3.2132 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=112 \[ -\frac{2049}{9317 \sqrt{1-2 x}}+\frac{305}{242 \sqrt{1-2 x} (5 x+3)}-\frac{5}{22 \sqrt{1-2 x} (5 x+3)^2}+\frac{54}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{9975 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

-2049/(9317*Sqrt[1 - 2*x]) - 5/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 305/(242*Sqrt[1 - 2*x]*(3 + 5*x)) + (54*Sqrt[3
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (9975*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.046175, 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} \[ -\frac{2049}{9317 \sqrt{1-2 x}}+\frac{305}{242 \sqrt{1-2 x} (5 x+3)}-\frac{5}{22 \sqrt{1-2 x} (5 x+3)^2}+\frac{54}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{9975 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2049/(9317*Sqrt[1 - 2*x]) - 5/(22*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 305/(242*Sqrt[1 - 2*x]*(3 + 5*x)) + (54*Sqrt[3
/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/7 - (9975*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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+5 x)^3} \, dx &=-\frac{5}{22 \sqrt{1-2 x} (3+5 x)^2}-\frac{1}{22} \int \frac{16-75 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{5}{22 \sqrt{1-2 x} (3+5 x)^2}+\frac{305}{242 \sqrt{1-2 x} (3+5 x)}+\frac{1}{242} \int \frac{348-2745 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx\\ &=-\frac{2049}{9317 \sqrt{1-2 x}}-\frac{5}{22 \sqrt{1-2 x} (3+5 x)^2}+\frac{305}{242 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{-25692+\frac{30735 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{9317}\\ &=-\frac{2049}{9317 \sqrt{1-2 x}}-\frac{5}{22 \sqrt{1-2 x} (3+5 x)^2}+\frac{305}{242 \sqrt{1-2 x} (3+5 x)}-\frac{81}{7} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{49875 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac{2049}{9317 \sqrt{1-2 x}}-\frac{5}{22 \sqrt{1-2 x} (3+5 x)^2}+\frac{305}{242 \sqrt{1-2 x} (3+5 x)}+\frac{81}{7} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{49875 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2662}\\ &=-\frac{2049}{9317 \sqrt{1-2 x}}-\frac{5}{22 \sqrt{1-2 x} (3+5 x)^2}+\frac{305}{242 \sqrt{1-2 x} (3+5 x)}+\frac{54}{7} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{9975 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}

Mathematica [C]  time = 0.0320884, size = 73, normalized size = 0.65 \[ \frac{\frac{35 \left (3990 (5 x+3)^2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+3355 x+1892\right )}{(5 x+3)^2}-143748 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )}{18634 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-143748*Hypergeometric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] + (35*(1892 + 3355*x + 3990*(3 + 5*x)^2*Hypergeometric
2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11]))/(3 + 5*x)^2)/(18634*Sqrt[1 - 2*x])

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Maple [A]  time = 0.011, size = 75, normalized size = 0.7 \begin{align*}{\frac{54\,\sqrt{21}}{49}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{16}{9317}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{1250}{1331\, \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{59}{10} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{627}{50}\sqrt{1-2\,x}} \right ) }-{\frac{9975\,\sqrt{55}}{14641}{\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+5*x)^3,x)

[Out]

54/49*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)+16/9317/(1-2*x)^(1/2)+1250/1331*(-59/10*(1-2*x)^(3/2)+627/5
0*(1-2*x)^(1/2))/(-10*x-6)^2-9975/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.85539, size = 161, normalized size = 1.44 \begin{align*} \frac{9975}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{27}{49} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{51225 \,{\left (2 \, x - 1\right )}^{2} + 215930 \, x - 109901}{9317 \,{\left (25 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 110 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 121 \, \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+5*x)^3,x, algorithm="maxima")

[Out]

9975/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/49*sqrt(21)*log(-(s
qrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/9317*(51225*(2*x - 1)^2 + 215930*x - 109901)/(2
5*(-2*x + 1)^(5/2) - 110*(-2*x + 1)^(3/2) + 121*sqrt(-2*x + 1))

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Fricas [A]  time = 1.59035, size = 428, normalized size = 3.82 \begin{align*} \frac{488775 \, \sqrt{11} \sqrt{5}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 790614 \, \sqrt{7} \sqrt{3}{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (102450 \, x^{2} + 5515 \, x - 29338\right )} \sqrt{-2 \, x + 1}}{1434818 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\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+5*x)^3,x, algorithm="fricas")

[Out]

1/1434818*(488775*sqrt(11)*sqrt(5)*(50*x^3 + 35*x^2 - 12*x - 9)*log((sqrt(11)*sqrt(5)*sqrt(-2*x + 1) + 5*x - 8
)/(5*x + 3)) + 790614*sqrt(7)*sqrt(3)*(50*x^3 + 35*x^2 - 12*x - 9)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x
+ 5)/(3*x + 2)) + 77*(102450*x^2 + 5515*x - 29338)*sqrt(-2*x + 1))/(50*x^3 + 35*x^2 - 12*x - 9)

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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+5*x)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.2516, size = 157, normalized size = 1.4 \begin{align*} \frac{9975}{29282} \, \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{27}{49} \, \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}{9317 \, \sqrt{-2 \, x + 1}} - \frac{25 \,{\left (295 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 627 \, \sqrt{-2 \, x + 1}\right )}}{5324 \,{\left (5 \, x + 3\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+5*x)^3,x, algorithm="giac")

[Out]

9975/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 27/49*sqrt(2
1)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 16/9317/sqrt(-2*x + 1) - 25/53
24*(295*(-2*x + 1)^(3/2) - 627*sqrt(-2*x + 1))/(5*x + 3)^2