3.1977 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=133 \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4}+\frac{100145 \sqrt{1-2 x}}{168 (3 x+2)}+\frac{4313 \sqrt{1-2 x}}{72 (3 x+2)^2}+\frac{301 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4) + (301*Sqrt[1 - 2*x])/(36*(2 + 3*x)^3) + (4313*Sqrt[1 - 2*x])/(72*(2 + 3*
x)^2) + (100145*Sqrt[1 - 2*x])/(168*(2 + 3*x)) + (3454265*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 12
10*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0572156, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {98, 149, 151, 156, 63, 206} \[ \frac{7 (1-2 x)^{3/2}}{12 (3 x+2)^4}+\frac{100145 \sqrt{1-2 x}}{168 (3 x+2)}+\frac{4313 \sqrt{1-2 x}}{72 (3 x+2)^2}+\frac{301 \sqrt{1-2 x}}{36 (3 x+2)^3}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*(1 - 2*x)^(3/2))/(12*(2 + 3*x)^4) + (301*Sqrt[1 - 2*x])/(36*(2 + 3*x)^3) + (4313*Sqrt[1 - 2*x])/(72*(2 + 3*
x)^2) + (100145*Sqrt[1 - 2*x])/(168*(2 + 3*x)) + (3454265*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 12
10*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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^{5/2}}{(2+3 x)^5 (3+5 x)} \, dx &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{(195-159 x) \sqrt{1-2 x}}{(2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^3}-\frac{1}{108} \int \frac{-15777+21621 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{4313 \sqrt{1-2 x}}{72 (2+3 x)^2}-\frac{\int \frac{-1197315+1358595 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{1512}\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{4313 \sqrt{1-2 x}}{72 (2+3 x)^2}+\frac{100145 \sqrt{1-2 x}}{168 (2+3 x)}-\frac{\int \frac{-51509115+31545675 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{10584}\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{4313 \sqrt{1-2 x}}{72 (2+3 x)^2}+\frac{100145 \sqrt{1-2 x}}{168 (2+3 x)}-\frac{3454265}{168} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+33275 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{4313 \sqrt{1-2 x}}{72 (2+3 x)^2}+\frac{100145 \sqrt{1-2 x}}{168 (2+3 x)}+\frac{3454265}{168} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-33275 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{7 (1-2 x)^{3/2}}{12 (2+3 x)^4}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^3}+\frac{4313 \sqrt{1-2 x}}{72 (2+3 x)^2}+\frac{100145 \sqrt{1-2 x}}{168 (2+3 x)}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.101816, size = 88, normalized size = 0.66 \[ \frac{\sqrt{1-2 x} \left (2703915 x^3+5498403 x^2+3730002 x+844322\right )}{168 (3 x+2)^4}+\frac{3454265 \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )}{84 \sqrt{21}}-1210 \sqrt{55} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[1 - 2*x]*(844322 + 3730002*x + 5498403*x^2 + 2703915*x^3))/(168*(2 + 3*x)^4) + (3454265*ArcTanh[Sqrt[3/7
]*Sqrt[1 - 2*x]])/(84*Sqrt[21]) - 1210*Sqrt[55]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.011, size = 84, normalized size = 0.6 \begin{align*} -162\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{100145\, \left ( 1-2\,x \right ) ^{7/2}}{504}}-{\frac{909931\, \left ( 1-2\,x \right ) ^{5/2}}{648}}+{\frac{2144065\, \left ( 1-2\,x \right ) ^{3/2}}{648}}-{\frac{5053615\,\sqrt{1-2\,x}}{1944}} \right ) }+{\frac{3454265\,\sqrt{21}}{1764}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-1210\,{\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)^(5/2)/(2+3*x)^5/(3+5*x),x)

[Out]

-162*(100145/504*(1-2*x)^(7/2)-909931/648*(1-2*x)^(5/2)+2144065/648*(1-2*x)^(3/2)-5053615/1944*(1-2*x)^(1/2))/
(-6*x-4)^4+3454265/1764*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1210*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))
*55^(1/2)

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Maxima [A]  time = 3.97156, size = 197, normalized size = 1.48 \begin{align*} 605 \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3454265}{3528} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2703915 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 19108551 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 45025365 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 35375305 \, \sqrt{-2 \, x + 1}}{84 \,{\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)^(5/2)/(2+3*x)^5/(3+5*x),x, algorithm="maxima")

[Out]

605*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3454265/3528*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/84*(2703915*(-2*x + 1)^(7/2) - 19108551*(-2*x +
1)^(5/2) + 45025365*(-2*x + 1)^(3/2) - 35375305*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1)^3 + 2646*(2*x
- 1)^2 + 8232*x - 1715)

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Fricas [A]  time = 1.3659, size = 459, normalized size = 3.45 \begin{align*} \frac{2134440 \, \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 ) + 3454265 \, \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 (2703915 \, x^{3} + 5498403 \, x^{2} + 3730002 \, x + 844322\right )} \sqrt{-2 \, x + 1}}{3528 \,{\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)^(5/2)/(2+3*x)^5/(3+5*x),x, algorithm="fricas")

[Out]

1/3528*(2134440*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)) + 3454265*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*(2703915*x^3 + 5498403*x^2 + 3730002*x + 844322)*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)**(5/2)/(2+3*x)**5/(3+5*x),x)

[Out]

Timed out

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Giac [A]  time = 2.39223, size = 188, normalized size = 1.41 \begin{align*} 605 \, \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{3454265}{3528} \, \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{2703915 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 19108551 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 45025365 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 35375305 \, \sqrt{-2 \, x + 1}}{1344 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

605*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 3454265/3528*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 1/1344*(2703915*(2*x - 1)^3*sq
rt(-2*x + 1) + 19108551*(2*x - 1)^2*sqrt(-2*x + 1) - 45025365*(-2*x + 1)^(3/2) + 35375305*sqrt(-2*x + 1))/(3*x
+ 2)^4