3.1917 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^2} \, dx\)

Optimal. Leaf size=154 \[ -\frac{46555 \sqrt{1-2 x}}{42 (5 x+3)}+\frac{6949 \sqrt{1-2 x}}{63 (3 x+2) (5 x+3)}+\frac{133 \sqrt{1-2 x}}{18 (3 x+2)^2 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(-46555*Sqrt[1 - 2*x])/(42*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)) + (133*Sqrt[1 - 2*x])/(18*
(2 + 3*x)^2*(3 + 5*x)) + (6949*Sqrt[1 - 2*x])/(63*(2 + 3*x)*(3 + 5*x)) - (321161*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(7*Sqrt[21]) + 1350*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0590627, antiderivative size = 154, 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 = {98, 151, 156, 63, 206} \[ -\frac{46555 \sqrt{1-2 x}}{42 (5 x+3)}+\frac{6949 \sqrt{1-2 x}}{63 (3 x+2) (5 x+3)}+\frac{133 \sqrt{1-2 x}}{18 (3 x+2)^2 (5 x+3)}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-46555*Sqrt[1 - 2*x])/(42*(3 + 5*x)) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)) + (133*Sqrt[1 - 2*x])/(18*
(2 + 3*x)^2*(3 + 5*x)) + (6949*Sqrt[1 - 2*x])/(63*(2 + 3*x)*(3 + 5*x)) - (321161*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*
x]])/(7*Sqrt[21]) + 1350*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)^4 (3+5 x)^2} \, dx &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{1}{9} \int \frac{155-233 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^2} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{133 \sqrt{1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac{1}{126} \int \frac{16912-23275 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{133 \sqrt{1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac{6949 \sqrt{1-2 x}}{63 (2+3 x) (3+5 x)}+\frac{1}{882} \int \frac{1275267-1459290 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{46555 \sqrt{1-2 x}}{42 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{133 \sqrt{1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac{6949 \sqrt{1-2 x}}{63 (2+3 x) (3+5 x)}-\frac{\int \frac{52679781-32262615 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{9702}\\ &=-\frac{46555 \sqrt{1-2 x}}{42 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{133 \sqrt{1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac{6949 \sqrt{1-2 x}}{63 (2+3 x) (3+5 x)}+\frac{321161}{14} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-37125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{46555 \sqrt{1-2 x}}{42 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{133 \sqrt{1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac{6949 \sqrt{1-2 x}}{63 (2+3 x) (3+5 x)}-\frac{321161}{14} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+37125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{46555 \sqrt{1-2 x}}{42 (3+5 x)}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)}+\frac{133 \sqrt{1-2 x}}{18 (2+3 x)^2 (3+5 x)}+\frac{6949 \sqrt{1-2 x}}{63 (2+3 x) (3+5 x)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.120513, size = 95, normalized size = 0.62 \[ -\frac{\sqrt{1-2 x} \left (418995 x^3+824092 x^2+539819 x+117752\right )}{14 (3 x+2)^3 (5 x+3)}-\frac{321161 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{7 \sqrt{21}}+1350 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*(117752 + 539819*x + 824092*x^2 + 418995*x^3))/(14*(2 + 3*x)^3*(3 + 5*x)) - (321161*ArcTanh[Sq
rt[3/7]*Sqrt[1 - 2*x]])/(7*Sqrt[21]) + 1350*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.011, size = 91, normalized size = 0.6 \begin{align*} 108\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{7001\, \left ( 1-2\,x \right ) ^{5/2}}{84}}-{\frac{10603\, \left ( 1-2\,x \right ) ^{3/2}}{27}}+{\frac{49973\,\sqrt{1-2\,x}}{108}} \right ) }-{\frac{321161\,\sqrt{21}}{147}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+110\,{\frac{\sqrt{1-2\,x}}{-2\,x-6/5}}+1350\,{\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)^4/(3+5*x)^2,x)

[Out]

108*(7001/84*(1-2*x)^(5/2)-10603/27*(1-2*x)^(3/2)+49973/108*(1-2*x)^(1/2))/(-6*x-4)^3-321161/147*arctanh(1/7*2
1^(1/2)*(1-2*x)^(1/2))*21^(1/2)+110*(1-2*x)^(1/2)/(-2*x-6/5)+1350*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2
)

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Maxima [A]  time = 1.62266, size = 197, normalized size = 1.28 \begin{align*} -675 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{321161}{294} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{418995 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 2905169 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 6712629 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 5168471 \, \sqrt{-2 \, x + 1}}{7 \,{\left (135 \,{\left (2 \, x - 1\right )}^{4} + 1242 \,{\left (2 \, x - 1\right )}^{3} + 4284 \,{\left (2 \, x - 1\right )}^{2} + 13132 \, x - 2793\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-675*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 321161/294*sqrt(21)*log(-(sq
rt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/7*(418995*(-2*x + 1)^(7/2) - 2905169*(-2*x + 1)^
(5/2) + 6712629*(-2*x + 1)^(3/2) - 5168471*sqrt(-2*x + 1))/(135*(2*x - 1)^4 + 1242*(2*x - 1)^3 + 4284*(2*x - 1
)^2 + 13132*x - 2793)

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Fricas [A]  time = 1.61769, size = 459, normalized size = 2.98 \begin{align*} \frac{198450 \, \sqrt{55}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{5 \, x - \sqrt{55} \sqrt{-2 \, x + 1} - 8}{5 \, x + 3}\right ) + 321161 \, \sqrt{21}{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (\frac{3 \, x + \sqrt{21} \sqrt{-2 \, x + 1} - 5}{3 \, x + 2}\right ) - 21 \,{\left (418995 \, x^{3} + 824092 \, x^{2} + 539819 \, x + 117752\right )} \sqrt{-2 \, x + 1}}{294 \,{\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/294*(198450*sqrt(55)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((5*x - sqrt(55)*sqrt(-2*x + 1) - 8)/(5*x
+ 3)) + 321161*sqrt(21)*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log((3*x + sqrt(21)*sqrt(-2*x + 1) - 5)/(3
*x + 2)) - 21*(418995*x^3 + 824092*x^2 + 539819*x + 117752)*sqrt(-2*x + 1))/(135*x^4 + 351*x^3 + 342*x^2 + 148
*x + 24)

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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)**4/(3+5*x)**2,x)

[Out]

Timed out

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Giac [A]  time = 2.2523, size = 188, normalized size = 1.22 \begin{align*} -675 \, \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{321161}{294} \, \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{275 \, \sqrt{-2 \, x + 1}}{5 \, x + 3} - \frac{63009 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 296884 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 349811 \, \sqrt{-2 \, x + 1}}{56 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-675*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 321161/294*sqrt(21
)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 275*sqrt(-2*x + 1)/(5*x + 3) -
1/56*(63009*(2*x - 1)^2*sqrt(-2*x + 1) - 296884*(-2*x + 1)^(3/2) + 349811*sqrt(-2*x + 1))/(3*x + 2)^3