3.2163 \(\int \frac{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)} \, dx\)

Optimal. Leaf size=67 \[ -\frac{125}{12} \sqrt{1-2 x}-\frac{2178}{49 \sqrt{1-2 x}}+\frac{1331}{84 (1-2 x)^{3/2}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

[Out]

1331/(84*(1 - 2*x)^(3/2)) - 2178/(49*Sqrt[1 - 2*x]) - (125*Sqrt[1 - 2*x])/12 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(147*Sqrt[21])

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Rubi [A]  time = 0.0255756, 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} \[ -\frac{125}{12} \sqrt{1-2 x}-\frac{2178}{49 \sqrt{1-2 x}}+\frac{1331}{84 (1-2 x)^{3/2}}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(84*(1 - 2*x)^(3/2)) - 2178/(49*Sqrt[1 - 2*x]) - (125*Sqrt[1 - 2*x])/12 + (2*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2
*x]])/(147*Sqrt[21])

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{(3+5 x)^3}{(1-2 x)^{5/2} (2+3 x)} \, dx &=\int \left (\frac{1331}{28 (1-2 x)^{5/2}}-\frac{2178}{49 (1-2 x)^{3/2}}+\frac{125}{12 \sqrt{1-2 x}}-\frac{1}{147 \sqrt{1-2 x} (2+3 x)}\right ) \, dx\\ &=\frac{1331}{84 (1-2 x)^{3/2}}-\frac{2178}{49 \sqrt{1-2 x}}-\frac{125}{12} \sqrt{1-2 x}-\frac{1}{147} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx\\ &=\frac{1331}{84 (1-2 x)^{3/2}}-\frac{2178}{49 \sqrt{1-2 x}}-\frac{125}{12} \sqrt{1-2 x}+\frac{1}{147} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{1331}{84 (1-2 x)^{3/2}}-\frac{2178}{49 \sqrt{1-2 x}}-\frac{125}{12} \sqrt{1-2 x}+\frac{2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{147 \sqrt{21}}\\ \end{align*}

Mathematica [C]  time = 0.0220156, size = 45, normalized size = 0.67 \[ -\frac{2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+35 \left (675 x^2-2115 x+632\right )}{567 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(35*(632 - 2115*x + 675*x^2) + 2*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7])/(567*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.009, size = 47, normalized size = 0.7 \begin{align*}{\frac{1331}{84} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\sqrt{21}}{3087}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{2178}{49}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{125}{12}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1331/84/(1-2*x)^(3/2)+2/3087*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2178/49/(1-2*x)^(1/2)-125/12*(1-2*x)
^(1/2)

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Maxima [A]  time = 1.55513, size = 81, normalized size = 1.21 \begin{align*} -\frac{1}{3087} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{125}{12} \, \sqrt{-2 \, x + 1} + \frac{121 \,{\left (432 \, x - 139\right )}}{588 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3087*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/12*sqrt(-2*x + 1) + 1
21/588*(432*x - 139)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.34875, size = 208, normalized size = 3.1 \begin{align*} \frac{\sqrt{21}{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (6125 \, x^{2} - 19193 \, x + 5736\right )} \sqrt{-2 \, x + 1}}{3087 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3087*(sqrt(21)*(4*x^2 - 4*x + 1)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) - 21*(6125*x^2 - 19193*x
+ 5736)*sqrt(-2*x + 1))/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 34.6786, size = 102, normalized size = 1.52 \begin{align*} - \frac{125 \sqrt{1 - 2 x}}{12} - \frac{2 \left (\begin{cases} - \frac{\sqrt{21} \operatorname{acoth}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 < - \frac{7}{3} \\- \frac{\sqrt{21} \operatorname{atanh}{\left (\frac{\sqrt{21} \sqrt{1 - 2 x}}{7} \right )}}{21} & \text{for}\: 2 x - 1 > - \frac{7}{3} \end{cases}\right )}{147} - \frac{2178}{49 \sqrt{1 - 2 x}} + \frac{1331}{84 \left (1 - 2 x\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-125*sqrt(1 - 2*x)/12 - 2*Piecewise((-sqrt(21)*acoth(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)
*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/3))/147 - 2178/(49*sqrt(1 - 2*x)) + 1331/(84*(1 - 2*x)**(3/2
))

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Giac [A]  time = 1.77186, size = 95, normalized size = 1.42 \begin{align*} -\frac{1}{3087} \, \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{125}{12} \, \sqrt{-2 \, x + 1} - \frac{121 \,{\left (432 \, x - 139\right )}}{588 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3087*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 125/12*sqrt(-2*x
+ 1) - 121/588*(432*x - 139)/((2*x - 1)*sqrt(-2*x + 1))