}}} + \frac {\sqrt {2} i}{231 \left (x - \frac {1}{2}\right )^{\frac {3}{2}}} + \frac {\sqrt {2} i}{20 \left (x - \frac {1}{2}\right )^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-50*sqrt(55)*I*atan(sqrt(110)*sqrt(x - 1/2)/11)/1331 + 18*sqrt(21)*I*atan(sqrt(42)*sqrt(x - 1/2)/7)/343 - 136*
sqrt(2)*I/(5929*sqrt(x - 1/2)) + sqrt(2)*I/(231*(x - 1/2)**(3/2)) + sqrt(2)*I/(20*(x - 1/2)**(5/2))

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Giac [A]
time = 2.29, size = 100, normalized size = 1.18 \begin {gather*} \frac {25}{1331} \, \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 {9}{343} \, \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 (408 \, x - 281\right )}}{17787 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 9/343*sqrt(21)*
log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/17787*(408*x - 281)/((2*x - 1)*
sqrt(-2*x + 1))

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Mupad [B]
time = 1.24, size = 51, normalized size = 0.60 \begin {gather*} \frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {544\,x}{5929}-\frac {1124}{17787}}{{\left (1-2\,x\right )}^{3/2}}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(18*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - ((544*x)/5929 - 1124/17787)/(1 - 2*x)^(3/2) - (50*55^(
1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331

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