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

Optimal. Leaf size=92 \[ \frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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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 {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (5 x+3)}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{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)^(3/2)*(2 + 3*x)*(3 + 5*x)^2),x]

[Out]

78/(847*Sqrt[1 - 2*x]) - 5/(11*Sqrt[1 - 2*x]*(3 + 5*x)) - (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 +
(300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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) (3+5 x)^2} \, dx &=-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {1}{11} \int \frac {3-45 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {2}{847} \int \frac {-\frac {699}{2}+\frac {585 x}{2}}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}+\frac {27}{7} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {750}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {27}{7} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {750}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {78}{847 \sqrt {1-2 x}}-\frac {5}{11 \sqrt {1-2 x} (3+5 x)}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \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 = 0.03, size = 73, normalized size = 0.79 \begin {gather*} \frac {2178 (5 x+3) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-35 \left (60 (5 x+3) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+11\right )}{847 \sqrt {1-2 x} (5 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2178*(3 + 5*x)*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] - 35*(11 + 60*(3 + 5*x)*Hypergeometric2F1[-1/2,
1, 1/2, (-5*(-1 + 2*x))/11]))/(847*Sqrt[1 - 2*x]*(3 + 5*x))

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IntegrateAlgebraic [A]  time = 0.19, size = 92, normalized size = 1.00 \begin {gather*} \frac {2 (195 (1-2 x)-44)}{847 (5 (1-2 x)-11) \sqrt {1-2 x}}-\frac {18}{7} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {300}{121} \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)*(3 + 5*x)^2),x]

[Out]

(2*(-44 + 195*(1 - 2*x)))/(847*(-11 + 5*(1 - 2*x))*Sqrt[1 - 2*x]) - (18*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/7 + (300*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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fricas [A]  time = 1.58, size = 116, normalized size = 1.26 \begin {gather*} \frac {7350 \, \sqrt {11} \sqrt {5} {\left (10 \, x^{2} + x - 3\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 11979 \, \sqrt {7} \sqrt {3} {\left (10 \, x^{2} + x - 3\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (390 \, x - 151\right )} \sqrt {-2 \, x + 1}}{65219 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/65219*(7350*sqrt(11)*sqrt(5)*(10*x^2 + x - 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) +
11979*sqrt(7)*sqrt(3)*(10*x^2 + x - 3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(390*x -
151)*sqrt(-2*x + 1))/(10*x^2 + x - 3)

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giac [A]  time = 1.33, size = 107, normalized size = 1.16 \begin {gather*} -\frac {150}{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}{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 {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \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)/(3+5*x)^2,x, algorithm="giac")

[Out]

-150/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 9/49*sqrt(21)
*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1
)^(3/2) - 11*sqrt(-2*x + 1))

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maple [A]  time = 0.01, size = 63, normalized size = 0.68 \begin {gather*} -\frac {18 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{49}+\frac {300 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {8}{847 \sqrt {-2 x +1}}+\frac {10 \sqrt {-2 x +1}}{121 \left (-2 x -\frac {6}{5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/847/(-2*x+1)^(1/2)+10/121*(-2*x+1)^(1/2)/(-2*x-6/5)+300/1331*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)-
18/49*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.08, size = 101, normalized size = 1.10 \begin {gather*} -\frac {150}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {9}{49} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {2 \, {\left (390 \, x - 151\right )}}{847 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 11 \, \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)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-150/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 9/49*sqrt(21)*log(-(sqr
t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/847*(390*x - 151)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-
2*x + 1))

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mupad [B]  time = 0.10, size = 64, normalized size = 0.70 \begin {gather*} \frac {\frac {156\,x}{847}-\frac {302}{4235}}{\frac {11\,\sqrt {1-2\,x}}{5}-{\left (1-2\,x\right )}^{3/2}}-\frac {18\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{49}+\frac {300\,\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)^(3/2)*(3*x + 2)*(5*x + 3)^2),x)

[Out]

((156*x)/847 - 302/4235)/((11*(1 - 2*x)^(1/2))/5 - (1 - 2*x)^(3/2)) - (18*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(
1/2))/7))/49 + (300*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331

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sympy [C]  time = 11.11, size = 376, normalized size = 4.09 \begin {gather*} \frac {30030 \sqrt {2} i \left (x - \frac {1}{2}\right )^{\frac {3}{2}}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {3388 \sqrt {2} i \sqrt {x - \frac {1}{2}}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {147000 \sqrt {55} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {239580 \sqrt {21} i \left (x - \frac {1}{2}\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {119790 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )^{2}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {73500 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )^{2}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {161700 \sqrt {55} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {110} \sqrt {x - \frac {1}{2}}}{11} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {263538 \sqrt {21} i \left (x - \frac {1}{2}\right ) \operatorname {atan}{\left (\frac {\sqrt {42} \sqrt {x - \frac {1}{2}}}{7} \right )}}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} - \frac {131769 \sqrt {21} i \pi \left (x - \frac {1}{2}\right )}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} + \frac {80850 \sqrt {55} i \pi \left (x - \frac {1}{2}\right )}{- 717409 x - 652190 \left (x - \frac {1}{2}\right )^{2} + \frac {717409}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

30030*sqrt(2)*I*(x - 1/2)**(3/2)/(-717409*x - 652190*(x - 1/2)**2 + 717409/2) + 3388*sqrt(2)*I*sqrt(x - 1/2)/(
-717409*x - 652190*(x - 1/2)**2 + 717409/2) - 147000*sqrt(55)*I*(x - 1/2)**2*atan(sqrt(110)*sqrt(x - 1/2)/11)/
(-717409*x - 652190*(x - 1/2)**2 + 717409/2) + 239580*sqrt(21)*I*(x - 1/2)**2*atan(sqrt(42)*sqrt(x - 1/2)/7)/(
-717409*x - 652190*(x - 1/2)**2 + 717409/2) - 119790*sqrt(21)*I*pi*(x - 1/2)**2/(-717409*x - 652190*(x - 1/2)*
*2 + 717409/2) + 73500*sqrt(55)*I*pi*(x - 1/2)**2/(-717409*x - 652190*(x - 1/2)**2 + 717409/2) - 161700*sqrt(5
5)*I*(x - 1/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-717409*x - 652190*(x - 1/2)**2 + 717409/2) + 263538*sqrt(21)
*I*(x - 1/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-717409*x - 652190*(x - 1/2)**2 + 717409/2) - 131769*sqrt(21)*I*p
i*(x - 1/2)/(-717409*x - 652190*(x - 1/2)**2 + 717409/2) + 80850*sqrt(55)*I*pi*(x - 1/2)/(-717409*x - 652190*(
x - 1/2)**2 + 717409/2)

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