3.1854 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=121 \[ \frac{995 \sqrt{1-2 x}}{22 (5 x+3)}-\frac{15 \sqrt{1-2 x}}{2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-15*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^2) + (995*Sqrt[1 - 2*x])/(22*(3 + 5*x
)) + 624*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (6665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.0412471, antiderivative size = 121, 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 = {99, 151, 156, 63, 206} \[ \frac{995 \sqrt{1-2 x}}{22 (5 x+3)}-\frac{15 \sqrt{1-2 x}}{2 (5 x+3)^2}+\frac{\sqrt{1-2 x}}{(3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*Sqrt[1 - 2*x])/(2*(3 + 5*x)^2) + Sqrt[1 - 2*x]/((2 + 3*x)*(3 + 5*x)^2) + (995*Sqrt[1 - 2*x])/(22*(3 + 5*x
)) + 624*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]] - (6665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

Rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^(p + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (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{\sqrt{1-2 x}}{(2+3 x)^2 (3+5 x)^3} \, dx &=\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}-\int \frac{-18+25 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{1}{22} \int \frac{-1298+1485 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}-\frac{1}{242} \int \frac{-53614+32835 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}-936 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{33325}{22} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}+936 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{33325}{22} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{15 \sqrt{1-2 x}}{2 (3+5 x)^2}+\frac{\sqrt{1-2 x}}{(2+3 x) (3+5 x)^2}+\frac{995 \sqrt{1-2 x}}{22 (3+5 x)}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0902104, size = 94, normalized size = 0.78 \[ \frac{\sqrt{1-2 x} \left (14925 x^2+18410 x+5662\right )}{22 (3 x+2) (5 x+3)^2}+624 \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{6665}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(5662 + 18410*x + 14925*x^2))/(22*(2 + 3*x)*(3 + 5*x)^2) + 624*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt
[1 - 2*x]] - (6665*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Maple [A]  time = 0.012, size = 82, normalized size = 0.7 \begin{align*} -6\,{\frac{\sqrt{1-2\,x}}{-2\,x-4/3}}+{\frac{624\,\sqrt{21}}{7}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+250\,{\frac{1}{ \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{133\, \left ( 1-2\,x \right ) ^{3/2}}{110}}+{\frac{131\,\sqrt{1-2\,x}}{50}} \right ) }-{\frac{6665\,\sqrt{55}}{121}{\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)^(1/2)/(2+3*x)^2/(3+5*x)^3,x)

[Out]

-6*(1-2*x)^(1/2)/(-2*x-4/3)+624/7*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+250*(-133/110*(1-2*x)^(3/2)+131
/50*(1-2*x)^(1/2))/(-10*x-6)^2-6665/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.47884, size = 173, normalized size = 1.43 \begin{align*} \frac{6665}{242} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{312}{7} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{14925 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 66670 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 74393 \, \sqrt{-2 \, x + 1}}{11 \,{\left (75 \,{\left (2 \, x - 1\right )}^{3} + 505 \,{\left (2 \, x - 1\right )}^{2} + 2266 \, x - 286\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6665/242*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 312/7*sqrt(21)*log(-(sqr
t(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/11*(14925*(-2*x + 1)^(5/2) - 66670*(-2*x + 1)^(3/
2) + 74393*sqrt(-2*x + 1))/(75*(2*x - 1)^3 + 505*(2*x - 1)^2 + 2266*x - 286)

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Fricas [A]  time = 1.6037, size = 428, normalized size = 3.54 \begin{align*} \frac{46655 \, \sqrt{11} \sqrt{5}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 75504 \, \sqrt{7} \sqrt{3}{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (14925 \, x^{2} + 18410 \, x + 5662\right )} \sqrt{-2 \, x + 1}}{1694 \,{\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1694*(46655*sqrt(11)*sqrt(5)*(75*x^3 + 140*x^2 + 87*x + 18)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/
(5*x + 3)) + 75504*sqrt(7)*sqrt(3)*(75*x^3 + 140*x^2 + 87*x + 18)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x +
5)/(3*x + 2)) + 77*(14925*x^2 + 18410*x + 5662)*sqrt(-2*x + 1))/(75*x^3 + 140*x^2 + 87*x + 18)

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Sympy [A]  time = 85.0684, size = 474, normalized size = 3.92 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

252*Piecewise((sqrt(21)*(-log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/4 + log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/4 - 1/(4*(sq
rt(21)*sqrt(1 - 2*x)/7 + 1)) - 1/(4*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)))/147, (x <= 1/2) & (x > -2/3))) + 1360*Pie
cewise((sqrt(55)*(-log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/4 + log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/4 - 1/(4*(sqrt(55
)*sqrt(1 - 2*x)/11 + 1)) - 1/(4*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)))/605, (x <= 1/2) & (x > -3/5))) + 440*Piecewi
se((sqrt(55)*(3*log(sqrt(55)*sqrt(1 - 2*x)/11 - 1)/16 - 3*log(sqrt(55)*sqrt(1 - 2*x)/11 + 1)/16 + 3/(16*(sqrt(
55)*sqrt(1 - 2*x)/11 + 1)) + 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 + 1)**2) + 3/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)
) - 1/(16*(sqrt(55)*sqrt(1 - 2*x)/11 - 1)**2))/6655, (x <= 1/2) & (x > -3/5))) - 1854*Piecewise((-sqrt(21)*aco
th(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 < -7/3), (-sqrt(21)*atanh(sqrt(21)*sqrt(1 - 2*x)/7)/21, 2*x - 1 > -7/
3)) + 3090*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(5
5)*sqrt(1 - 2*x)/11)/55, 2*x - 1 > -11/5))

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Giac [A]  time = 1.46266, size = 166, normalized size = 1.37 \begin{align*} \frac{6665}{242} \, \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{312}{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{9 \, \sqrt{-2 \, x + 1}}{3 \, x + 2} - \frac{5 \,{\left (665 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 1441 \, \sqrt{-2 \, x + 1}\right )}}{44 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6665/242*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 312/7*sqrt(21)
*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 9*sqrt(-2*x + 1)/(3*x + 2) - 5/4
4*(665*(-2*x + 1)^(3/2) - 1441*sqrt(-2*x + 1))/(5*x + 3)^2