3.1907 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx\)

Optimal. Leaf size=153 \[ \frac{2788127 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{120077 \sqrt{1-2 x}}{882 (3 x+2)^2}+\frac{5732 \sqrt{1-2 x}}{315 (3 x+2)^3}+\frac{41 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{7 \sqrt{1-2 x}}{15 (3 x+2)^5}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*Sqrt[1 - 2*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (5732*Sqrt[1 - 2*x])/(315*(2 + 3*x)
^3) + (120077*Sqrt[1 - 2*x])/(882*(2 + 3*x)^2) + (2788127*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (96169877*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21]) - 2750*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0723188, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {98, 151, 156, 63, 206} \[ \frac{2788127 \sqrt{1-2 x}}{2058 (3 x+2)}+\frac{120077 \sqrt{1-2 x}}{882 (3 x+2)^2}+\frac{5732 \sqrt{1-2 x}}{315 (3 x+2)^3}+\frac{41 \sqrt{1-2 x}}{15 (3 x+2)^4}+\frac{7 \sqrt{1-2 x}}{15 (3 x+2)^5}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(7*Sqrt[1 - 2*x])/(15*(2 + 3*x)^5) + (41*Sqrt[1 - 2*x])/(15*(2 + 3*x)^4) + (5732*Sqrt[1 - 2*x])/(315*(2 + 3*x)
^3) + (120077*Sqrt[1 - 2*x])/(882*(2 + 3*x)^2) + (2788127*Sqrt[1 - 2*x])/(2058*(2 + 3*x)) + (96169877*ArcTanh[
Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21]) - 2750*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 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)^{3/2}}{(2+3 x)^6 (3+5 x)} \, dx &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{186-295 x}{\sqrt{1-2 x} (2+3 x)^5 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{1}{420} \int \frac{26712-40180 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{5732 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{\int \frac{2928660-4012400 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx}{8820}\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{5732 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{120077 \sqrt{1-2 x}}{882 (2+3 x)^2}+\frac{\int \frac{222229980-252161700 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{123480}\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{5732 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{120077 \sqrt{1-2 x}}{882 (2+3 x)^2}+\frac{2788127 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{\int \frac{9560404980-5855066700 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{864360}\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{5732 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{120077 \sqrt{1-2 x}}{882 (2+3 x)^2}+\frac{2788127 \sqrt{1-2 x}}{2058 (2+3 x)}-\frac{96169877 \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx}{2058}+75625 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{5732 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{120077 \sqrt{1-2 x}}{882 (2+3 x)^2}+\frac{2788127 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{96169877 \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2058}-75625 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 \sqrt{1-2 x}}{15 (2+3 x)^5}+\frac{41 \sqrt{1-2 x}}{15 (2+3 x)^4}+\frac{5732 \sqrt{1-2 x}}{315 (2+3 x)^3}+\frac{120077 \sqrt{1-2 x}}{882 (2+3 x)^2}+\frac{2788127 \sqrt{1-2 x}}{2058 (2+3 x)}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.110522, size = 93, normalized size = 0.61 \[ \frac{\sqrt{1-2 x} \left (1129191435 x^4+3049001415 x^3+3088510878 x^2+1391064622 x+235067382\right )}{10290 (3 x+2)^5}+\frac{96169877 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{1029 \sqrt{21}}-2750 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^6*(3 + 5*x)),x]

[Out]

(Sqrt[1 - 2*x]*(235067382 + 1391064622*x + 3088510878*x^2 + 3049001415*x^3 + 1129191435*x^4))/(10290*(2 + 3*x)
^5) + (96169877*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(1029*Sqrt[21]) - 2750*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 -
2*x]]

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Maple [A]  time = 0.011, size = 93, normalized size = 0.6 \begin{align*} -486\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{5}} \left ({\frac{2788127\, \left ( 1-2\,x \right ) ^{9/2}}{6174}}-{\frac{2406977\, \left ( 1-2\,x \right ) ^{7/2}}{567}}+{\frac{127289798\, \left ( 1-2\,x \right ) ^{5/2}}{8505}}-{\frac{17098361\, \left ( 1-2\,x \right ) ^{3/2}}{729}}+{\frac{20099611\,\sqrt{1-2\,x}}{1458}} \right ) }+{\frac{96169877\,\sqrt{21}}{21609}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-2750\,{\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)^(3/2)/(2+3*x)^6/(3+5*x),x)

[Out]

-486*(2788127/6174*(1-2*x)^(9/2)-2406977/567*(1-2*x)^(7/2)[phone]/8505*(1-2*x)^(5/2)-17098361/729*(1-2*x)^(
3/2)[phone]/1458*(1-2*x)^(1/2))/(-6*x-4)^5+96169877/21609*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2750*
arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 3.66781, size = 221, normalized size = 1.44 \begin{align*} 1375 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{96169877}{43218} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{1129191435 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 10614768570 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 37423200612 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 58647378230 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 34470832865 \, \sqrt{-2 \, x + 1}}{5145 \,{\left (243 \,{\left (2 \, x - 1\right )}^{5} + 2835 \,{\left (2 \, x - 1\right )}^{4} + 13230 \,{\left (2 \, x - 1\right )}^{3} + 30870 \,{\left (2 \, x - 1\right )}^{2} + 72030 \, x - 19208\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x),x, algorithm="maxima")

[Out]

1375*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 96169877/43218*sqrt(21)*log(
-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/5145*(1129191435*(-2*x + 1)^(9/2) - 10614768
570*(-2*x + 1)^(7/2) + 37423200612*(-2*x + 1)^(5/2) - 58647378230*(-2*x + 1)^(3/2) + 34470832865*sqrt(-2*x + 1
))/(243*(2*x - 1)^5 + 2835*(2*x - 1)^4 + 13230*(2*x - 1)^3 + 30870*(2*x - 1)^2 + 72030*x - 19208)

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Fricas [A]  time = 1.63132, size = 559, normalized size = 3.65 \begin{align*} \frac{297123750 \, \sqrt{55}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{5 \, x + \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 480849385 \, \sqrt{21}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \log \left (\frac{3 \, x - \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \,{\left (1129191435 \, x^{4} + 3049001415 \, x^{3} + 3088510878 \, x^{2} + 1391064622 \, x + 235067382\right )} \sqrt{-2 \, x + 1}}{216090 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x),x, algorithm="fricas")

[Out]

1/216090*(297123750*sqrt(55)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((5*x + sqrt(55)*sqrt(-2
*x + 1) - 8)/(5*x + 3)) + 480849385*sqrt(21)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*log((3*x -
sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1129191435*x^4 + 3049001415*x^3 + 3088510878*x^2 + 1391064622*x
+ 235067382)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**(3/2)/(2+3*x)**6/(3+5*x),x)

[Out]

Timed out

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Giac [A]  time = 2.21302, size = 209, normalized size = 1.37 \begin{align*} 1375 \, \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{96169877}{43218} \, \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{1129191435 \,{\left (2 \, x - 1\right )}^{4} \sqrt{-2 \, x + 1} + 10614768570 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 37423200612 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 58647378230 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 34470832865 \, \sqrt{-2 \, x + 1}}{164640 \,{\left (3 \, x + 2\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^6/(3+5*x),x, algorithm="giac")

[Out]

1375*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 96169877/43218*sqr
t(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/164640*(1129191435*(2*x -
1)^4*sqrt(-2*x + 1) + 10614768570*(2*x - 1)^3*sqrt(-2*x + 1) + 37423200612*(2*x - 1)^2*sqrt(-2*x + 1) - 58647
378230*(-2*x + 1)^(3/2) + 34470832865*sqrt(-2*x + 1))/(3*x + 2)^5