3.2196 \(\int \frac {1}{(1-2 x)^{5/2} (3+5 x)^3} \, dx\)

Optimal. Leaf size=96 \[ \frac {175}{14641 \sqrt {1-2 x}}-\frac {7}{242 (1-2 x)^{3/2} (5 x+3)}+\frac {35}{3993 (1-2 x)^{3/2}}-\frac {1}{22 (1-2 x)^{3/2} (5 x+3)^2}-\frac {175 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

[Out]

35/3993/(1-2*x)^(3/2)-1/22/(1-2*x)^(3/2)/(3+5*x)^2-7/242/(1-2*x)^(3/2)/(3+5*x)-175/161051*arctanh(1/11*55^(1/2
)*(1-2*x)^(1/2))*55^(1/2)+175/14641/(1-2*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 110, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {51, 63, 206} \[ -\frac {875 \sqrt {1-2 x}}{29282 (5 x+3)}-\frac {875 \sqrt {1-2 x}}{7986 (5 x+3)^2}+\frac {70}{363 \sqrt {1-2 x} (5 x+3)^2}+\frac {2}{33 (1-2 x)^{3/2} (5 x+3)^2}-\frac {175 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 70/(363*Sqrt[1 - 2*x]*(3 + 5*x)^2) - (875*Sqrt[1 - 2*x])/(7986*(3 + 5*x)^
2) - (875*Sqrt[1 - 2*x])/(29282*(3 + 5*x)) - (175*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, 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 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)^{5/2} (3+5 x)^3} \, dx &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {35}{33} \int \frac {1}{(1-2 x)^{3/2} (3+5 x)^3} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}+\frac {875}{363} \int \frac {1}{\sqrt {1-2 x} (3+5 x)^3} \, dx\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}+\frac {875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)^2} \, dx}{2662}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{29282 (3+5 x)}+\frac {875 \int \frac {1}{\sqrt {1-2 x} (3+5 x)} \, dx}{29282}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{29282 (3+5 x)}-\frac {875 \operatorname {Subst}\left (\int \frac {1}{\frac {11}{2}-\frac {5 x^2}{2}} \, dx,x,\sqrt {1-2 x}\right )}{29282}\\ &=\frac {2}{33 (1-2 x)^{3/2} (3+5 x)^2}+\frac {70}{363 \sqrt {1-2 x} (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{7986 (3+5 x)^2}-\frac {875 \sqrt {1-2 x}}{29282 (3+5 x)}-\frac {175 \sqrt {\frac {5}{11}} \tanh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )}{14641}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 30, normalized size = 0.31 \[ \frac {8 \, _2F_1\left (-\frac {3}{2},3;-\frac {1}{2};-\frac {5}{11} (2 x-1)\right )}{3993 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*Hypergeometric2F1[-3/2, 3, -1/2, (-5*(-1 + 2*x))/11])/(3993*(1 - 2*x)^(3/2))

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fricas [A]  time = 1.00, size = 105, normalized size = 1.09 \[ \frac {525 \, \sqrt {11} \sqrt {5} {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac {\sqrt {11} \sqrt {5} \sqrt {-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) - 11 \, {\left (52500 \, x^{3} + 17500 \, x^{2} - 22995 \, x - 4764\right )} \sqrt {-2 \, x + 1}}{966306 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/966306*(525*sqrt(11)*sqrt(5)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*
x - 8)/(5*x + 3)) - 11*(52500*x^3 + 17500*x^2 - 22995*x - 4764)*sqrt(-2*x + 1))/(100*x^4 + 20*x^3 - 59*x^2 - 6
*x + 9)

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giac [A]  time = 1.26, size = 89, normalized size = 0.93 \[ \frac {175}{322102} \, \sqrt {55} \log \left (\frac {{\left | -2 \, \sqrt {55} + 10 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {16 \, {\left (45 \, x - 28\right )}}{43923 \, {\left (2 \, x - 1\right )} \sqrt {-2 \, x + 1}} + \frac {25 \, {\left (5 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 13 \, \sqrt {-2 \, x + 1}\right )}}{5324 \, {\left (5 \, x + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

175/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 16/43923*(45
*x - 28)/((2*x - 1)*sqrt(-2*x + 1)) + 25/5324*(5*(-2*x + 1)^(3/2) - 13*sqrt(-2*x + 1))/(5*x + 3)^2

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maple [A]  time = 0.01, size = 66, normalized size = 0.69 \[ -\frac {175 \sqrt {55}\, \arctanh \left (\frac {\sqrt {55}\, \sqrt {-2 x +1}}{11}\right )}{161051}+\frac {8}{3993 \left (-2 x +1\right )^{\frac {3}{2}}}+\frac {120}{14641 \sqrt {-2 x +1}}+\frac {\frac {125 \left (-2 x +1\right )^{\frac {3}{2}}}{1331}-\frac {325 \sqrt {-2 x +1}}{1331}}{\left (-10 x -6\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3993/(-2*x+1)^(3/2)+120/14641/(-2*x+1)^(1/2)+5000/14641*(11/40*(-2*x+1)^(3/2)-143/200*(-2*x+1)^(1/2))/(-10*x
-6)^2-175/161051*arctanh(1/11*55^(1/2)*(-2*x+1)^(1/2))*55^(1/2)

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maxima [A]  time = 1.11, size = 92, normalized size = 0.96 \[ \frac {175}{322102} \, \sqrt {55} \log \left (-\frac {\sqrt {55} - 5 \, \sqrt {-2 \, x + 1}}{\sqrt {55} + 5 \, \sqrt {-2 \, x + 1}}\right ) - \frac {13125 \, {\left (2 \, x - 1\right )}^{3} + 48125 \, {\left (2 \, x - 1\right )}^{2} + 67760 \, x - 44528}{43923 \, {\left (25 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 110 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 121 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

175/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 1/43923*(13125*(2*x -
1)^3 + 48125*(2*x - 1)^2 + 67760*x - 44528)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2)
)

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mupad [B]  time = 1.21, size = 72, normalized size = 0.75 \[ -\frac {175\,\sqrt {55}\,\mathrm {atanh}\left (\frac {\sqrt {55}\,\sqrt {1-2\,x}}{11}\right )}{161051}-\frac {\frac {112\,x}{1815}+\frac {175\,{\left (2\,x-1\right )}^2}{3993}+\frac {175\,{\left (2\,x-1\right )}^3}{14641}-\frac {368}{9075}}{\frac {121\,{\left (1-2\,x\right )}^{3/2}}{25}-\frac {22\,{\left (1-2\,x\right )}^{5/2}}{5}+{\left (1-2\,x\right )}^{7/2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (175*55^(1/2)*atanh((55^(1/2)*(1 - 2*x)^(1/2))/11))/161051 - ((112*x)/1815 + (175*(2*x - 1)^2)/3993 + (175*(
2*x - 1)^3)/14641 - 368/9075)/((121*(1 - 2*x)^(3/2))/25 - (22*(1 - 2*x)^(5/2))/5 + (1 - 2*x)^(7/2))

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sympy [C]  time = 5.94, size = 1027, normalized size = 10.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((105000*sqrt(55)*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5)
))/(-96630600*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) + 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x
+ 3/5)**(153/2)) + 52500*sqrt(55)*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)/(-96630600*I*sqrt(-1 + 11
/(10*(x + 3/5)))*(x + 3/5)**(155/2) + 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 115500*sq
rt(55)*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/(-96630600*I*sqrt
(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) + 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 5
7750*sqrt(55)*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)/(-96630600*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x
+ 3/5)**(155/2) + 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 577500*sqrt(2)*I*(x + 3/5)**7
7/(-96630600*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) + 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x +
3/5)**(153/2)) + 847000*sqrt(2)*I*(x + 3/5)**76/(-96630600*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)
+ 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 139755*sqrt(2)*I*(x + 3/5)**75/(-96630600*I*s
qrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) + 106293660*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2))
- 43923*sqrt(2)*I*(x + 3/5)**74/(-96630600*I*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) + 106293660*I*sqr
t(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)), 11/(10*Abs(x + 3/5)) > 1), (-105000*sqrt(55)*sqrt(1 - 11/(10*(x
+ 3/5)))*(x + 3/5)**(155/2)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3
/5)**(155/2) - 106293660*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 115500*sqrt(55)*sqrt(1 - 11/(10*(
x + 3/5)))*(x + 3/5)**(153/2)*asin(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*I*sqrt(1 - 11/(10*(x + 3/5)))*(x +
3/5)**(155/2) - 106293660*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 577500*sqrt(2)*(x + 3/5)**77/(96
630600*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(
153/2)) - 847000*sqrt(2)*(x + 3/5)**76/(96630600*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*
I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 139755*sqrt(2)*(x + 3/5)**75/(96630600*I*sqrt(1 - 11/(10*(
x + 3/5)))*(x + 3/5)**(155/2) - 106293660*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 43923*sqrt(2)*(x
+ 3/5)**74/(96630600*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*I*sqrt(1 - 11/(10*(x + 3/5)
))*(x + 3/5)**(153/2)), True))

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