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

Optimal. Leaf size=119 \[ \frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (5 x+3)}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{121} \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {103, 151, 152, 156, 63, 206} \begin {gather*} \frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (5 x+3)}+\frac {3}{7 \sqrt {1-2 x} (3 x+2) (5 x+3)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{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)^2*(3 + 5*x)^2),x]

[Out]

4644/(5929*Sqrt[1 - 2*x]) - 340/(77*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(7*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) - (1314
*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (3150*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 151

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] && IntegerQ[m]

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)^2} \, dx &=\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {1}{7} \int \frac {23-75 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1}{77} \int \frac {369-3060 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {2 \int \frac {-\frac {56277}{2}+17415 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{5929}\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}+\frac {1971}{49} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-\frac {7875}{121} \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1971}{49} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+\frac {7875}{121} \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=\frac {4644}{5929 \sqrt {1-2 x}}-\frac {340}{77 \sqrt {1-2 x} (3+5 x)}+\frac {3}{7 \sqrt {1-2 x} (2+3 x) (3+5 x)}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{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 = 0.05, size = 93, normalized size = 0.78 \begin {gather*} \frac {158994 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {3}{7}-\frac {6 x}{7}\right )-7 \left (22050 \left (15 x^2+19 x+6\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\frac {5}{11} (2 x-1)\right )+11220 x+7117\right )}{5929 \sqrt {1-2 x} (3 x+2) (5 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(158994*(6 + 19*x + 15*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] - 7*(7117 + 11220*x + 22050*(6 + 19
*x + 15*x^2)*Hypergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11]))/(5929*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x))

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IntegrateAlgebraic [A]  time = 0.24, size = 110, normalized size = 0.92 \begin {gather*} \frac {4 \left (17415 (1-2 x)^2-39678 (1-2 x)+308\right )}{5929 \left (15 (1-2 x)^2-68 (1-2 x)+77\right ) \sqrt {1-2 x}}-\frac {1314}{49} \sqrt {\frac {3}{7}} \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )+\frac {3150}{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)^2*(3 + 5*x)^2),x]

[Out]

(4*(308 - 39678*(1 - 2*x) + 17415*(1 - 2*x)^2))/(5929*(77 - 68*(1 - 2*x) + 15*(1 - 2*x)^2)*Sqrt[1 - 2*x]) - (1
314*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (3150*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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fricas [A]  time = 1.51, size = 142, normalized size = 1.19 \begin {gather*} \frac {540225 \, \sqrt {11} \sqrt {5} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (-\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 874467 \, \sqrt {7} \sqrt {3} {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\right )} \log \left (\frac {\sqrt {7} \sqrt {3} \sqrt {-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \, {\left (69660 \, x^{2} + 9696 \, x - 21955\right )} \sqrt {-2 \, x + 1}}{456533 \, {\left (30 \, x^{3} + 23 \, x^{2} - 7 \, x - 6\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)^2,x, algorithm="fricas")

[Out]

1/456533*(540225*sqrt(11)*sqrt(5)*(30*x^3 + 23*x^2 - 7*x - 6)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)
/(5*x + 3)) + 874467*sqrt(7)*sqrt(3)*(30*x^3 + 23*x^2 - 7*x - 6)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5
)/(3*x + 2)) - 77*(69660*x^2 + 9696*x - 21955)*sqrt(-2*x + 1))/(30*x^3 + 23*x^2 - 7*x - 6)

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giac [A]  time = 1.23, size = 132, normalized size = 1.11 \begin {gather*} -\frac {1575}{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 {657}{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 (17415 \, {\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \, {\left (15 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 77 \, \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)^2,x, algorithm="giac")

[Out]

-1575/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 657/343*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/5929*(17415*(2*x - 1)^2 +
79356*x - 39370)/(15*(2*x - 1)^2*sqrt(-2*x + 1) - 68*(-2*x + 1)^(3/2) + 77*sqrt(-2*x + 1))

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maple [A]  time = 0.02, size = 79, normalized size = 0.66 \begin {gather*} -\frac {1314 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{343}+\frac {3150 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{1331}+\frac {16}{5929 \sqrt {-2 x +1}}+\frac {50 \sqrt {-2 x +1}}{121 \left (-2 x -\frac {6}{5}\right )}+\frac {18 \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)^2,x)

[Out]

16/5929/(-2*x+1)^(1/2)+50/121*(-2*x+1)^(1/2)/(-2*x-6/5)+3150/1331*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/
2)+18/49*(-2*x+1)^(1/2)/(-2*x-4/3)-1314/343*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*21^(1/2)

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maxima [A]  time = 1.23, size = 119, normalized size = 1.00 \begin {gather*} -\frac {1575}{1331} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {657}{343} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {4 \, {\left (17415 \, {\left (2 \, x - 1\right )}^{2} + 79356 \, x - 39370\right )}}{5929 \, {\left (15 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 68 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 77 \, \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)^2,x, algorithm="maxima")

[Out]

-1575/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 657/343*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/5929*(17415*(2*x - 1)^2 + 79356*x - 39370)/(1
5*(-2*x + 1)^(5/2) - 68*(-2*x + 1)^(3/2) + 77*sqrt(-2*x + 1))

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mupad [B]  time = 1.27, size = 80, normalized size = 0.67 \begin {gather*} \frac {3150\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{1331}-\frac {1314\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{343}+\frac {\frac {105808\,x}{29645}+\frac {4644\,{\left (2\,x-1\right )}^2}{5929}-\frac {31496}{17787}}{\frac {77\,\sqrt {1-2\,x}}{15}-\frac {68\,{\left (1-2\,x\right )}^{3/2}}{15}+{\left (1-2\,x\right )}^{5/2}} \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)^2),x)

[Out]

(3150*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/1331 - (1314*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7)
)/343 + ((105808*x)/29645 + (4644*(2*x - 1)^2)/5929 - 31496/17787)/((77*(1 - 2*x)^(1/2))/15 - (68*(1 - 2*x)^(3
/2))/15 + (1 - 2*x)^(5/2))

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sympy [C]  time = 16.08, size = 894, normalized size = 7.51

result too large to display

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)**2,x)

[Out]

64827000*sqrt(55)*(x - 1/2)**(7/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I
*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) - 104936040*sqrt(21)*(x - 1/2)**(7/2)*atan(sqrt(42)*sqrt(x -
1/2)/7)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) - 32413500*
sqrt(55)*pi*(x - 1/2)**(7/2)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2
)**(3/2)) + 52468020*sqrt(21)*pi*(x - 1/2)**(7/2)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2)
- 35153041*I*(x - 1/2)**(3/2)) + 146941200*sqrt(55)*(x - 1/2)**(5/2)*atan(sqrt(110)*sqrt(x - 1/2)/11)/(-273919
80*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) - 237855024*sqrt(21)*(x - 1
/2)**(5/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 351530
41*I*(x - 1/2)**(3/2)) - 73470600*sqrt(55)*pi*(x - 1/2)**(5/2)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x -
1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) + 118927512*sqrt(21)*pi*(x - 1/2)**(5/2)/(-27391980*I*(x - 1/2)**(
7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) + 83194650*sqrt(55)*(x - 1/2)**(3/2)*atan(sq
rt(110)*sqrt(x - 1/2)/11)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**
(3/2)) - 134667918*sqrt(21)*(x - 1/2)**(3/2)*atan(sqrt(42)*sqrt(x - 1/2)/7)/(-27391980*I*(x - 1/2)**(7/2) - 62
088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) - 41597325*sqrt(55)*pi*(x - 1/2)**(3/2)/(-27391980*I*
(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) + 67333959*sqrt(21)*pi*(x - 1/2)
**(3/2)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)) - 10727640*
sqrt(2)*(x - 1/2)**3/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x - 1/2)**(3/2)
) - 12220824*sqrt(2)*(x - 1/2)**2/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 35153041*I*(x
- 1/2)**(3/2)) - 47432*sqrt(2)*(x - 1/2)/(-27391980*I*(x - 1/2)**(7/2) - 62088488*I*(x - 1/2)**(5/2) - 3515304
1*I*(x - 1/2)**(3/2))

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