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

Optimal. Leaf size=92 \[ -\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 92, 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, 152, 156, 63, 206} \begin {gather*} -\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (3 x+2)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{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/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)),x]

[Out]

-58/(539*Sqrt[1 - 2*x]) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)) + (228*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49
- (50*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 152

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 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}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {1}{7} \int \frac {8-45 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}-\frac {2}{539} \int \frac {-482+\frac {435 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}-\frac {342}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx+\frac {125}{11} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {342}{49} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )-\frac {125}{11} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {58}{539 \sqrt {1-2 x}}+\frac {3}{7 \sqrt {1-2 x} (2+3 x)}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{11} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 70, normalized size = 0.76 \begin {gather*} \frac {-2508 (3 x+2) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )+2450 (3 x+2) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+231}{539 \sqrt {1-2 x} (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(231 - 2508*(2 + 3*x)*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] + 2450*(2 + 3*x)*Hypergeometric2F1[-1/2,
1, 1/2, (-5*(-1 + 2*x))/11])/(539*Sqrt[1 - 2*x]*(2 + 3*x))

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IntegrateAlgebraic [A]  time = 0.18, size = 92, normalized size = 1.00 \begin {gather*} -\frac {2 (87 (1-2 x)+28)}{539 (3 (1-2 x)-7) \sqrt {1-2 x}}+\frac {228}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )-\frac {50}{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/((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)),x]

[Out]

(-2*(28 + 87*(1 - 2*x)))/(539*(-7 + 3*(1 - 2*x))*Sqrt[1 - 2*x]) + (228*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/49 - (50*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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fricas [A]  time = 1.35, size = 116, normalized size = 1.26 \begin {gather*} \frac {8575 \, \sqrt {11} \sqrt {5} {\left (6 \, x^{2} + x - 2\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 13794 \, \sqrt {7} \sqrt {3} {\left (6 \, x^{2} + x - 2\right )} \log \left (-\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \, {\left (174 \, x - 115\right )} \sqrt {-2 \, x + 1}}{41503 \, {\left (6 \, x^{2} + x - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/41503*(8575*sqrt(11)*sqrt(5)*(6*x^2 + x - 2)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 13
794*sqrt(7)*sqrt(3)*(6*x^2 + x - 2)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(174*x - 1
15)*sqrt(-2*x + 1))/(6*x^2 + x - 2)

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giac [A]  time = 1.23, size = 107, normalized size = 1.16 \begin {gather*} \frac {25}{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 {114}{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 {2 \, {\left (174 \, x - 115\right )}}{539 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 114/343*sqrt(21)
*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/539*(174*x - 115)/(3*(-2*x + 1
)^(3/2) - 7*sqrt(-2*x + 1))

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maple [A]  time = 0.01, size = 63, normalized size = 0.68 \begin {gather*} \frac {228 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{343}-\frac {50 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{121}+\frac {8}{539 \sqrt {-2 x +1}}-\frac {6 \sqrt {-2 x +1}}{49 \left (-2 x -\frac {4}{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/539/(-2*x+1)^(1/2)-50/121*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-6/49*(-2*x+1)^(1/2)/(-2*x-4/3)+228/
343*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.19, size = 101, normalized size = 1.10 \begin {gather*} \frac {25}{121} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {114}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {2 \, {\left (174 \, x - 115\right )}}{539 \, {\left (3 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 7 \, \sqrt {-2 \, x + 1}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 114/343*sqrt(21)*log(-(sqr
t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/539*(174*x - 115)/(3*(-2*x + 1)^(3/2) - 7*sqrt(-2
*x + 1))

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mupad [B]  time = 1.28, size = 65, normalized size = 0.71 \begin {gather*} \frac {228\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}-\frac {\frac {116\,x}{539}-\frac {230}{1617}}{\frac {7\,\sqrt {1-2\,x}}{3}-{\left (1-2\,x\right )}^{3/2}}-\frac {50\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{121} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(228*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/343 - ((116*x)/539 - 230/1617)/((7*(1 - 2*x)^(1/2))/3 - (1
- 2*x)^(3/2)) - (50*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/121

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sympy [C]  time = 12.02, size = 376, normalized size = 4.09 \begin {gather*} - \frac {13398 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} + \frac {2156 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} + \frac {102900 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} - \frac {165528 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} - \frac {51450 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} + \frac {82764 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} + \frac {120050 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} - \frac {193116 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} - \frac {60025 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} + \frac {96558 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{- 290521 x - 249018 \left (x - \frac {1}{2}\right )^{2} + \frac {290521}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-13398*sqrt(2)*I*(x - 1/2)**(3/2)/(-290521*x - 249018*(x - 1/2)**2 + 290521/2) + 2156*sqrt(2)*I*sqrt(x - 1/2)/
(-290521*x - 249018*(x - 1/2)**2 + 290521/2) + 102900*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)
/(-290521*x - 249018*(x - 1/2)**2 + 290521/2) - 165528*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)*sqrt(x - 1/2)/7)/
(-290521*x - 249018*(x - 1/2)**2 + 290521/2) - 51450*sqrt(55)*I*pi*(x - 1/2)**2/(-290521*x - 249018*(x - 1/2)*
*2 + 290521/2) + 82764*sqrt(21)*I*pi*(x - 1/2)**2/(-290521*x - 249018*(x - 1/2)**2 + 290521/2) + 120050*sqrt(5
5)*I*(x - 1/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-290521*x - 249018*(x - 1/2)**2 + 290521/2) - 193116*sqrt(21)
*I*(x - 1/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-290521*x - 249018*(x - 1/2)**2 + 290521/2) - 60025*sqrt(55)*I*pi
*(x - 1/2)/(-290521*x - 249018*(x - 1/2)**2 + 290521/2) + 96558*sqrt(21)*I*pi*(x - 1/2)/(-290521*x - 249018*(x
- 1/2)**2 + 290521/2)

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