3.1837 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{337955 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{14555 \sqrt{1-2 x}}{1176 (3 x+2)^2}+\frac{139 \sqrt{1-2 x}}{84 (3 x+2)^3}+\frac{\sqrt{1-2 x}}{4 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

Sqrt[1 - 2*x]/(4*(2 + 3*x)^4) + (139*Sqrt[1 - 2*x])/(84*(2 + 3*x)^3) + (14555*Sqrt[1 - 2*x])/(1176*(2 + 3*x)^2
) + (337955*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (11656955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 25
0*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.058789, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{337955 \sqrt{1-2 x}}{2744 (3 x+2)}+\frac{14555 \sqrt{1-2 x}}{1176 (3 x+2)^2}+\frac{139 \sqrt{1-2 x}}{84 (3 x+2)^3}+\frac{\sqrt{1-2 x}}{4 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(4*(2 + 3*x)^4) + (139*Sqrt[1 - 2*x])/(84*(2 + 3*x)^3) + (14555*Sqrt[1 - 2*x])/(1176*(2 + 3*x)^2
) + (337955*Sqrt[1 - 2*x])/(2744*(2 + 3*x)) + (11656955*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 25
0*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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)^5 (3+5 x)} \, dx &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}-\frac{1}{4} \int \frac{-23+35 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}-\frac{1}{84} \int \frac{-2535+3475 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}-\frac{\int \frac{-192405+218325 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{1176}\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}-\frac{\int \frac{-8277405+5069325 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{8232}\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}-\frac{11656955 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2744}+6875 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}+\frac{11656955 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2744}-6875 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{\sqrt{1-2 x}}{4 (2+3 x)^4}+\frac{139 \sqrt{1-2 x}}{84 (2+3 x)^3}+\frac{14555 \sqrt{1-2 x}}{1176 (2+3 x)^2}+\frac{337955 \sqrt{1-2 x}}{2744 (2+3 x)}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.185854, size = 88, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (9124785 x^3+18555225 x^2+12587542 x+2849254\right )}{2744 (3 x+2)^4}+\frac{11656955 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1372 \sqrt{21}}-250 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(2849254 + 12587542*x + 18555225*x^2 + 9124785*x^3))/(2744*(2 + 3*x)^4) + (11656955*ArcTanh[Sqr
t[3/7]*Sqrt[1 - 2*x]])/(1372*Sqrt[21]) - 250*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.012, size = 84, normalized size = 0.6 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{337955\, \left ( 1-2\,x \right ) ^{7/2}}{8232}}-{\frac{3070705\, \left ( 1-2\,x \right ) ^{5/2}}{10584}}+{\frac{3100927\, \left ( 1-2\,x \right ) ^{3/2}}{4536}}-{\frac{116015\,\sqrt{1-2\,x}}{216}} \right ) }+{\frac{11656955\,\sqrt{21}}{28812}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-250\,{\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)^(1/2)/(2+3*x)^5/(3+5*x),x)

[Out]

-162*(337955/8232*(1-2*x)^(7/2)-3070705/10584*(1-2*x)^(5/2)+3100927/4536*(1-2*x)^(3/2)-116015/216*(1-2*x)^(1/2
))/(-6*x-4)^4+11656955/28812*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-250*arctanh(1/11*55^(1/2)*(1-2*x)^(1
/2))*55^(1/2)

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Maxima [A]  time = 1.83038, size = 197, normalized size = 1.48 \begin{align*} 125 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11656955}{57624} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{9124785 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 64484805 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 151945423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 119379435 \, \sqrt{-2 \, x + 1}}{1372 \,{\left (81 \,{\left (2 \, x - 1\right )}^{4} + 756 \,{\left (2 \, x - 1\right )}^{3} + 2646 \,{\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 11656955/57624*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/1372*(9124785*(-2*x + 1)^(7/2) - 64484805*(-2
*x + 1)^(5/2) + 151945423*(-2*x + 1)^(3/2) - 119379435*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 264
6*(2*x - 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.66639, size = 466, normalized size = 3.5 \begin{align*} \frac{7203000 \, \sqrt{55}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 11656955 \, \sqrt{21}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (9124785 \, x^{3} + 18555225 \, x^{2} + 12587542 \, x + 2849254\right )} \sqrt{-2 \, x + 1}}{57624 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/57624*(7203000*sqrt(55)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*
x + 3)) + 11656955*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(
3*x + 2)) + 21*(9124785*x^3 + 18555225*x^2 + 12587542*x + 2849254)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2
+ 96*x + 16)

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

[Out]

Timed out

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Giac [A]  time = 2.15542, size = 188, normalized size = 1.41 \begin{align*} 125 \, \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{11656955}{57624} \, \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{9124785 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 64484805 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 151945423 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 119379435 \, \sqrt{-2 \, x + 1}}{21952 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 11656955/57624*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/21952*(9124785*(2*x - 1)^3
*sqrt(-2*x + 1) + 64484805*(2*x - 1)^2*sqrt(-2*x + 1) - 151945423*(-2*x + 1)^(3/2) + 119379435*sqrt(-2*x + 1))
/(3*x + 2)^4