3.2187 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=105 \[ \frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{218}{2541 (1-2 x)^{3/2}}-\frac{54}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1400 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

218/(2541*(1 - 2*x)^(3/2)) + 3274/(65219*Sqrt[1 - 2*x]) - 5/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) - (54*Sqrt[3/7]*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (1400*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.0442246, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {103, 152, 156, 63, 206} \[ \frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (5 x+3)}+\frac{218}{2541 (1-2 x)^{3/2}}-\frac{54}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1400 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

218/(2541*(1 - 2*x)^(3/2)) + 3274/(65219*Sqrt[1 - 2*x]) - 5/(11*(1 - 2*x)^(3/2)*(3 + 5*x)) - (54*Sqrt[3/7]*Arc
Tanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/49 + (1400*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 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)^2} \, dx &=-\frac{5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac{1}{11} \int \frac{-17-75 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)} \, dx\\ &=\frac{218}{2541 (1-2 x)^{3/2}}-\frac{5}{11 (1-2 x)^{3/2} (3+5 x)}+\frac{2 \int \frac{\frac{3}{2}+\frac{4905 x}{2}}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{2541}\\ &=\frac{218}{2541 (1-2 x)^{3/2}}+\frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac{4 \int \frac{\frac{58701}{4}-\frac{73665 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{195657}\\ &=\frac{218}{2541 (1-2 x)^{3/2}}+\frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (3+5 x)}+\frac{81}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{3500 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{1331}\\ &=\frac{218}{2541 (1-2 x)^{3/2}}+\frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac{81}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{3500 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{1331}\\ &=\frac{218}{2541 (1-2 x)^{3/2}}+\frac{3274}{65219 \sqrt{1-2 x}}-\frac{5}{11 (1-2 x)^{3/2} (3+5 x)}-\frac{54}{49} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1400 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}

Mathematica [C]  time = 0.023879, size = 73, normalized size = 0.7 \[ -\frac{35 \left (56 (5 x+3) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+33\right )-2178 (5 x+3) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{2541 (1-2 x)^{3/2} (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-2178*(3 + 5*x)*Hypergeometric2F1[-3/2, 1, -1/2, 3/7 - (6*x)/7] + 35*(33 + 56*(3 + 5*x)*Hypergeometric2F1[-3
/2, 1, -1/2, (-5*(-1 + 2*x))/11]))/(2541*(1 - 2*x)^(3/2)*(3 + 5*x))

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Maple [A]  time = 0.013, size = 72, normalized size = 0.7 \begin{align*} -{\frac{54\,\sqrt{21}}{343}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{8}{2541} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}}+{\frac{824}{65219}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{50}{1331}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{1400\,\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+5*x)^2,x)

[Out]

-54/343*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+8/2541/(1-2*x)^(3/2)+824/65219/(1-2*x)^(1/2)+50/1331*(1-2
*x)^(1/2)/(-2*x-6/5)+1400/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.63153, size = 149, normalized size = 1.42 \begin{align*} -\frac{700}{14641} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{27}{343} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (24555 \,{\left (2 \, x - 1\right )}^{2} + 24112 \, x - 15444\right )}}{195657 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 11 \,{\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)^2,x, algorithm="maxima")

[Out]

-700/14641*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 27/343*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/195657*(24555*(2*x - 1)^2 + 24112*x - 15444)/(
5*(-2*x + 1)^(5/2) - 11*(-2*x + 1)^(3/2))

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Fricas [A]  time = 1.11494, size = 421, normalized size = 4.01 \begin{align*} \frac{720300 \, \sqrt{11} \sqrt{5}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 1185921 \, \sqrt{7} \sqrt{3}{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (98220 \, x^{2} - 74108 \, x + 9111\right )} \sqrt{-2 \, x + 1}}{15065589 \,{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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)^2,x, algorithm="fricas")

[Out]

1/15065589*(720300*sqrt(11)*sqrt(5)*(20*x^3 - 8*x^2 - 7*x + 3)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8
)/(5*x + 3)) + 1185921*sqrt(7)*sqrt(3)*(20*x^3 - 8*x^2 - 7*x + 3)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x -
5)/(3*x + 2)) - 77*(98220*x^2 - 74108*x + 9111)*sqrt(-2*x + 1))/(20*x^3 - 8*x^2 - 7*x + 3)

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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)**2,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.85664, size = 157, normalized size = 1.5 \begin{align*} -\frac{700}{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{27}{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 (309 \, x - 193\right )}}{195657 \,{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}} - \frac{125 \, \sqrt{-2 \, x + 1}}{1331 \,{\left (5 \, x + 3\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)^2,x, algorithm="giac")

[Out]

-700/14641*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 27/343*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/195657*(309*x - 193)/((2*x
- 1)*sqrt(-2*x + 1)) - 125/1331*sqrt(-2*x + 1)/(5*x + 3)