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)) + 175/(14641*Sqrt[1 - 2*x]) - 1/(22*(1 - 2*x)^(3/2)*(3 + 5*x)^2) - 7/(242*(1 - 2*x)^
(3/2)*(3 + 5*x)) - (175*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi [A]  time = 0.028567, 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*}

Mathematica [C]  time = 0.0057136, size = 30, normalized size = 0.31 \[ \frac{8 \, _2F_1\left (-\frac{3}{2},3;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )}{3993 (1-2 x)^{3/2}} \]

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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Maple [A]  time = 0.011, size = 66, normalized size = 0.7 \begin{align*}{\frac{8}{3993} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{120}{14641}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{5000}{14641\, \left ( -10\,x-6 \right ) ^{2}} \left ({\frac{11}{40} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{143}{200}\sqrt{1-2\,x}} \right ) }-{\frac{175\,\sqrt{55}}{161051}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 3.45388, size = 124, normalized size = 1.29 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.03877, size = 309, normalized size = 3.22 \begin{align*} \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 )}} \end{align*}

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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Sympy [C]  time = 6.36311, size = 983, normalized size = 10.24 \begin{align*} \text{result too large to display} \end{align*}

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)*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5))
)/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)
**(153/2)) + 52500*sqrt(55)*I*pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)/(96630600*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 115500*sqrt(55)*sq
rt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*acosh(sqrt(110)/(10*sqrt(x + 3/5)))/(96630600*sqrt(-1 + 11/(10*(
x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 57750*sqrt(55)*I*
pi*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) -
106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 577500*sqrt(2)*(x + 3/5)**77/(96630600*sqrt(-1 +
11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 847000*s
qrt(2)*(x + 3/5)**76/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x
+ 3/5)))*(x + 3/5)**(153/2)) + 139755*sqrt(2)*(x + 3/5)**75/(96630600*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)*
*(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 43923*sqrt(2)*(x + 3/5)**74/(96630600*
sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)),
11/(10*Abs(x + 3/5)) > 1), (105000*sqrt(55)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2)*asin(sqrt(110)/(1
0*sqrt(x + 3/5)))/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/
5)))*(x + 3/5)**(153/2)) - 115500*sqrt(55)*I*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)*asin(sqrt(110)/(10
*sqrt(x + 3/5)))/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5
)))*(x + 3/5)**(153/2)) - 577500*sqrt(2)*I*(x + 3/5)**77/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155
/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) + 847000*sqrt(2)*I*(x + 3/5)**76/(96630600*sqr
t(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)) - 1397
55*sqrt(2)*I*(x + 3/5)**75/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(155/2) - 106293660*sqrt(1 - 11/(1
0*(x + 3/5)))*(x + 3/5)**(153/2)) - 43923*sqrt(2)*I*(x + 3/5)**74/(96630600*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3
/5)**(155/2) - 106293660*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5)**(153/2)), True))

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Giac [A]  time = 1.72678, size = 120, normalized size = 1.25 \begin{align*} \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}} \end{align*}

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