3.2169 \(\int \frac{(2+3 x)^6}{(1-2 x)^{5/2} (3+5 x)} \, dx\)

Optimal. Leaf size=106 \[ \frac{729 (1-2 x)^{7/2}}{1120}-\frac{43011 (1-2 x)^{5/2}}{4000}+\frac{169209 (1-2 x)^{3/2}}{2000}-\frac{5992353 \sqrt{1-2 x}}{10000}-\frac{2739541}{3872 \sqrt{1-2 x}}+\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}} \]

[Out]

117649/(1056*(1 - 2*x)^(3/2)) - 2739541/(3872*Sqrt[1 - 2*x]) - (5992353*Sqrt[1 - 2*x])/10000 + (169209*(1 - 2*
x)^(3/2))/2000 - (43011*(1 - 2*x)^(5/2))/4000 + (729*(1 - 2*x)^(7/2))/1120 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(75625*Sqrt[55])

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Rubi [A]  time = 0.0665858, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {87, 43, 63, 206} \[ \frac{729 (1-2 x)^{7/2}}{1120}-\frac{43011 (1-2 x)^{5/2}}{4000}+\frac{169209 (1-2 x)^{3/2}}{2000}-\frac{5992353 \sqrt{1-2 x}}{10000}-\frac{2739541}{3872 \sqrt{1-2 x}}+\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)),x]

[Out]

117649/(1056*(1 - 2*x)^(3/2)) - 2739541/(3872*Sqrt[1 - 2*x]) - (5992353*Sqrt[1 - 2*x])/10000 + (169209*(1 - 2*
x)^(3/2))/2000 - (43011*(1 - 2*x)^(5/2))/4000 + (729*(1 - 2*x)^(7/2))/1120 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]])/(75625*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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)^6}{(1-2 x)^{5/2} (3+5 x)} \, dx &=\int \left (\frac{117649}{352 (1-2 x)^{5/2}}-\frac{2739541}{3872 (1-2 x)^{3/2}}+\frac{3946293}{10000 \sqrt{1-2 x}}+\frac{639819 x}{2000 \sqrt{1-2 x}}+\frac{8019 x^2}{50 \sqrt{1-2 x}}+\frac{729 x^3}{20 \sqrt{1-2 x}}+\frac{1}{75625 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2739541}{3872 \sqrt{1-2 x}}-\frac{3946293 \sqrt{1-2 x}}{10000}+\frac{\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{75625}+\frac{729}{20} \int \frac{x^3}{\sqrt{1-2 x}} \, dx+\frac{8019}{50} \int \frac{x^2}{\sqrt{1-2 x}} \, dx+\frac{639819 \int \frac{x}{\sqrt{1-2 x}} \, dx}{2000}\\ &=\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2739541}{3872 \sqrt{1-2 x}}-\frac{3946293 \sqrt{1-2 x}}{10000}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{75625}+\frac{729}{20} \int \left (\frac{1}{8 \sqrt{1-2 x}}-\frac{3}{8} \sqrt{1-2 x}+\frac{3}{8} (1-2 x)^{3/2}-\frac{1}{8} (1-2 x)^{5/2}\right ) \, dx+\frac{8019}{50} \int \left (\frac{1}{4 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}+\frac{1}{4} (1-2 x)^{3/2}\right ) \, dx+\frac{639819 \int \left (\frac{1}{2 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}\right ) \, dx}{2000}\\ &=\frac{117649}{1056 (1-2 x)^{3/2}}-\frac{2739541}{3872 \sqrt{1-2 x}}-\frac{5992353 \sqrt{1-2 x}}{10000}+\frac{169209 (1-2 x)^{3/2}}{2000}-\frac{43011 (1-2 x)^{5/2}}{4000}+\frac{729 (1-2 x)^{7/2}}{1120}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{75625 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0322961, size = 60, normalized size = 0.57 \[ \frac{14 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{5}{11} (1-2 x)\right )-99 \left (759375 x^5+4374000 x^4+14029875 x^3+58833450 x^2-123370605 x+40864276\right )}{3609375 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6/((1 - 2*x)^(5/2)*(3 + 5*x)),x]

[Out]

(-99*(40864276 - 123370605*x + 58833450*x^2 + 14029875*x^3 + 4374000*x^4 + 759375*x^5) + 14*Hypergeometric2F1[
-3/2, 1, -1/2, (5*(1 - 2*x))/11])/(3609375*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.01, size = 74, normalized size = 0.7 \begin{align*}{\frac{117649}{1056} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{169209}{2000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{43011}{4000} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{729}{1120} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{2\,\sqrt{55}}{4159375}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{2739541}{3872}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{5992353}{10000}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6/(1-2*x)^(5/2)/(3+5*x),x)

[Out]

117649/1056/(1-2*x)^(3/2)+169209/2000*(1-2*x)^(3/2)-43011/4000*(1-2*x)^(5/2)+729/1120*(1-2*x)^(7/2)-2/4159375*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-2739541/3872/(1-2*x)^(1/2)-5992353/10000*(1-2*x)^(1/2)

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Maxima [A]  time = 2.02845, size = 117, normalized size = 1.1 \begin{align*} \frac{729}{1120} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{43011}{4000} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{169209}{2000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{4159375} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{5992353}{10000} \, \sqrt{-2 \, x + 1} + \frac{16807 \,{\left (489 \, x - 206\right )}}{5808 \,{\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)^6/(1-2*x)^(5/2)/(3+5*x),x, algorithm="maxima")

[Out]

729/1120*(-2*x + 1)^(7/2) - 43011/4000*(-2*x + 1)^(5/2) + 169209/2000*(-2*x + 1)^(3/2) + 1/4159375*sqrt(55)*lo
g(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 5992353/10000*sqrt(-2*x + 1) + 16807/5808*(4
89*x - 206)/(-2*x + 1)^(3/2)

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Fricas [A]  time = 1.30264, size = 304, normalized size = 2.87 \begin{align*} \frac{21 \, \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 (33078375 \, x^{5} + 190531440 \, x^{4} + 611141355 \, x^{3} + 2562785082 \, x^{2} - 5374023537 \, x + 1780047848\right )} \sqrt{-2 \, x + 1}}{87346875 \,{\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)^6/(1-2*x)^(5/2)/(3+5*x),x, algorithm="fricas")

[Out]

1/87346875*(21*sqrt(55)*(4*x^2 - 4*x + 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) - 55*(33078375*x^
5 + 190531440*x^4 + 611141355*x^3 + 2562785082*x^2 - 5374023537*x + 1780047848)*sqrt(-2*x + 1))/(4*x^2 - 4*x +
1)

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Sympy [A]  time = 84.1344, size = 138, normalized size = 1.3 \begin{align*} \frac{729 \left (1 - 2 x\right )^{\frac{7}{2}}}{1120} - \frac{43011 \left (1 - 2 x\right )^{\frac{5}{2}}}{4000} + \frac{169209 \left (1 - 2 x\right )^{\frac{3}{2}}}{2000} - \frac{5992353 \sqrt{1 - 2 x}}{10000} + \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 )}{75625} - \frac{2739541}{3872 \sqrt{1 - 2 x}} + \frac{117649}{1056 \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)**6/(1-2*x)**(5/2)/(3+5*x),x)

[Out]

729*(1 - 2*x)**(7/2)/1120 - 43011*(1 - 2*x)**(5/2)/4000 + 169209*(1 - 2*x)**(3/2)/2000 - 5992353*sqrt(1 - 2*x)
/10000 + 2*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(5
5)*sqrt(1 - 2*x)/11)/55, 2*x - 1 > -11/5))/75625 - 2739541/(3872*sqrt(1 - 2*x)) + 117649/(1056*(1 - 2*x)**(3/2
))

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Giac [A]  time = 2.48085, size = 150, normalized size = 1.42 \begin{align*} -\frac{729}{1120} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{43011}{4000} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{169209}{2000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{4159375} \, \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{5992353}{10000} \, \sqrt{-2 \, x + 1} - \frac{16807 \,{\left (489 \, x - 206\right )}}{5808 \,{\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)^6/(1-2*x)^(5/2)/(3+5*x),x, algorithm="giac")

[Out]

-729/1120*(2*x - 1)^3*sqrt(-2*x + 1) - 43011/4000*(2*x - 1)^2*sqrt(-2*x + 1) + 169209/2000*(-2*x + 1)^(3/2) +
1/4159375*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 5992353/10000
*sqrt(-2*x + 1) - 16807/5808*(489*x - 206)/((2*x - 1)*sqrt(-2*x + 1))