3.2197 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=125 \[ -\frac{65167}{717409 \sqrt{1-2 x}}+\frac{295}{242 (1-2 x)^{3/2} (5 x+3)}-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{5}{22 (1-2 x)^{3/2} (5 x+3)^2}+\frac{162}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{47075 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

[Out]

-5969/(27951*(1 - 2*x)^(3/2)) - 65167/(717409*Sqrt[1 - 2*x]) - 5/(22*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 295/(242*(
1 - 2*x)^(3/2)*(3 + 5*x)) + (162*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (47075*Sqrt[5/11]*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]])/14641

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Rubi [A]  time = 0.0563536, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{65167}{717409 \sqrt{1-2 x}}+\frac{295}{242 (1-2 x)^{3/2} (5 x+3)}-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{5}{22 (1-2 x)^{3/2} (5 x+3)^2}+\frac{162}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{47075 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-5969/(27951*(1 - 2*x)^(3/2)) - 65167/(717409*Sqrt[1 - 2*x]) - 5/(22*(1 - 2*x)^(3/2)*(3 + 5*x)^2) + 295/(242*(
1 - 2*x)^(3/2)*(3 + 5*x)) + (162*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 - (47075*Sqrt[5/11]*ArcTanh[Sq
rt[5/11]*Sqrt[1 - 2*x]])/14641

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+5 x)^3} \, dx &=-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}-\frac{1}{22} \int \frac{-4-105 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac{295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac{1}{242} \int \frac{-772-4425 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac{295}{242 (1-2 x)^{3/2} (3+5 x)}-\frac{\int \frac{-18276+\frac{268605 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{27951}\\ &=-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{65167}{717409 \sqrt{1-2 x}}-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac{295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac{2 \int \frac{1290129-\frac{2932515 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{2152227}\\ &=-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{65167}{717409 \sqrt{1-2 x}}-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac{295}{242 (1-2 x)^{3/2} (3+5 x)}-\frac{243}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{235375 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{29282}\\ &=-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{65167}{717409 \sqrt{1-2 x}}-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac{295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac{243}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{235375 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{29282}\\ &=-\frac{5969}{27951 (1-2 x)^{3/2}}-\frac{65167}{717409 \sqrt{1-2 x}}-\frac{5}{22 (1-2 x)^{3/2} (3+5 x)^2}+\frac{295}{242 (1-2 x)^{3/2} (3+5 x)}+\frac{162}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{47075 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{14641}\\ \end{align*}

Mathematica [C]  time = 0.0364612, size = 73, normalized size = 0.58 \[ \frac{\frac{35 \left (3766 (5 x+3)^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+9735 x+5478\right )}{(5 x+3)^2}-143748 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{55902 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-143748*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7] + (35*(5478 + 9735*x + 3766*(3 + 5*x)^2*Hypergeometri
c2F1[-3/2, 1, -1/2, (-5*(-1 + 2*x))/11]))/(3 + 5*x)^2)/(55902*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.015, size = 84, normalized size = 0.7 \begin{align*}{\frac{162\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{16}{27951} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{2208}{717409}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{31250}{14641\, \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{11}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{583}{250}\sqrt{1-2\,x}} \right ) }-{\frac{47075\,\sqrt{55}}{161051}{\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+5*x)^3,x)

[Out]

162/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+16/27951/(1-2*x)^(3/2)+2208/717409/(1-2*x)^(1/2)+31250/14
641*(-11/10*(1-2*x)^(3/2)+583/250*(1-2*x)^(1/2))/(-10*x-6)^2-47075/161051*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
*55^(1/2)

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Maxima [A]  time = 1.82957, size = 173, normalized size = 1.38 \begin{align*} \frac{47075}{322102} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{81}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4887525 \,{\left (2 \, x - 1\right )}^{3} + 10014785 \,{\left (2 \, x - 1\right )}^{2} - 1331968 \, x + 815056}{2152227 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 121 \,{\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+5*x)^3,x, algorithm="maxima")

[Out]

47075/322102*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 81/343*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/2152227*(4887525*(2*x - 1)^3 + 10014785*(2*x
- 1)^2 - 1331968*x + 815056)/(25*(-2*x + 1)^(7/2) - 110*(-2*x + 1)^(5/2) + 121*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.06335, size = 501, normalized size = 4.01 \begin{align*} \frac{48440175 \, \sqrt{11} \sqrt{5}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 78270786 \, \sqrt{7} \sqrt{3}{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (19550100 \, x^{3} - 9295580 \, x^{2} - 6032979 \, x + 2971158\right )} \sqrt{-2 \, x + 1}}{331442958 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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+5*x)^3,x, algorithm="fricas")

[Out]

1/331442958*(48440175*sqrt(11)*sqrt(5)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x +
1) + 5*x - 8)/(5*x + 3)) + 78270786*sqrt(7)*sqrt(3)*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(-(sqrt(7)*sqrt(
3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(19550100*x^3 - 9295580*x^2 - 6032979*x + 2971158)*sqrt(-2*x + 1)
)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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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+5*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.31875, size = 173, normalized size = 1.38 \begin{align*} \frac{47075}{322102} \, \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{81}{343} \, \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{16 \,{\left (828 \, x - 491\right )}}{2152227 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{125 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 53 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\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+5*x)^3,x, algorithm="giac")

[Out]

47075/322102*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 81/343*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/2152227*(828*x - 491)/((
2*x - 1)*sqrt(-2*x + 1)) - 125/5324*(25*(-2*x + 1)^(3/2) - 53*sqrt(-2*x + 1))/(5*x + 3)^2