3.2036 \(\int \frac{(2+3 x)^5}{\sqrt{1-2 x} (3+5 x)} \, dx\)

Optimal. Leaf size=93 \[ -\frac{27}{80} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{4774713 \sqrt{1-2 x}}{50000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

[Out]

(-4774713*Sqrt[1 - 2*x])/50000 + (268707*(1 - 2*x)^(3/2))/5000 - (51057*(1 - 2*x)^(5/2))/2500 + (5751*(1 - 2*x
)^(7/2))/1400 - (27*(1 - 2*x)^(9/2))/80 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

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Rubi [A]  time = 0.0272446, antiderivative size = 93, 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 = {88, 63, 206} \[ -\frac{27}{80} (1-2 x)^{9/2}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{4774713 \sqrt{1-2 x}}{50000}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4774713*Sqrt[1 - 2*x])/50000 + (268707*(1 - 2*x)^(3/2))/5000 - (51057*(1 - 2*x)^(5/2))/2500 + (5751*(1 - 2*x
)^(7/2))/1400 - (27*(1 - 2*x)^(9/2))/80 - (2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(3125*Sqrt[55])

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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)^5}{\sqrt{1-2 x} (3+5 x)} \, dx &=\int \left (\frac{4774713}{50000 \sqrt{1-2 x}}-\frac{806121 \sqrt{1-2 x}}{5000}+\frac{51057}{500} (1-2 x)^{3/2}-\frac{5751}{200} (1-2 x)^{5/2}+\frac{243}{80} (1-2 x)^{7/2}+\frac{1}{3125 \sqrt{1-2 x} (3+5 x)}\right ) \, dx\\ &=-\frac{4774713 \sqrt{1-2 x}}{50000}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{27}{80} (1-2 x)^{9/2}+\frac{\int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{3125}\\ &=-\frac{4774713 \sqrt{1-2 x}}{50000}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{27}{80} (1-2 x)^{9/2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{3125}\\ &=-\frac{4774713 \sqrt{1-2 x}}{50000}+\frac{268707 (1-2 x)^{3/2}}{5000}-\frac{51057 (1-2 x)^{5/2}}{2500}+\frac{5751 (1-2 x)^{7/2}}{1400}-\frac{27}{80} (1-2 x)^{9/2}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0494919, size = 61, normalized size = 0.66 \[ -\frac{3 \sqrt{1-2 x} \left (39375 x^4+160875 x^3+295290 x^2+348095 x+425872\right )}{21875}-\frac{2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{3125 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*Sqrt[1 - 2*x]*(425872 + 348095*x + 295290*x^2 + 160875*x^3 + 39375*x^4))/21875 - (2*ArcTanh[Sqrt[5/11]*Sqr
t[1 - 2*x]])/(3125*Sqrt[55])

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Maple [A]  time = 0.006, size = 65, normalized size = 0.7 \begin{align*}{\frac{268707}{5000} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{51057}{2500} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}+{\frac{5751}{1400} \left ( 1-2\,x \right ) ^{{\frac{7}{2}}}}-{\frac{27}{80} \left ( 1-2\,x \right ) ^{{\frac{9}{2}}}}-{\frac{2\,\sqrt{55}}{171875}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) }-{\frac{4774713}{50000}\sqrt{1-2\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

268707/5000*(1-2*x)^(3/2)-51057/2500*(1-2*x)^(5/2)+5751/1400*(1-2*x)^(7/2)-27/80*(1-2*x)^(9/2)-2/171875*arctan
h(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)-4774713/50000*(1-2*x)^(1/2)

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Maxima [A]  time = 1.61031, size = 111, normalized size = 1.19 \begin{align*} -\frac{27}{80} \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} + \frac{5751}{1400} \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - \frac{51057}{2500} \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + \frac{268707}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{171875} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4774713}{50000} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/80*(-2*x + 1)^(9/2) + 5751/1400*(-2*x + 1)^(7/2) - 51057/2500*(-2*x + 1)^(5/2) + 268707/5000*(-2*x + 1)^(3
/2) + 1/171875*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4774713/50000*sqrt
(-2*x + 1)

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Fricas [A]  time = 1.68734, size = 208, normalized size = 2.24 \begin{align*} -\frac{3}{21875} \,{\left (39375 \, x^{4} + 160875 \, x^{3} + 295290 \, x^{2} + 348095 \, x + 425872\right )} \sqrt{-2 \, x + 1} + \frac{1}{171875} \, \sqrt{55} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/21875*(39375*x^4 + 160875*x^3 + 295290*x^2 + 348095*x + 425872)*sqrt(-2*x + 1) + 1/171875*sqrt(55)*log((5*x
+ sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3))

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Sympy [A]  time = 50.9721, size = 126, normalized size = 1.35 \begin{align*} - \frac{27 \left (1 - 2 x\right )^{\frac{9}{2}}}{80} + \frac{5751 \left (1 - 2 x\right )^{\frac{7}{2}}}{1400} - \frac{51057 \left (1 - 2 x\right )^{\frac{5}{2}}}{2500} + \frac{268707 \left (1 - 2 x\right )^{\frac{3}{2}}}{5000} - \frac{4774713 \sqrt{1 - 2 x}}{50000} + \frac{2 \left (\begin{cases} - \frac{\sqrt{55} \operatorname{acoth}{\left (\frac{\sqrt{55}}{5 \sqrt{1 - 2 x}} \right )}}{55} & \text{for}\: \frac{1}{1 - 2 x} > \frac{5}{11} \\- \frac{\sqrt{55} \operatorname{atanh}{\left (\frac{\sqrt{55}}{5 \sqrt{1 - 2 x}} \right )}}{55} & \text{for}\: \frac{1}{1 - 2 x} < \frac{5}{11} \end{cases}\right )}{3125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*(1 - 2*x)**(9/2)/80 + 5751*(1 - 2*x)**(7/2)/1400 - 51057*(1 - 2*x)**(5/2)/2500 + 268707*(1 - 2*x)**(3/2)/5
000 - 4774713*sqrt(1 - 2*x)/50000 + 2*Piecewise((-sqrt(55)*acoth(sqrt(55)/(5*sqrt(1 - 2*x)))/55, 1/(1 - 2*x) >
5/11), (-sqrt(55)*atanh(sqrt(55)/(5*sqrt(1 - 2*x)))/55, 1/(1 - 2*x) < 5/11))/3125

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Giac [A]  time = 1.49304, size = 143, normalized size = 1.54 \begin{align*} -\frac{27}{80} \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} - \frac{5751}{1400} \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} - \frac{51057}{2500} \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} + \frac{268707}{5000} \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + \frac{1}{171875} \, \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{4774713}{50000} \, \sqrt{-2 \, x + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/80*(2*x - 1)^4*sqrt(-2*x + 1) - 5751/1400*(2*x - 1)^3*sqrt(-2*x + 1) - 51057/2500*(2*x - 1)^2*sqrt(-2*x +
1) + 268707/5000*(-2*x + 1)^(3/2) + 1/171875*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) +
5*sqrt(-2*x + 1))) - 4774713/50000*sqrt(-2*x + 1)