3.19.49 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=154 \[ \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac {6649 \sqrt {1-2 x}}{27 (3 x+2) (5 x+3)}+\frac {917 \sqrt {1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac {44545 \sqrt {1-2 x}}{18 (5 x+3)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {98, 149, 151, 156, 63, 206} \begin {gather*} \frac {7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)}+\frac {6649 \sqrt {1-2 x}}{27 (3 x+2) (5 x+3)}+\frac {917 \sqrt {1-2 x}}{54 (3 x+2)^2 (5 x+3)}-\frac {44545 \sqrt {1-2 x}}{18 (5 x+3)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

(-44545*Sqrt[1 - 2*x])/(18*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)) + (917*Sqrt[1 - 2*x])/(5
4*(2 + 3*x)^2*(3 + 5*x)) + (6649*Sqrt[1 - 2*x])/(27*(2 + 3*x)*(3 + 5*x)) - (307295*ArcTanh[Sqrt[3/7]*Sqrt[1 -
2*x]])/(3*Sqrt[21]) + 3014*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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 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 {(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^2} \, dx &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {1}{9} \int \frac {(197-163 x) \sqrt {1-2 x}}{(2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}-\frac {1}{54} \int \frac {-16180+22273 x}{\sqrt {1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {1}{378} \int \frac {-1220205+1396290 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}+\frac {\int \frac {-50405355+30869685 x}{\sqrt {1-2 x} (2+3 x) (3+5 x)} \, dx}{4158}\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}+\frac {307295}{6} \int \frac {1}{\sqrt {1-2 x} (2+3 x)} \, dx-82885 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {307295}{6} \operatorname {Subst}\left (\int \frac {1}{\frac {7}{2}-\frac {3 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )+82885 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )\\ &=-\frac {44545 \sqrt {1-2 x}}{18 (3+5 x)}+\frac {7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)}+\frac {917 \sqrt {1-2 x}}{54 (2+3 x)^2 (3+5 x)}+\frac {6649 \sqrt {1-2 x}}{27 (2+3 x) (3+5 x)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.17, size = 95, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {1-2 x} \left (400905 x^3+788512 x^2+516513 x+112668\right )}{6 (3 x+2)^3 (5 x+3)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

-1/6*(Sqrt[1 - 2*x]*(112668 + 516513*x + 788512*x^2 + 400905*x^3))/((2 + 3*x)^3*(3 + 5*x)) - (307295*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]) + 3014*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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IntegrateAlgebraic [A]  time = 0.40, size = 115, normalized size = 0.75 \begin {gather*} \frac {\sqrt {1-2 x} \left (400905 (1-2 x)^3-2779739 (1-2 x)^2+6422815 (1-2 x)-4945325\right )}{3 (3 (1-2 x)-7)^3 (5 (1-2 x)-11)}-\frac {307295 \tanh ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3 \sqrt {21}}+3014 \sqrt {55} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^2),x]

[Out]

((-4945325 + 6422815*(1 - 2*x) - 2779739*(1 - 2*x)^2 + 400905*(1 - 2*x)^3)*Sqrt[1 - 2*x])/(3*(-7 + 3*(1 - 2*x)
)^3*(-11 + 5*(1 - 2*x))) - (307295*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(3*Sqrt[21]) + 3014*Sqrt[55]*ArcTanh[Sqrt
[5/11]*Sqrt[1 - 2*x]]

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fricas [A]  time = 1.07, size = 150, normalized size = 0.97 \begin {gather*} \frac {189882 \, \sqrt {55} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {5 \, x - \sqrt {55} \sqrt {-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 307295 \, \sqrt {21} {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \, {\left (400905 \, x^{3} + 788512 \, x^{2} + 516513 \, x + 112668\right )} \sqrt {-2 \, x + 1}}{126 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^2,x, algorithm="fricas")

[Out]

1/126*(189882*sqrt(55)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) + 307295*sqrt(21)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3
*x + 2)) - 21*(400905*x^3 + 788512*x^2 + 516513*x + 112668)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148
*x + 24)

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giac [A]  time = 1.02, size = 139, normalized size = 0.90 \begin {gather*} -1507 \, \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 {307295}{126} \, \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 {605 \, \sqrt {-2 \, x + 1}}{5 \, x + 3} - \frac {60579 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 285460 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 336385 \, \sqrt {-2 \, x + 1}}{24 \, {\left (3 \, x + 2\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^2,x, algorithm="giac")

[Out]

-1507*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 307295/126*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 605*sqrt(-2*x + 1)/(5*x + 3) -
1/24*(60579*(2*x - 1)^2*sqrt(-2*x + 1) - 285460*(-2*x + 1)^(3/2) + 336385*sqrt(-2*x + 1))/(3*x + 2)^3

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maple [A]  time = 0.01, size = 91, normalized size = 0.59 \begin {gather*} -\frac {307295 \sqrt {21}\, \arctanh \left (\frac {\sqrt {21}\, \sqrt {-2 x +1}}{7}\right )}{63}+3014 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )+\frac {242 \sqrt {-2 x +1}}{-2 x -\frac {6}{5}}+\frac {20193 \left (-2 x +1\right )^{\frac {5}{2}}-\frac {285460 \left (-2 x +1\right )^{\frac {3}{2}}}{3}+\frac {336385 \sqrt {-2 x +1}}{3}}{\left (-6 x -4\right )^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x+1)^(5/2)/(3*x+2)^4/(5*x+3)^2,x)

[Out]

242*(-2*x+1)^(1/2)/(-2*x-6/5)+3014*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)+108*(6731/36*(-2*x+1)^(5/2)-
71365/81*(-2*x+1)^(3/2)+336385/324*(-2*x+1)^(1/2))/(-6*x-4)^3-307295/63*arctanh(1/7*21^(1/2)*(-2*x+1)^(1/2))*2
1^(1/2)

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maxima [A]  time = 1.33, size = 146, normalized size = 0.95 \begin {gather*} -1507 \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) + \frac {307295}{126} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {400905 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 2779739 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 6422815 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 4945325 \, \sqrt {-2 \, x + 1}}{3 \, {\left (135 \, {\left (2 \, x - 1\right )}^{4} + 1242 \, {\left (2 \, x - 1\right )}^{3} + 4284 \, {\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1507*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 307295/126*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/3*(400905*(-2*x + 1)^(7/2) - 2779739*(-2*x + 1)
^(5/2) + 6422815*(-2*x + 1)^(3/2) - 4945325*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 4284*(2*x -
1)^2 + 13132*x - 2793)

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mupad [B]  time = 0.10, size = 108, normalized size = 0.70 \begin {gather*} 3014\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )-\frac {307295\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{63}-\frac {\frac {989065\,\sqrt {1-2\,x}}{81}-\frac {1284563\,{\left (1-2\,x\right )}^{3/2}}{81}+\frac {2779739\,{\left (1-2\,x\right )}^{5/2}}{405}-\frac {8909\,{\left (1-2\,x\right )}^{7/2}}{9}}{\frac {13132\,x}{135}+\frac {476\,{\left (2\,x-1\right )}^2}{15}+\frac {46\,{\left (2\,x-1\right )}^3}{5}+{\left (2\,x-1\right )}^4-\frac {931}{45}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(5/2)/((3*x + 2)^4*(5*x + 3)^2),x)

[Out]

3014*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11) - (307295*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2))/7))/63
- ((989065*(1 - 2*x)^(1/2))/81 - (1284563*(1 - 2*x)^(3/2))/81 + (2779739*(1 - 2*x)^(5/2))/405 - (8909*(1 - 2*x
)^(7/2))/9)/((13132*x)/135 + (476*(2*x - 1)^2)/15 + (46*(2*x - 1)^3)/5 + (2*x - 1)^4 - 931/45)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)/(2+3*x)**4/(3+5*x)**2,x)

[Out]

Timed out

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