3.1986 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=106 \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)}-\frac{748 \sqrt{1-2 x}}{15 (5 x+3)}-\frac{910}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1562}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-748*Sqrt[1 - 2*x])/(15*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5*x)) - (910*Sqrt[7/3]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/3 + (1562*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

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Rubi [A]  time = 0.0345304, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 149, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{3 (3 x+2) (5 x+3)}-\frac{748 \sqrt{1-2 x}}{15 (5 x+3)}-\frac{910}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1562}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-748*Sqrt[1 - 2*x])/(15*(3 + 5*x)) + (7*(1 - 2*x)^(3/2))/(3*(2 + 3*x)*(3 + 5*x)) - (910*Sqrt[7/3]*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/3 + (1562*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/5

Rule 98

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

Rule 149

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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{(1-2 x)^{5/2}}{(2+3 x)^2 (3+5 x)^2} \, dx &=\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)}+\frac{1}{3} \int \frac{(131-31 x) \sqrt{1-2 x}}{(2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{748 \sqrt{1-2 x}}{15 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)}+\frac{1}{15} \int \frac{-3771+2306 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{748 \sqrt{1-2 x}}{15 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)}+\frac{3185}{3} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{8591}{5} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{748 \sqrt{1-2 x}}{15 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)}-\frac{3185}{3} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{8591}{5} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{748 \sqrt{1-2 x}}{15 (3+5 x)}+\frac{7 (1-2 x)^{3/2}}{3 (2+3 x) (3+5 x)}-\frac{910}{3} \sqrt{\frac{7}{3}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{1562}{5} \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0625756, size = 105, normalized size = 0.99 \[ \frac{-22750 \sqrt{21} \left (15 x^2+19 x+6\right ) \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+14058 \sqrt{55} \left (15 x^2+19 x+6\right ) \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-15 \sqrt{1-2 x} (2314 x+1461)}{225 (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[1 - 2*x]*(1461 + 2314*x) - 22750*Sqrt[21]*(6 + 19*x + 15*x^2)*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] + 140
58*Sqrt[55]*(6 + 19*x + 15*x^2)*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(225*(2 + 3*x)*(3 + 5*x))

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Maple [A]  time = 0.012, size = 70, normalized size = 0.7 \begin{align*}{\frac{98}{9}\sqrt{1-2\,x} \left ( -2\,x-{\frac{4}{3}} \right ) ^{-1}}-{\frac{910\,\sqrt{21}}{9}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{242}{25}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{1562\,\sqrt{55}}{25}{\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((1-2*x)^(5/2)/(2+3*x)^2/(3+5*x)^2,x)

[Out]

98/9*(1-2*x)^(1/2)/(-2*x-4/3)-910/9*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+242/25*(1-2*x)^(1/2)/(-2*x-6/
5)+1562/25*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.85715, size = 149, normalized size = 1.41 \begin{align*} -\frac{781}{25} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{455}{9} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{4 \,{\left (1157 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2618 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-781/25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 455/9*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/15*(1157*(-2*x + 1)^(3/2) - 2618*sqrt(-2*x + 1))/(
15*(2*x - 1)^2 + 136*x + 9)

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Fricas [A]  time = 1.3721, size = 363, normalized size = 3.42 \begin{align*} \frac{7029 \, \sqrt{11} \sqrt{5}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 11375 \, \sqrt{7} \sqrt{3}{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 15 \,{\left (2314 \, x + 1461\right )} \sqrt{-2 \, x + 1}}{225 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/225*(7029*sqrt(11)*sqrt(5)*(15*x^2 + 19*x + 6)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x + 1) - 5*x + 8)/(5*x + 3)) +
11375*sqrt(7)*sqrt(3)*(15*x^2 + 19*x + 6)*log((sqrt(7)*sqrt(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 15*(231
4*x + 1461)*sqrt(-2*x + 1))/(15*x^2 + 19*x + 6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 2.43142, size = 157, normalized size = 1.48 \begin{align*} -\frac{781}{25} \, \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{455}{9} \, \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{4 \,{\left (1157 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2618 \, \sqrt{-2 \, x + 1}\right )}}{15 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-781/25*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 455/9*sqrt(21)*
log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 4/15*(1157*(-2*x + 1)^(3/2) - 261
8*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)