3.2172 \(\int \frac{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{27}{20} \sqrt{1-2 x}-\frac{784}{121 \sqrt{1-2 x}}+\frac{343}{132 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

[Out]

343/(132*(1 - 2*x)^(3/2)) - 784/(121*Sqrt[1 - 2*x]) - (27*Sqrt[1 - 2*x])/20 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(605*Sqrt[55])

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Rubi [A]  time = 0.0266733, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {87, 63, 206} \[ -\frac{27}{20} \sqrt{1-2 x}-\frac{784}{121 \sqrt{1-2 x}}+\frac{343}{132 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(132*(1 - 2*x)^(3/2)) - 784/(121*Sqrt[1 - 2*x]) - (27*Sqrt[1 - 2*x])/20 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2
*x]])/(605*Sqrt[55])

Rule 87

Int[(((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_))/((a_.) + (b_.)*(x_)), x_Symbol] :> Int[ExpandIntegr
and[(e + f*x)^FractionalPart[p], ((c + d*x)^n*(e + f*x)^IntegerPart[p])/(a + b*x), x], x] /; FreeQ[{a, b, c, d
, e, f}, x] && IGtQ[n, 0] && LtQ[p, -1] && FractionQ[p]

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{(2+3 x)^3}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac{343}{44 (1-2 x)^{5/2}}-\frac{784}{121 (1-2 x)^{3/2}}+\frac{27}{20 \sqrt{1-2 x}}+\frac{1}{605 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{343}{132 (1-2 x)^{3/2}}-\frac{784}{121 \sqrt{1-2 x}}-\frac{27}{20} \sqrt{1-2 x}+\frac{1}{605} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{343}{132 (1-2 x)^{3/2}}-\frac{784}{121 \sqrt{1-2 x}}-\frac{27}{20} \sqrt{1-2 x}-\frac{1}{605} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{343}{132 (1-2 x)^{3/2}}-\frac{784}{121 \sqrt{1-2 x}}-\frac{27}{20} \sqrt{1-2 x}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{605 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0218716, size = 45, normalized size = 0.67 \[ \frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )-99 \left (225 x^2-765 x+218\right )}{4125 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-99*(218 - 765*x + 225*x^2) + 2*Hypergeometric2F1[-3/2, 1, -1/2, (5*(1 - 2*x))/11])/(4125*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.008, size = 47, normalized size = 0.7 \begin{align*}{\frac{343}{132} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,\sqrt{55}}{33275}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{784}{121}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{27}{20}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/132/(1-2*x)^(3/2)-2/33275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-784/121/(1-2*x)^(1/2)-27/20*(1-2*x
)^(1/2)

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Maxima [A]  time = 3.56866, size = 81, normalized size = 1.21 \begin{align*} \frac{1}{33275} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27}{20} \, \sqrt{-2 \, x + 1} + \frac{49 \,{\left (384 \, x - 115\right )}}{1452 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33275*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/20*sqrt(-2*x + 1) + 49
/1452*(384*x - 115)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.37173, size = 212, normalized size = 3.16 \begin{align*} \frac{3 \, \sqrt{55}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) - 55 \,{\left (9801 \, x^{2} - 33321 \, x + 9494\right )} \sqrt{-2 \, x + 1}}{99825 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99825*(3*sqrt(55)*(4*x^2 - 4*x + 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(9801*x^2 - 3332
1*x + 9494)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 33.572, size = 102, normalized size = 1.52 \begin{align*} - \frac{27 \sqrt{1 - 2 x}}{20} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 < - \frac{11}{5} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55} \sqrt{1 - 2 x}}{11} \right )}}{55} & \text{for}\: 2 x - 1 > - \frac{11}{5} \end{cases}\right )}{605} - \frac{784}{121 \sqrt{1 - 2 x}} + \frac{343}{132 \left (1 - 2 x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*sqrt(1 - 2*x)/20 + 2*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55
)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 > -11/5))/605 - 784/(121*sqrt(1 - 2*x)) + 343/(132*(1 - 2*x)**(
3/2))

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Giac [A]  time = 2.02544, size = 95, normalized size = 1.42 \begin{align*} \frac{1}{33275} \, \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{27}{20} \, \sqrt{-2 \, x + 1} - \frac{49 \,{\left (384 \, x - 115\right )}}{1452 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/33275*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/20*sqrt(-2*x
+ 1) - 49/1452*(384*x - 115)/((2*x - 1)*sqrt(-2*x + 1))