3.2189 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=159 \[ \frac{172105}{65219 \sqrt{1-2 x}}+\frac{24}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{745}{22 (1-2 x)^{3/2} (5 x+3)}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}+\frac{15185}{2541 (1-2 x)^{3/2}}-\frac{4455}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{117500 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

15185/(2541*(1 - 2*x)^(3/2)) + 172105/(65219*Sqrt[1 - 2*x]) - 745/(22*(1 - 2*x)^(3/2)*(3 + 5*x)) + 3/(14*(1 -
2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) + 24/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.0714527, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {103, 151, 152, 156, 63, 206} \[ \frac{172105}{65219 \sqrt{1-2 x}}+\frac{24}{7 (1-2 x)^{3/2} (3 x+2) (5 x+3)}-\frac{745}{22 (1-2 x)^{3/2} (5 x+3)}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)}+\frac{15185}{2541 (1-2 x)^{3/2}}-\frac{4455}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{117500 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

15185/(2541*(1 - 2*x)^(3/2)) + 172105/(65219*Sqrt[1 - 2*x]) - 745/(22*(1 - 2*x)^(3/2)*(3 + 5*x)) + 3/(14*(1 -
2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)) + 24/(7*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)) - (4455*Sqrt[3/7]*ArcTanh[Sqrt[
3/7]*Sqrt[1 - 2*x]])/49 + (117500*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^2} \, dx &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{1}{14} \int \frac{22-135 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac{1}{98} \int \frac{245-11760 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{\int \frac{-98245-547575 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx}{1078}\\ &=\frac{15185}{2541 (1-2 x)^{3/2}}-\frac{745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac{\int \frac{-\frac{4091745}{2}+\frac{33482925 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{124509}\\ &=\frac{15185}{2541 (1-2 x)^{3/2}}+\frac{172105}{65219 \sqrt{1-2 x}}-\frac{745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{2 \int \frac{\frac{618657585}{4}-\frac{379491525 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{9587193}\\ &=\frac{15185}{2541 (1-2 x)^{3/2}}+\frac{172105}{65219 \sqrt{1-2 x}}-\frac{745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}+\frac{13365}{98} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{293750 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac{15185}{2541 (1-2 x)^{3/2}}+\frac{172105}{65219 \sqrt{1-2 x}}-\frac{745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{13365}{98} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{293750 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1331}\\ &=\frac{15185}{2541 (1-2 x)^{3/2}}+\frac{172105}{65219 \sqrt{1-2 x}}-\frac{745}{22 (1-2 x)^{3/2} (3+5 x)}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)}+\frac{24}{7 (1-2 x)^{3/2} (2+3 x) (3+5 x)}-\frac{4455}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{117500 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}

Mathematica [C]  time = 0.0430872, size = 78, normalized size = 0.49 \[ \frac{359370 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-329000 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )-\frac{33 \left (46935 x^2+60996 x+19771\right )}{(3 x+2)^2 (5 x+3)}}{5082 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-33*(19771 + 60996*x + 46935*x^2))/((2 + 3*x)^2*(3 + 5*x)) + 359370*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (
6*x)/7] - 329000*Hypergeometric2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11])/(5082*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.016, size = 100, normalized size = 0.6 \begin{align*}{\frac{4374}{2401\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{151}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{119}{6}\sqrt{1-2\,x}} \right ) }-{\frac{4455\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{32}{124509} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{5408}{3195731}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{1250}{1331}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{117500\,\sqrt{55}}{14641}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4374/2401*(151/18*(1-2*x)^(3/2)-119/6*(1-2*x)^(1/2))/(-6*x-4)^2-4455/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)+32/124509/(1-2*x)^(3/2)+5408/3195731/(1-2*x)^(1/2)+1250/1331*(1-2*x)^(1/2)/(-2*x-6/5)+117500/14641*arc
tanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.58346, size = 197, normalized size = 1.24 \begin{align*} -\frac{58750}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4455}{686} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{23234175 \,{\left (2 \, x - 1\right )}^{4} + 106925310 \,{\left (2 \, x - 1\right )}^{3} + 122999835 \,{\left (2 \, x - 1\right )}^{2} + 285824 \, x - 170016}{195657 \,{\left (45 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 309 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 707 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 539 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-58750/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4455/686*sqrt(21)*lo
g(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/195657*(23234175*(2*x - 1)^4 + 106925310*(
2*x - 1)^3 + 122999835*(2*x - 1)^2 + 285824*x - 170016)/(45*(-2*x + 1)^(9/2) - 309*(-2*x + 1)^(7/2) + 707*(-2*
x + 1)^(5/2) - 539*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.13346, size = 568, normalized size = 3.57 \begin{align*} \frac{120907500 \, \sqrt{11} \sqrt{5}{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 195676965 \, \sqrt{7} \sqrt{3}{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (92936700 \, x^{4} + 27977220 \, x^{3} - 58371045 \, x^{2} - 9008764 \, x + 9784671\right )} \sqrt{-2 \, x + 1}}{30131178 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30131178*(120907500*sqrt(11)*sqrt(5)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*log(-(sqrt(11)*sqrt(5)
*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) + 195676965*sqrt(7)*sqrt(3)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x +
12)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(92936700*x^4 + 27977220*x^3 - 58371045*x^
2 - 9008764*x + 9784671)*sqrt(-2*x + 1))/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.05527, size = 194, normalized size = 1.22 \begin{align*} -\frac{58750}{14641} \, \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{4455}{686} \, \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{64 \,{\left (507 \, x - 292\right )}}{9587193 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{3125 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\right )}} + \frac{243 \,{\left (151 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 357 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-58750/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 4455/686*s
qrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/9587193*(507*x - 292)/
((2*x - 1)*sqrt(-2*x + 1)) - 3125/1331*sqrt(-2*x + 1)/(5*x + 3) + 243/9604*(151*(-2*x + 1)^(3/2) - 357*sqrt(-2
*x + 1))/(3*x + 2)^2