3.1836 \(\int \frac{\sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=113 \[ \frac{1207 \sqrt{1-2 x}}{49 (3 x+2)}+\frac{52 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

Sqrt[1 - 2*x]/(3*(2 + 3*x)^3) + (52*Sqrt[1 - 2*x])/(21*(2 + 3*x)^2) + (1207*Sqrt[1 - 2*x])/(49*(2 + 3*x)) + (8
3264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0446905, antiderivative size = 113, 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} \[ \frac{1207 \sqrt{1-2 x}}{49 (3 x+2)}+\frac{52 \sqrt{1-2 x}}{21 (3 x+2)^2}+\frac{\sqrt{1-2 x}}{3 (3 x+2)^3}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - 2*x]/(3*(2 + 3*x)^3) + (52*Sqrt[1 - 2*x])/(21*(2 + 3*x)^2) + (1207*Sqrt[1 - 2*x])/(49*(2 + 3*x)) + (8
3264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*Sqrt[21]) - 50*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)^4 (3+5 x)} \, dx &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3}-\frac{1}{3} \int \frac{-18+25 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3}+\frac{52 \sqrt{1-2 x}}{21 (2+3 x)^2}-\frac{1}{42} \int \frac{-1374+1560 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3}+\frac{52 \sqrt{1-2 x}}{21 (2+3 x)^2}+\frac{1207 \sqrt{1-2 x}}{49 (2+3 x)}-\frac{1}{294} \int \frac{-59124+36210 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3}+\frac{52 \sqrt{1-2 x}}{21 (2+3 x)^2}+\frac{1207 \sqrt{1-2 x}}{49 (2+3 x)}-\frac{41632}{49} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+1375 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{\sqrt{1-2 x}}{3 (2+3 x)^3}+\frac{52 \sqrt{1-2 x}}{21 (2+3 x)^2}+\frac{1207 \sqrt{1-2 x}}{49 (2+3 x)}+\frac{41632}{49} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-1375 \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}}{3 (2+3 x)^3}+\frac{52 \sqrt{1-2 x}}{21 (2+3 x)^2}+\frac{1207 \sqrt{1-2 x}}{49 (2+3 x)}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0828374, size = 83, normalized size = 0.73 \[ \frac{\sqrt{1-2 x} \left (10863 x^2+14848 x+5087\right )}{49 (3 x+2)^3}+\frac{83264 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{49 \sqrt{21}}-50 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(5087 + 14848*x + 10863*x^2))/(49*(2 + 3*x)^3) + (83264*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(49*S
qrt[21]) - 50*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.012, size = 75, normalized size = 0.7 \begin{align*} -54\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1207\, \left ( 1-2\,x \right ) ^{5/2}}{147}}-{\frac{7346\, \left ( 1-2\,x \right ) ^{3/2}}{189}}+{\frac{1243\,\sqrt{1-2\,x}}{27}} \right ) }+{\frac{83264\,\sqrt{21}}{1029}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-50\,{\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)^4/(3+5*x),x)

[Out]

-54*(1207/147*(1-2*x)^(5/2)-7346/189*(1-2*x)^(3/2)+1243/27*(1-2*x)^(1/2))/(-6*x-4)^3+83264/1029*arctanh(1/7*21
^(1/2)*(1-2*x)^(1/2))*21^(1/2)-50*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 3.56564, size = 173, normalized size = 1.53 \begin{align*} 25 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{41632}{1029} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (10863 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 51422 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 60907 \, \sqrt{-2 \, x + 1}\right )}}{49 \,{\left (27 \,{\left (2 \, x - 1\right )}^{3} + 189 \,{\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 41632/1029*sqrt(21)*log(-(sqrt
(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/49*(10863*(-2*x + 1)^(5/2) - 51422*(-2*x + 1)^(3/2
) + 60907*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x - 1)^2 + 882*x - 98)

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Fricas [A]  time = 1.70025, size = 378, normalized size = 3.35 \begin{align*} \frac{25725 \, \sqrt{55}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 41632 \, \sqrt{21}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (10863 \, x^{2} + 14848 \, x + 5087\right )} \sqrt{-2 \, x + 1}}{1029 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1029*(25725*sqrt(55)*(27*x^3 + 54*x^2 + 36*x + 8)*log((5*x + sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x + 3)) + 41632
*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(10863*x^2 + 14
848*x + 5087)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)

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Sympy [A]  time = 116.878, size = 566, normalized size = 5.01 \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)**4/(3+5*x),x)

[Out]

660*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))) - 264*Piec
ewise((sqrt(21)*(3*log(sqrt(21)*sqrt(1 - 2*x)/7 - 1)/16 - 3*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/16 + 3/(16*(sqrt
(21)*sqrt(1 - 2*x)/7 + 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) + 3/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1))
- 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2))/1029, (x <= 1/2) & (x > -2/3))) + 112*Piecewise((sqrt(21)*(-5*log(
sqrt(21)*sqrt(1 - 2*x)/7 - 1)/32 + 5*log(sqrt(21)*sqrt(1 - 2*x)/7 + 1)/32 - 5/(32*(sqrt(21)*sqrt(1 - 2*x)/7 +
1)) - 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 + 1)**3) - 5/(32*(sqrt(21)*sq
rt(1 - 2*x)/7 - 1)) + 1/(16*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**2) - 1/(48*(sqrt(21)*sqrt(1 - 2*x)/7 - 1)**3))/720
3, (x <= 1/2) & (x > -2/3))) - 1650*Piecewise((-sqrt(21)*acoth(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)) + 2750*Piecewise((-sqrt(55)*acoth(sqrt(55)*sqr
t(1 - 2*x)/11)/55, 2*x - 1 < -11/5), (-sqrt(55)*atanh(sqrt(55)*sqrt(1 - 2*x)/11)/55, 2*x - 1 > -11/5))

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Giac [A]  time = 2.73274, size = 166, normalized size = 1.47 \begin{align*} 25 \, \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{41632}{1029} \, \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{10863 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 51422 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 60907 \, \sqrt{-2 \, x + 1}}{196 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

25*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 41632/1029*sqrt(21)*
log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/196*(10863*(2*x - 1)^2*sqrt(-2*
x + 1) - 51422*(-2*x + 1)^(3/2) + 60907*sqrt(-2*x + 1))/(3*x + 2)^3