3.2106 \(\int \frac{(2+3 x)^4}{(1-2 x)^{3/2} (3+5 x)} \, dx\)

Optimal. Leaf size=80 \[ \frac{81}{200} (1-2 x)^{5/2}-\frac{963}{200} (1-2 x)^{3/2}+\frac{34371 \sqrt{1-2 x}}{1000}+\frac{2401}{88 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}} \]

[Out]

2401/(88*Sqrt[1 - 2*x]) + (34371*Sqrt[1 - 2*x])/1000 - (963*(1 - 2*x)^(3/2))/200 + (81*(1 - 2*x)^(5/2))/200 -
(2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1375*Sqrt[55])

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Rubi [A]  time = 0.0452434, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {87, 43, 63, 206} \[ \frac{81}{200} (1-2 x)^{5/2}-\frac{963}{200} (1-2 x)^{3/2}+\frac{34371 \sqrt{1-2 x}}{1000}+\frac{2401}{88 \sqrt{1-2 x}}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)),x]

[Out]

2401/(88*Sqrt[1 - 2*x]) + (34371*Sqrt[1 - 2*x])/1000 - (963*(1 - 2*x)^(3/2))/200 + (81*(1 - 2*x)^(5/2))/200 -
(2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1375*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)^4}{(1-2 x)^{3/2} (3+5 x)} \, dx &=\int \left (\frac{2401}{88 (1-2 x)^{3/2}}-\frac{21951}{1000 \sqrt{1-2 x}}-\frac{2079 x}{100 \sqrt{1-2 x}}-\frac{81 x^2}{10 \sqrt{1-2 x}}+\frac{1}{1375 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=\frac{2401}{88 \sqrt{1-2 x}}+\frac{21951 \sqrt{1-2 x}}{1000}+\frac{\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1375}-\frac{81}{10} \int \frac{x^2}{\sqrt{1-2 x}} \, dx-\frac{2079}{100} \int \frac{x}{\sqrt{1-2 x}} \, dx\\ &=\frac{2401}{88 \sqrt{1-2 x}}+\frac{21951 \sqrt{1-2 x}}{1000}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1375}-\frac{81}{10} \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{2079}{100} \int \left (\frac{1}{2 \sqrt{1-2 x}}-\frac{1}{2} \sqrt{1-2 x}\right ) \, dx\\ &=\frac{2401}{88 \sqrt{1-2 x}}+\frac{34371 \sqrt{1-2 x}}{1000}-\frac{963}{200} (1-2 x)^{3/2}+\frac{81}{200} (1-2 x)^{5/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1375 \sqrt{55}}\\ \end{align*}

Mathematica [C]  time = 0.0210152, size = 50, normalized size = 0.62 \[ \frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{5}{11} (1-2 x)\right )-33 \left (675 x^3+3000 x^2+10815 x-11926\right )}{6875 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)^(3/2)*(3 + 5*x)),x]

[Out]

(-33*(-11926 + 10815*x + 3000*x^2 + 675*x^3) + 2*Hypergeometric2F1[-1/2, 1, 1/2, (5*(1 - 2*x))/11])/(6875*Sqrt
[1 - 2*x])

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Maple [A]  time = 0.007, size = 56, normalized size = 0.7 \begin{align*} -{\frac{963}{200} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{81}{200} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{2\,\sqrt{55}}{75625}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }+{\frac{2401}{88}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{34371}{1000}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x)

[Out]

-963/200*(1-2*x)^(3/2)+81/200*(1-2*x)^(5/2)-2/75625*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)+2401/88/(1-2
*x)^(1/2)+34371/1000*(1-2*x)^(1/2)

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Maxima [A]  time = 1.52235, size = 99, normalized size = 1.24 \begin{align*} \frac{81}{200} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - \frac{963}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{75625} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{34371}{1000} \, \sqrt{-2 \, x + 1} + \frac{2401}{88 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x, algorithm="maxima")

[Out]

81/200*(-2*x + 1)^(5/2) - 963/200*(-2*x + 1)^(3/2) + 1/75625*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt
(55) + 5*sqrt(-2*x + 1))) + 34371/1000*sqrt(-2*x + 1) + 2401/88/sqrt(-2*x + 1)

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Fricas [A]  time = 1.6103, size = 205, normalized size = 2.56 \begin{align*} \frac{\sqrt{55}{\left (2 \, x - 1\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 55 \,{\left (4455 \, x^{3} + 19800 \, x^{2} + 71379 \, x - 78712\right )} \sqrt{-2 \, x + 1}}{75625 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x, algorithm="fricas")

[Out]

1/75625*(sqrt(55)*(2*x - 1)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 55*(4455*x^3 + 19800*x^2 + 71
379*x - 78712)*sqrt(-2*x + 1))/(2*x - 1)

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Sympy [A]  time = 56.608, size = 114, normalized size = 1.42 \begin{align*} \frac{81 \left (1 - 2 x\right )^{\frac{5}{2}}}{200} - \frac{963 \left (1 - 2 x\right )^{\frac{3}{2}}}{200} + \frac{34371 \sqrt{1 - 2 x}}{1000} + \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 )}{1375} + \frac{2401}{88 \sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)**(3/2)/(3+5*x),x)

[Out]

81*(1 - 2*x)**(5/2)/200 - 963*(1 - 2*x)**(3/2)/200 + 34371*sqrt(1 - 2*x)/1000 + 2*Piecewise((-sqrt(55)*acoth(s
qrt(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))/1375 + 2401/(88*sqrt(1 - 2*x))

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Giac [A]  time = 2.02529, size = 112, normalized size = 1.4 \begin{align*} \frac{81}{200} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - \frac{963}{200} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{75625} \, \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{34371}{1000} \, \sqrt{-2 \, x + 1} + \frac{2401}{88 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)^(3/2)/(3+5*x),x, algorithm="giac")

[Out]

81/200*(2*x - 1)^2*sqrt(-2*x + 1) - 963/200*(-2*x + 1)^(3/2) + 1/75625*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*s
qrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 34371/1000*sqrt(-2*x + 1) + 2401/88/sqrt(-2*x + 1)