3.1999 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=172 \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{113875 \sqrt{1-2 x}}{6 (5 x+3)}+\frac{1256 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac{581 \sqrt{1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac{169975 \sqrt{1-2 x}}{54 (5 x+3)^2}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-169975*Sqrt[1 - 2*x])/(54*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)^2) + (581*Sqrt[1 - 2*x
])/(27*(2 + 3*x)^2*(3 + 5*x)^2) + (1256*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)^2) + (113875*Sqrt[1 - 2*x])/(6*(
3 + 5*x)) + (785570*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]]

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Rubi [A]  time = 0.0701064, antiderivative size = 172, 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 = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{9 (3 x+2)^3 (5 x+3)^2}+\frac{113875 \sqrt{1-2 x}}{6 (5 x+3)}+\frac{1256 \sqrt{1-2 x}}{3 (3 x+2) (5 x+3)^2}+\frac{581 \sqrt{1-2 x}}{27 (3 x+2)^2 (5 x+3)^2}-\frac{169975 \sqrt{1-2 x}}{54 (5 x+3)^2}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(-169975*Sqrt[1 - 2*x])/(54*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^3*(3 + 5*x)^2) + (581*Sqrt[1 - 2*x
])/(27*(2 + 3*x)^2*(3 + 5*x)^2) + (1256*Sqrt[1 - 2*x])/(3*(2 + 3*x)*(3 + 5*x)^2) + (113875*Sqrt[1 - 2*x])/(6*(
3 + 5*x)) + (785570*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*
x]]

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 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)^{5/2}}{(2+3 x)^4 (3+5 x)^3} \, dx &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{1}{9} \int \frac{(232-233 x) \sqrt{1-2 x}}{(2+3 x)^3 (3+5 x)^3} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}-\frac{1}{54} \int \frac{-26260+39738 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac{1256 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}-\frac{1}{378} \int \frac{-2861390+3956400 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{169975 \sqrt{1-2 x}}{54 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac{1256 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{\int \frac{-205876440+235585350 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx}{8316}\\ &=-\frac{169975 \sqrt{1-2 x}}{54 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac{1256 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{113875 \sqrt{1-2 x}}{6 (3+5 x)}-\frac{\int \frac{-8504523720+5208414750 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{91476}\\ &=-\frac{169975 \sqrt{1-2 x}}{54 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac{1256 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{113875 \sqrt{1-2 x}}{6 (3+5 x)}-392785 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{1271325}{2} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{169975 \sqrt{1-2 x}}{54 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac{1256 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{113875 \sqrt{1-2 x}}{6 (3+5 x)}+392785 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{1271325}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{169975 \sqrt{1-2 x}}{54 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{9 (2+3 x)^3 (3+5 x)^2}+\frac{581 \sqrt{1-2 x}}{27 (2+3 x)^2 (3+5 x)^2}+\frac{1256 \sqrt{1-2 x}}{3 (2+3 x) (3+5 x)^2}+\frac{113875 \sqrt{1-2 x}}{6 (3+5 x)}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.118964, size = 98, normalized size = 0.57 \[ \frac{\sqrt{1-2 x} \left (5124375 x^4+13153400 x^3+12649336 x^2+5401374 x+864074\right )}{2 (3 x+2)^3 (5 x+3)^2}+\frac{785570 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{\sqrt{21}}-23115 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

(Sqrt[1 - 2*x]*(864074 + 5401374*x + 12649336*x^2 + 13153400*x^3 + 5124375*x^4))/(2*(2 + 3*x)^3*(3 + 5*x)^2) +
(785570*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/Sqrt[21] - 23115*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.015, size = 103, normalized size = 0.6 \begin{align*} -108\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{6883\, \left ( 1-2\,x \right ) ^{5/2}}{6}}-{\frac{145600\, \left ( 1-2\,x \right ) ^{3/2}}{27}}+{\frac{342265\,\sqrt{1-2\,x}}{54}} \right ) }+{\frac{785570\,\sqrt{21}}{21}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+5500\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{273\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{2981\,\sqrt{1-2\,x}}{100}} \right ) }-23115\,{\it Artanh} \left ( 1/11\,\sqrt{55}\sqrt{1-2\,x} \right ) \sqrt{55} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^3,x)

[Out]

-108*(6883/6*(1-2*x)^(5/2)-145600/27*(1-2*x)^(3/2)+342265/54*(1-2*x)^(1/2))/(-6*x-4)^3+785570/21*arctanh(1/7*2
1^(1/2)*(1-2*x)^(1/2))*21^(1/2)+5500*(-273/20*(1-2*x)^(3/2)+2981/100*(1-2*x)^(1/2))/(-10*x-6)^2-23115*arctanh(
1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 2.22993, size = 220, normalized size = 1.28 \begin{align*} \frac{23115}{2} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{392785}{21} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{5124375 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 46804300 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 160263994 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 243823580 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 139064695 \, \sqrt{-2 \, x + 1}}{675 \,{\left (2 \, x - 1\right )}^{5} + 7695 \,{\left (2 \, x - 1\right )}^{4} + 35082 \,{\left (2 \, x - 1\right )}^{3} + 79954 \,{\left (2 \, x - 1\right )}^{2} + 182182 \, x - 49588} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

23115/2*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 392785/21*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + (5124375*(-2*x + 1)^(9/2) - 46804300*(-2*x + 1)^
(7/2) + 160263994*(-2*x + 1)^(5/2) - 243823580*(-2*x + 1)^(3/2) + 139064695*sqrt(-2*x + 1))/(675*(2*x - 1)^5 +
7695*(2*x - 1)^4 + 35082*(2*x - 1)^3 + 79954*(2*x - 1)^2 + 182182*x - 49588)

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Fricas [A]  time = 1.46875, size = 536, normalized size = 3.12 \begin{align*} \frac{485415 \, \sqrt{55}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 785570 \, \sqrt{21}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (5124375 \, x^{4} + 13153400 \, x^{3} + 12649336 \, x^{2} + 5401374 \, x + 864074\right )} \sqrt{-2 \, x + 1}}{42 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/42*(485415*sqrt(55)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((5*x + sqrt(55)*sqrt(-2*x +
1) - 8)/(5*x + 3)) + 785570*sqrt(21)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log((3*x - sqrt(2
1)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(5124375*x^4 + 13153400*x^3 + 12649336*x^2 + 5401374*x + 864074)*sqrt(-
2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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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)**4/(3+5*x)**3,x)

[Out]

Timed out

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Giac [A]  time = 2.30123, size = 204, normalized size = 1.19 \begin{align*} \frac{23115}{2} \, \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{392785}{21} \, \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{55 \,{\left (1365 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 2981 \, \sqrt{-2 \, x + 1}\right )}}{4 \,{\left (5 \, x + 3\right )}^{2}} + \frac{61947 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 291200 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 342265 \, \sqrt{-2 \, x + 1}}{4 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

23115/2*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 392785/21*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 55/4*(1365*(-2*x + 1)^(3/2) -
2981*sqrt(-2*x + 1))/(5*x + 3)^2 + 1/4*(61947*(2*x - 1)^2*sqrt(-2*x + 1) - 291200*(-2*x + 1)^(3/2) + 342265*s
qrt(-2*x + 1))/(3*x + 2)^3