3.1998 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=145 \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}+\frac{2311 \sqrt{1-2 x}}{5 x+3}+\frac{931 \sqrt{1-2 x}}{18 (3 x+2) (5 x+3)^2}-\frac{6899 \sqrt{1-2 x}}{18 (5 x+3)^2}+4555 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-14073 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-6899*Sqrt[1 - 2*x])/(18*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5*x)^2) + (931*Sqrt[1 - 2*x])
/(18*(2 + 3*x)*(3 + 5*x)^2) + (2311*Sqrt[1 - 2*x])/(3 + 5*x) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
- 14073*Sqrt[11/5]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0575274, antiderivative size = 145, 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 = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{6 (3 x+2)^2 (5 x+3)^2}+\frac{2311 \sqrt{1-2 x}}{5 x+3}+\frac{931 \sqrt{1-2 x}}{18 (3 x+2) (5 x+3)^2}-\frac{6899 \sqrt{1-2 x}}{18 (5 x+3)^2}+4555 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-14073 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6899*Sqrt[1 - 2*x])/(18*(3 + 5*x)^2) + (7*(1 - 2*x)^(3/2))/(6*(2 + 3*x)^2*(3 + 5*x)^2) + (931*Sqrt[1 - 2*x])
/(18*(2 + 3*x)*(3 + 5*x)^2) + (2311*Sqrt[1 - 2*x])/(3 + 5*x) + 4555*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]]
- 14073*Sqrt[11/5]*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)^3 (3+5 x)^3} \, dx &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{1}{6} \int \frac{(199-167 x) \sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{931 \sqrt{1-2 x}}{18 (2+3 x) (3+5 x)^2}-\frac{1}{18} \int \frac{-16591+22941 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{6899 \sqrt{1-2 x}}{18 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{931 \sqrt{1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac{1}{396} \int \frac{-1193742+1366002 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{6899 \sqrt{1-2 x}}{18 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{931 \sqrt{1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac{2311 \sqrt{1-2 x}}{3+5 x}-\frac{\int \frac{-49312098+30200148 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{4356}\\ &=-\frac{6899 \sqrt{1-2 x}}{18 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{931 \sqrt{1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac{2311 \sqrt{1-2 x}}{3+5 x}-\frac{95655}{2} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{154803}{2} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{6899 \sqrt{1-2 x}}{18 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{931 \sqrt{1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac{2311 \sqrt{1-2 x}}{3+5 x}+\frac{95655}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{154803}{2} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{6899 \sqrt{1-2 x}}{18 (3+5 x)^2}+\frac{7 (1-2 x)^{3/2}}{6 (2+3 x)^2 (3+5 x)^2}+\frac{931 \sqrt{1-2 x}}{18 (2+3 x) (3+5 x)^2}+\frac{2311 \sqrt{1-2 x}}{3+5 x}+4555 \sqrt{21} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-14073 \sqrt{\frac{11}{5}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0770248, size = 119, normalized size = 0.82 \[ \frac{5 \sqrt{1-2 x} \left (207990 x^3+395215 x^2+249939 x+52607\right )+45550 \sqrt{21} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-28146 \sqrt{55} \left (15 x^2+19 x+6\right )^2 \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{10 (3 x+2)^2 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[1 - 2*x]*(52607 + 249939*x + 395215*x^2 + 207990*x^3) + 45550*Sqrt[21]*(6 + 19*x + 15*x^2)^2*ArcTanh[S
qrt[3/7]*Sqrt[1 - 2*x]] - 28146*Sqrt[55]*(6 + 19*x + 15*x^2)^2*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/(10*(2 + 3*x
)^2*(3 + 5*x)^2)

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Maple [A]  time = 0.013, size = 94, normalized size = 0.7 \begin{align*} -252\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{67\, \left ( 1-2\,x \right ) ^{3/2}}{4}}-{\frac{1421\,\sqrt{1-2\,x}}{36}} \right ) }+4555\,{\it Artanh} \left ( 1/7\,\sqrt{21}\sqrt{1-2\,x} \right ) \sqrt{21}+1100\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{207\, \left ( 1-2\,x \right ) ^{3/2}}{20}}+{\frac{451\,\sqrt{1-2\,x}}{20}} \right ) }-{\frac{14073\,\sqrt{55}}{5}{\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)^3/(3+5*x)^3,x)

[Out]

-252*(67/4*(1-2*x)^(3/2)-1421/36*(1-2*x)^(1/2))/(-6*x-4)^2+4555*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+1
100*(-207/20*(1-2*x)^(3/2)+451/20*(1-2*x)^(1/2))/(-10*x-6)^2-14073/5*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(
1/2)

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Maxima [A]  time = 4.23579, size = 197, normalized size = 1.36 \begin{align*} \frac{14073}{10} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{4555}{2} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (103995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 707200 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 1602293 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1209516 \, \sqrt{-2 \, x + 1}\right )}}{225 \,{\left (2 \, x - 1\right )}^{4} + 2040 \,{\left (2 \, x - 1\right )}^{3} + 6934 \,{\left (2 \, x - 1\right )}^{2} + 20944 \, x - 4543} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14073/10*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4555/2*sqrt(21)*log(-(sq
rt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2*(103995*(-2*x + 1)^(7/2) - 707200*(-2*x + 1)^(5/
2) + 1602293*(-2*x + 1)^(3/2) - 1209516*sqrt(-2*x + 1))/(225*(2*x - 1)^4 + 2040*(2*x - 1)^3 + 6934*(2*x - 1)^2
+ 20944*x - 4543)

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Fricas [A]  time = 1.48173, size = 474, normalized size = 3.27 \begin{align*} \frac{14073 \, \sqrt{11} \sqrt{5}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 22775 \, \sqrt{21}{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 5 \,{\left (207990 \, x^{3} + 395215 \, x^{2} + 249939 \, x + 52607\right )} \sqrt{-2 \, x + 1}}{10 \,{\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(14073*sqrt(11)*sqrt(5)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) +
5*x - 8)/(5*x + 3)) + 22775*sqrt(21)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*log((3*x - sqrt(21)*sqrt(-2*x
+ 1) - 5)/(3*x + 2)) + 5*(207990*x^3 + 395215*x^2 + 249939*x + 52607)*sqrt(-2*x + 1))/(225*x^4 + 570*x^3 + 54
1*x^2 + 228*x + 36)

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

[Out]

Timed out

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Giac [A]  time = 2.13147, size = 200, normalized size = 1.38 \begin{align*} \frac{14073}{10} \, \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{4555}{2} \, \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{2 \,{\left (103995 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 707200 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 1602293 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 1209516 \, \sqrt{-2 \, x + 1}\right )}}{{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

14073/10*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 4555/2*sqrt(21
)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2*(103995*(2*x - 1)^3*sqrt(-2*x
+ 1) + 707200*(2*x - 1)^2*sqrt(-2*x + 1) - 1602293*(-2*x + 1)^(3/2) + 1209516*sqrt(-2*x + 1))/(15*(2*x - 1)^2
+ 136*x + 9)^2