3.1924 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=93 \[ \frac{71 \sqrt{1-2 x}}{10 (5 x+3)}-\frac{11 \sqrt{1-2 x}}{10 (5 x+3)^2}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

[Out]

(-11*Sqrt[1 - 2*x])/(10*(3 + 5*x)^2) + (71*Sqrt[1 - 2*x])/(10*(3 + 5*x)) + 14*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]] - (2379*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55])

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Rubi [A]  time = 0.0350219, antiderivative size = 93, 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, 151, 156, 63, 206} \[ \frac{71 \sqrt{1-2 x}}{10 (5 x+3)}-\frac{11 \sqrt{1-2 x}}{10 (5 x+3)^2}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-11*Sqrt[1 - 2*x])/(10*(3 + 5*x)^2) + (71*Sqrt[1 - 2*x])/(10*(3 + 5*x)) + 14*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[
1 - 2*x]] - (2379*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55])

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 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 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)^{3/2}}{(2+3 x) (3+5 x)^3} \, dx &=-\frac{11 \sqrt{1-2 x}}{10 (3+5 x)^2}-\frac{1}{10} \int \frac{92-107 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{11 \sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{71 \sqrt{1-2 x}}{10 (3+5 x)}+\frac{1}{110} \int \frac{3828-2343 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{11 \sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{71 \sqrt{1-2 x}}{10 (3+5 x)}-147 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{2379}{10} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{11 \sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{71 \sqrt{1-2 x}}{10 (3+5 x)}+147 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{2379}{10} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{11 \sqrt{1-2 x}}{10 (3+5 x)^2}+\frac{71 \sqrt{1-2 x}}{10 (3+5 x)}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}}\\ \end{align*}

Mathematica [A]  time = 0.0625302, size = 78, normalized size = 0.84 \[ \frac{\sqrt{1-2 x} (355 x+202)}{10 (5 x+3)^2}+14 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{2379 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{5 \sqrt{55}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(202 + 355*x))/(10*(3 + 5*x)^2) + 14*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (2379*ArcTanh[
Sqrt[5/11]*Sqrt[1 - 2*x]])/(5*Sqrt[55])

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Maple [A]  time = 0.009, size = 66, normalized size = 0.7 \begin{align*} 14\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+50\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{71\, \left ( 1-2\,x \right ) ^{3/2}}{50}}+{\frac{759\,\sqrt{1-2\,x}}{250}} \right ) }-{\frac{2379\,\sqrt{55}}{275}{\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)^(3/2)/(2+3*x)/(3+5*x)^3,x)

[Out]

14*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+50*(-71/50*(1-2*x)^(3/2)+759/250*(1-2*x)^(1/2))/(-10*x-6)^2-23
79/275*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.59353, size = 149, normalized size = 1.6 \begin{align*} \frac{2379}{550} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - 7 \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 759 \, \sqrt{-2 \, x + 1}}{5 \,{\left (25 \,{\left (2 \, x - 1\right )}^{2} + 220 \, x + 11\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2379/550*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 7*sqrt(21)*log(-(sqrt(21
) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/5*(355*(-2*x + 1)^(3/2) - 759*sqrt(-2*x + 1))/(25*(2*
x - 1)^2 + 220*x + 11)

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Fricas [A]  time = 1.53335, size = 317, normalized size = 3.41 \begin{align*} \frac{2379 \, \sqrt{55}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 3850 \, \sqrt{21}{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 55 \,{\left (355 \, x + 202\right )} \sqrt{-2 \, x + 1}}{550 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/550*(2379*sqrt(55)*(25*x^2 + 30*x + 9)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 3850*sqrt(21)*(2
5*x^2 + 30*x + 9)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 55*(355*x + 202)*sqrt(-2*x + 1))/(25*x^
2 + 30*x + 9)

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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)**(3/2)/(2+3*x)/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.61933, size = 144, normalized size = 1.55 \begin{align*} \frac{2379}{550} \, \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 ) - 7 \, \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{355 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 759 \, \sqrt{-2 \, x + 1}}{20 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2379/550*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 7*sqrt(21)*log
(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/20*(355*(-2*x + 1)^(3/2) - 759*sqr
t(-2*x + 1))/(5*x + 3)^2