3.2134 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=166 \[ -\frac{15987390}{456533 \sqrt{1-2 x}}+\frac{1176400}{5929 \sqrt{1-2 x} (5 x+3)}-\frac{35825}{1078 \sqrt{1-2 x} (5 x+3)^2}+\frac{435}{98 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

[Out]

-15987390/(456533*Sqrt[1 - 2*x]) - 35825/(1078*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3
+ 5*x)^2) + 435/(98*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2) + 1176400/(5929*Sqrt[1 - 2*x]*(3 + 5*x)) + (414315*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1561125*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/1331

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Rubi [A]  time = 0.0730113, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ -\frac{15987390}{456533 \sqrt{1-2 x}}+\frac{1176400}{5929 \sqrt{1-2 x} (5 x+3)}-\frac{35825}{1078 \sqrt{1-2 x} (5 x+3)^2}+\frac{435}{98 \sqrt{1-2 x} (3 x+2) (5 x+3)^2}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)^2}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-15987390/(456533*Sqrt[1 - 2*x]) - 35825/(1078*Sqrt[1 - 2*x]*(3 + 5*x)^2) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3
+ 5*x)^2) + 435/(98*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)^2) + 1176400/(5929*Sqrt[1 - 2*x]*(3 + 5*x)) + (414315*S
qrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 - (1561125*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 (3+5 x)^3} \, dx &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{1}{14} \int \frac{55-135 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^3} \, dx\\ &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1}{98} \int \frac{5195-15225 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^3} \, dx\\ &=-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}-\frac{\int \frac{296270-1074750 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx}{2156}\\ &=-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}+\frac{\int \frac{5187810-42350400 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{23716}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}-\frac{\int \frac{-391579365+239810850 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{913066}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}-\frac{1242945}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx+\frac{7805625 \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx}{2662}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}+\frac{1242945}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )-\frac{7805625 \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )}{2662}\\ &=-\frac{15987390}{456533 \sqrt{1-2 x}}-\frac{35825}{1078 \sqrt{1-2 x} (3+5 x)^2}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^2}+\frac{435}{98 \sqrt{1-2 x} (2+3 x) (3+5 x)^2}+\frac{1176400}{5929 \sqrt{1-2 x} (3+5 x)}+\frac{414315}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )-\frac{1561125 \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1331}\\ \end{align*}

Mathematica [C]  time = 0.0547425, size = 83, normalized size = 0.5 \[ \frac{-1102906530 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )[phone] \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )+\frac{77 \left (105876000 x^3+201146925 x^2+127185805 x+26765111\right )}{(3 x+2)^2 (5 x+3)^2}}{913066 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(26765111 + 127185805*x + 201146925*x^2 + 105876000*x^3))/((2 + 3*x)^2*(3 + 5*x)^2) - 1102906530*Hypergeo
metric2F1[-1/2, 1, 1/2, 3/7 - (6*x)/7] + 1070931750*Hypergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11])/(9130
66*Sqrt[1 - 2*x])

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Maple [A]  time = 0.015, size = 103, normalized size = 0.6 \begin{align*} -{\frac{8748}{343\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{217}{36} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{511}{36}\sqrt{1-2\,x}} \right ) }+{\frac{414315\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{64}{456533}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{312500}{1331\, \left ( -10\,x-6 \right ) ^{2}} \left ( -{\frac{191}{100} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2079}{500}\sqrt{1-2\,x}} \right ) }-{\frac{1561125\,\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/(3+5*x)^3,x)

[Out]

-8748/343*(217/36*(1-2*x)^(3/2)-511/36*(1-2*x)^(1/2))/(-6*x-4)^2+414315/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2
))*21^(1/2)+64/456533/(1-2*x)^(1/2)+312500/1331*(-191/100*(1-2*x)^(3/2)+2079/500*(1-2*x)^(1/2))/(-10*x-6)^2-15
61125/14641*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2))*55^(1/2)

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Maxima [A]  time = 1.76648, size = 209, normalized size = 1.26 \begin{align*} \frac{1561125}{29282} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) - \frac{414315}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{2 \,{\left (1798581375 \,{\left (2 \, x - 1\right )}^{4} + 12230911800 \,{\left (2 \, x - 1\right )}^{3} + 27711289905 \,{\left (2 \, x - 1\right )}^{2} + 41836111240 \, x - 20918245348\right )}}{456533 \,{\left (225 \,{\left (-2 \, x + 1\right )}^{\frac{9}{2}} - 2040 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} + 6934 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} - 10472 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 5929 \, \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)^3,x, algorithm="maxima")

[Out]

1561125/29282*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 414315/4802*sqrt(21
)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/456533*(1798581375*(2*x - 1)^4 + 12230
911800*(2*x - 1)^3 + 27711289905*(2*x - 1)^2 + 41836111240*x - 20918245348)/(225*(-2*x + 1)^(9/2) - 2040*(-2*x
+ 1)^(7/2) + 6934*(-2*x + 1)^(5/2) - 10472*(-2*x + 1)^(3/2) + 5929*sqrt(-2*x + 1))

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Fricas [A]  time = 1.6253, size = 599, normalized size = 3.61 \begin{align*} \frac{3748261125 \, \sqrt{11} \sqrt{5}{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} + 5 \, x - 8}{5 \, x + 3}\right ) + 6065985915 \, \sqrt{7} \sqrt{3}{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \log \left (-\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} - 3 \, x + 5}{3 \, x + 2}\right ) + 77 \,{\left (7194325500 \, x^{4} + 10073172600 \, x^{3} + 1810042755 \, x^{2} - 2503057145 \, x - 909821467\right )} \sqrt{-2 \, x + 1}}{70306082 \,{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\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)^3,x, algorithm="fricas")

[Out]

1/70306082*(3748261125*sqrt(11)*sqrt(5)*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*log((sqrt(11)*sqrt
(5)*sqrt(-2*x + 1) + 5*x - 8)/(5*x + 3)) + 6065985915*sqrt(7)*sqrt(3)*(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 -
156*x - 36)*log(-(sqrt(7)*sqrt(3)*sqrt(-2*x + 1) - 3*x + 5)/(3*x + 2)) + 77*(7194325500*x^4 + 10073172600*x^3
+ 1810042755*x^2 - 2503057145*x - 909821467)*sqrt(-2*x + 1))/(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 3
6)

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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)**3,x)

[Out]

Exception raised: ValueError

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Giac [A]  time = 2.38461, size = 212, normalized size = 1.28 \begin{align*} \frac{1561125}{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{414315}{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{64}{456533 \, \sqrt{-2 \, x + 1}} + \frac{2 \,{\left (256941225 \,{\left (2 \, x - 1\right )}^{3} \sqrt{-2 \, x + 1} + 1747282440 \,{\left (2 \, x - 1\right )}^{2} \sqrt{-2 \, x + 1} - 3958787399 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} + 2988341532 \, \sqrt{-2 \, x + 1}\right )}}{65219 \,{\left (15 \,{\left (2 \, x - 1\right )}^{2} + 136 \, x + 9\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)^3,x, algorithm="giac")

[Out]

1561125/29282*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) - 414315/48
02*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) + 64/456533/sqrt(-2*x +
1) + 2/65219*(256941225*(2*x - 1)^3*sqrt(-2*x + 1) + 1747282440*(2*x - 1)^2*sqrt(-2*x + 1) - 3958787399*(-2*x
+ 1)^(3/2) + 2988341532*sqrt(-2*x + 1))/(15*(2*x - 1)^2 + 136*x + 9)^2