3.2113 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2525}{3773 \sqrt{1-2 x}}+\frac{225}{98 \sqrt{1-2 x} (3 x+2)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{8025}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-2525/(3773*Sqrt[1 - 2*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 225/(98*Sqrt[1 - 2*x]*(2 + 3*x)) + (8025*Sqrt[
3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.0456548, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {103, 151, 152, 156, 63, 206} \[ -\frac{2525}{3773 \sqrt{1-2 x}}+\frac{225}{98 \sqrt{1-2 x} (3 x+2)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2}+\frac{8025}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2525/(3773*Sqrt[1 - 2*x]) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 225/(98*Sqrt[1 - 2*x]*(2 + 3*x)) + (8025*Sqrt[
3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (250*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/11

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 152

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] && IntegersQ[2*m, 2*n, 2*p]

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}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{1}{14} \int \frac{25-75 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{225}{98 \sqrt{1-2 x} (2+3 x)}+\frac{1}{98} \int \frac{425-3375 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{2525}{3773 \sqrt{1-2 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{225}{98 \sqrt{1-2 x} (2+3 x)}-\frac{\int \frac{-\frac{63025}{2}+\frac{37875 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{3773}\\ &=-\frac{2525}{3773 \sqrt{1-2 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{225}{98 \sqrt{1-2 x} (2+3 x)}-\frac{24075}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{625}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{2525}{3773 \sqrt{1-2 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{225}{98 \sqrt{1-2 x} (2+3 x)}+\frac{24075}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{625}{11} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{2525}{3773 \sqrt{1-2 x}}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2}+\frac{225}{98 \sqrt{1-2 x} (2+3 x)}+\frac{8025}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0272322, size = 80, normalized size = 0.71 \[ \frac{7 \left (24500 (3 x+2)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+7425 x+5181\right )-176550 (3 x+2)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{7546 \sqrt{1-2 x} (3 x+2)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-176550*(2 + 3*x)^2*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] + 7*(5181 + 7425*x + 24500*(2 + 3*x)^2*Hyp
ergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11]))/(7546*Sqrt[1 - 2*x]*(2 + 3*x)^2)

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Maple [A]  time = 0.011, size = 75, normalized size = 0.7 \begin{align*} -{\frac{486}{343\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{77}{18} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{553}{54}\sqrt{1-2\,x}} \right ) }+{\frac{8025\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{16}{3773}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{250\,\sqrt{55}}{121}{\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/(1-2*x)^(3/2)/(2+3*x)^3/(3+5*x),x)

[Out]

-486/343*(77/18*(1-2*x)^(3/2)-553/54*(1-2*x)^(1/2))/(-6*x-4)^2+8025/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)+16/3773/(1-2*x)^(1/2)-250/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.69116, size = 161, normalized size = 1.44 \begin{align*} \frac{125}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{8025}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{22725 \,{\left (2 \, x - 1\right )}^{2} + 108150 \, x - 54859}{3773 \,{\left (9 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 42 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 49 \, \sqrt{-2 \, x + 1}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8025/4802*sqrt(21)*log(-(
sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/3773*(22725*(2*x - 1)^2 + 108150*x - 54859)/(9
*(-2*x + 1)^(5/2) - 42*(-2*x + 1)^(3/2) + 49*sqrt(-2*x + 1))

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Fricas [A]  time = 1.67523, size = 421, normalized size = 3.76 \begin{align*} \frac{600250 \, \sqrt{11} \sqrt{5}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 971025 \, \sqrt{7} \sqrt{3}{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (45450 \, x^{2} + 8625 \, x - 16067\right )} \sqrt{-2 \, x + 1}}{581042 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/581042*(600250*sqrt(11)*sqrt(5)*(18*x^3 + 15*x^2 - 4*x - 4)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8)/
(5*x + 3)) + 971025*sqrt(7)*sqrt(3)*(18*x^3 + 15*x^2 - 4*x - 4)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5
)/(3*x + 2)) + 77*(45450*x^2 + 8625*x - 16067)*sqrt(-2*x + 1))/(18*x^3 + 15*x^2 - 4*x - 4)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.12663, size = 157, normalized size = 1.4 \begin{align*} \frac{125}{121} \, \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{8025}{4802} \, \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{16}{3773 \, \sqrt{-2 \, x + 1}} - \frac{9 \,{\left (33 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 79 \, \sqrt{-2 \, x + 1}\right )}}{196 \,{\left (3 \, x + 2\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

125/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 8025/4802*sqrt(
21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/3773/sqrt(-2*x + 1) - 9/19
6*(33*(-2*x + 1)^(3/2) - 79*sqrt(-2*x + 1))/(3*x + 2)^2