3.2114 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=132 \[ -\frac{12790}{3773 \sqrt{1-2 x}}+\frac{565}{49 \sqrt{1-2 x} (3 x+2)}+\frac{8}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{40140}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

-12790/(3773*Sqrt[1 - 2*x]) + 1/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 8/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 565/(49*Sqrt
[1 - 2*x]*(2 + 3*x)) + (40140*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*Sqrt[5/11]*ArcTanh[Sqrt[
5/11]*Sqrt[1 - 2*x]])/11

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Rubi [A]  time = 0.058251, antiderivative size = 132, 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 = {103, 151, 152, 156, 63, 206} \[ -\frac{12790}{3773 \sqrt{1-2 x}}+\frac{565}{49 \sqrt{1-2 x} (3 x+2)}+\frac{8}{7 \sqrt{1-2 x} (3 x+2)^2}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3}+\frac{40140}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-12790/(3773*Sqrt[1 - 2*x]) + 1/(7*Sqrt[1 - 2*x]*(2 + 3*x)^3) + 8/(7*Sqrt[1 - 2*x]*(2 + 3*x)^2) + 565/(49*Sqrt
[1 - 2*x]*(2 + 3*x)) + (40140*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1250*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)^4 (3+5 x)} \, dx &=\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{1}{21} \int \frac{42-105 x}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)} \, dx\\ &=\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{8}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{1}{294} \int \frac{2310-8400 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)} \, dx\\ &=\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{8}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{565}{49 \sqrt{1-2 x} (2+3 x)}+\frac{\int \frac{43680-355950 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{2058}\\ &=-\frac{12790}{3773 \sqrt{1-2 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{8}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{565}{49 \sqrt{1-2 x} (2+3 x)}-\frac{\int \frac{-3293220+2014425 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{79233}\\ &=-\frac{12790}{3773 \sqrt{1-2 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{8}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{565}{49 \sqrt{1-2 x} (2+3 x)}-\frac{60210}{343} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{3125}{11} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=-\frac{12790}{3773 \sqrt{1-2 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{8}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{565}{49 \sqrt{1-2 x} (2+3 x)}+\frac{60210}{343} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{3125}{11} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=-\frac{12790}{3773 \sqrt{1-2 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3}+\frac{8}{7 \sqrt{1-2 x} (2+3 x)^2}+\frac{565}{49 \sqrt{1-2 x} (2+3 x)}+\frac{40140}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1250}{11} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0319875, size = 85, normalized size = 0.64 \[ \frac{-441540 (3 x+2)^3 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )+428750 (3 x+2)^3 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+231 \left (1695 x^2+2316 x+793\right )}{3773 \sqrt{1-2 x} (3 x+2)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(231*(793 + 2316*x + 1695*x^2) - 441540*(2 + 3*x)^3*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] + 428750*(2
+ 3*x)^3*Hypergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11])/(3773*Sqrt[1 - 2*x]*(2 + 3*x)^3)

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Maple [A]  time = 0.013, size = 84, normalized size = 0.6 \begin{align*} -{\frac{486}{2401\, \left ( -6\,x-4 \right ) ^{3}} \left ({\frac{1357}{3} \left ( 1-2\,x \right ) ^{{\frac{5}{2}}}}-{\frac{57806}{27} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{68453}{27}\sqrt{1-2\,x}} \right ) }+{\frac{40140\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{32}{26411}{\frac{1}{\sqrt{1-2\,x}}}}-{\frac{1250\,\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)^4/(3+5*x),x)

[Out]

-486/2401*(1357/3*(1-2*x)^(5/2)-57806/27*(1-2*x)^(3/2)+68453/27*(1-2*x)^(1/2))/(-6*x-4)^3+40140/2401*arctanh(1
/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+32/26411/(1-2*x)^(1/2)-1250/121*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1
/2)

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Maxima [A]  time = 2.30102, size = 185, normalized size = 1.4 \begin{align*} \frac{625}{121} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{20070}{2401} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) + \frac{2 \,{\left (172665 \,{\left (2 \, x - 1\right )}^{3} + 817110 \,{\left (2 \, x - 1\right )}^{2} + 1934226 \, x - 967897\right )}}{3773 \,{\left (27 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 189 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 441 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 343 \, \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)^4/(3+5*x),x, algorithm="maxima")

[Out]

625/121*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 20070/2401*sqrt(21)*log(-
(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 2/3773*(172665*(2*x - 1)^3 + 817110*(2*x - 1)^2
+ 1934226*x - 967897)/(27*(-2*x + 1)^(7/2) - 189*(-2*x + 1)^(5/2) + 441*(-2*x + 1)^(3/2) - 343*sqrt(-2*x + 1)
)

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Fricas [A]  time = 1.63094, size = 485, normalized size = 3.67 \begin{align*} \frac{1500625 \, \sqrt{11} \sqrt{5}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 2428470 \, \sqrt{7} \sqrt{3}{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (345330 \, x^{3} + 299115 \, x^{2} - 74556 \, x - 80863\right )} \sqrt{-2 \, x + 1}}{290521 \,{\left (54 \, x^{4} + 81 \, x^{3} + 18 \, x^{2} - 20 \, x - 8\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/290521*(1500625*sqrt(11)*sqrt(5)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1)
+ 5*x - 8)/(5*x + 3)) + 2428470*sqrt(7)*sqrt(3)*(54*x^4 + 81*x^3 + 18*x^2 - 20*x - 8)*log(-(sqrt(7)*sqrt(3)*sq
rt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(345330*x^3 + 299115*x^2 - 74556*x - 80863)*sqrt(-2*x + 1))/(54*x^4 +
81*x^3 + 18*x^2 - 20*x - 8)

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

[Out]

Exception raised: ValueError

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Giac [A]  time = 1.63166, size = 178, normalized size = 1.35 \begin{align*} \frac{625}{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{20070}{2401} \, \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{32}{26411 \, \sqrt{-2 \, x + 1}} + \frac{9 \,{\left (12213 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 57806 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 68453 \, \sqrt{-2 \, x + 1}\right )}}{9604 \,{\left (3 \, x + 2\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

625/121*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 20070/2401*sqrt
(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 32/26411/sqrt(-2*x + 1) + 9/
9604*(12213*(2*x - 1)^2*sqrt(-2*x + 1) - 57806*(-2*x + 1)^(3/2) + 68453*sqrt(-2*x + 1))/(3*x + 2)^3