3.2046 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^5 (3+5 x)} \, dx\)

Optimal. Leaf size=137 \[ \frac{3135 \sqrt{1-2 x}}{56 (3 x+2)}+\frac{45 \sqrt{1-2 x}}{8 (3 x+2)^2}+\frac{3 \sqrt{1-2 x}}{4 (3 x+2)^3}+\frac{3 \sqrt{1-2 x}}{28 (3 x+2)^4}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

(3*Sqrt[1 - 2*x])/(28*(2 + 3*x)^4) + (3*Sqrt[1 - 2*x])/(4*(2 + 3*x)^3) + (45*Sqrt[1 - 2*x])/(8*(2 + 3*x)^2) +
(3135*Sqrt[1 - 2*x])/(56*(2 + 3*x)) + (36045*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Rubi [A]  time = 0.0566626, antiderivative size = 137, 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 = {103, 151, 156, 63, 206} \[ \frac{3135 \sqrt{1-2 x}}{56 (3 x+2)}+\frac{45 \sqrt{1-2 x}}{8 (3 x+2)^2}+\frac{3 \sqrt{1-2 x}}{4 (3 x+2)^3}+\frac{3 \sqrt{1-2 x}}{28 (3 x+2)^4}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 - 2*x])/(28*(2 + 3*x)^4) + (3*Sqrt[1 - 2*x])/(4*(2 + 3*x)^3) + (45*Sqrt[1 - 2*x])/(8*(2 + 3*x)^2) +
(3135*Sqrt[1 - 2*x])/(56*(2 + 3*x)) + (36045*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*
ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

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}{\sqrt{1-2 x} (2+3 x)^5 (3+5 x)} \, dx &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{1}{28} \int \frac{77-105 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{3 \sqrt{1-2 x}}{4 (2+3 x)^3}+\frac{1}{588} \int \frac{8085-11025 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{3 \sqrt{1-2 x}}{4 (2+3 x)^3}+\frac{45 \sqrt{1-2 x}}{8 (2+3 x)^2}+\frac{\int \frac{612255-694575 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)} \, dx}{8232}\\ &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{3 \sqrt{1-2 x}}{4 (2+3 x)^3}+\frac{45 \sqrt{1-2 x}}{8 (2+3 x)^2}+\frac{3135 \sqrt{1-2 x}}{56 (2+3 x)}+\frac{\int \frac{26337255-16129575 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{57624}\\ &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{3 \sqrt{1-2 x}}{4 (2+3 x)^3}+\frac{45 \sqrt{1-2 x}}{8 (2+3 x)^2}+\frac{3135 \sqrt{1-2 x}}{56 (2+3 x)}-\frac{108135}{56} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+3125 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{3 \sqrt{1-2 x}}{4 (2+3 x)^3}+\frac{45 \sqrt{1-2 x}}{8 (2+3 x)^2}+\frac{3135 \sqrt{1-2 x}}{56 (2+3 x)}+\frac{108135}{56} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-3125 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{3 \sqrt{1-2 x}}{28 (2+3 x)^4}+\frac{3 \sqrt{1-2 x}}{4 (2+3 x)^3}+\frac{45 \sqrt{1-2 x}}{8 (2+3 x)^2}+\frac{3135 \sqrt{1-2 x}}{56 (2+3 x)}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [A]  time = 0.0883101, size = 92, normalized size = 0.67 \[ \frac{3 \sqrt{1-2 x} \left (28215 x^3+57375 x^2+38922 x+8810\right )}{56 (3 x+2)^4}+\frac{36045}{28} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-1250 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 - 2*x]*(8810 + 38922*x + 57375*x^2 + 28215*x^3))/(56*(2 + 3*x)^4) + (36045*Sqrt[3/7]*ArcTanh[Sqrt[3/
7]*Sqrt[1 - 2*x]])/28 - 1250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]]

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Maple [A]  time = 0.01, size = 84, normalized size = 0.6 \begin{align*} -486\,{\frac{1}{ \left ( -6\,x-4 \right ) ^{4}} \left ({\frac{1045\, \left ( 1-2\,x \right ) ^{7/2}}{168}}-{\frac{1055\, \left ( 1-2\,x \right ) ^{5/2}}{24}}+{\frac{22373\, \left ( 1-2\,x \right ) ^{3/2}}{216}}-{\frac{369133\,\sqrt{1-2\,x}}{4536}} \right ) }+{\frac{36045\,\sqrt{21}}{196}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }-{\frac{1250\,\sqrt{55}}{11}{\it Artanh} \left ({\frac{\sqrt{55}}{11}\sqrt{1-2\,x}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*x)^5/(3+5*x)/(1-2*x)^(1/2),x)

[Out]

-486*(1045/168*(1-2*x)^(7/2)-1055/24*(1-2*x)^(5/2)+22373/216*(1-2*x)^(3/2)-369133/4536*(1-2*x)^(1/2))/(-6*x-4)
^4+36045/196*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-1250/11*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2
)

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Maxima [A]  time = 2.61335, size = 197, normalized size = 1.44 \begin{align*} \frac{625}{11} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{36045}{392} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{3 \,{\left (28215 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 199395 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 469833 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 369133 \, \sqrt{-2 \, x + 1}\right )}}{28 \,{\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+3*x)^5/(3+5*x)/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

625/11*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 36045/392*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 3/28*(28215*(-2*x + 1)^(7/2) - 199395*(-2*x + 1)^
(5/2) + 469833*(-2*x + 1)^(3/2) - 369133*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 = 1.6424, size = 489, normalized size = 3.57 \begin{align*} \frac{245000 \, \sqrt{11} \sqrt{5}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 396495 \, \sqrt{7} \sqrt{3}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 231 \,{\left (28215 \, x^{3} + 57375 \, x^{2} + 38922 \, x + 8810\right )} \sqrt{-2 \, x + 1}}{4312 \,{\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+3*x)^5/(3+5*x)/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/4312*(245000*sqrt(11)*sqrt(5)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1)
+ 5*x - 8)/(5*x + 3)) + 396495*sqrt(7)*sqrt(3)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log(-(sqrt(7)*sqrt(3)*
sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 231*(28215*x^3 + 57375*x^2 + 38922*x + 8810)*sqrt(-2*x + 1))/(81*x^4 +
216*x^3 + 216*x^2 + 96*x + 16)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.54075, size = 188, normalized size = 1.37 \begin{align*} \frac{625}{11} \, \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{36045}{392} \, \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{3 \,{\left (28215 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 199395 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 469833 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 369133 \, \sqrt{-2 \, x + 1}\right )}}{448 \,{\left (3 \, x + 2\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/11*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 36045/392*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 3/448*(28215*(2*x - 1)^3*sqrt(
-2*x + 1) + 199395*(2*x - 1)^2*sqrt(-2*x + 1) - 469833*(-2*x + 1)^(3/2) + 369133*sqrt(-2*x + 1))/(3*x + 2)^4