3.2124 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^2} \, dx\)

Optimal. Leaf size=146 \[ \frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (5 x+3)}+\frac{165}{49 \sqrt{1-2 x} (3 x+2) (5 x+3)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)}-\frac{70065}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{24000}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

[Out]

245865/(41503*Sqrt[1 - 2*x]) - 36175/(1078*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x
)) + 165/(49*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) - (70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (24
000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

________________________________________________________________________________________

Rubi [A]  time = 0.0583304, antiderivative size = 146, 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} \[ \frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (5 x+3)}+\frac{165}{49 \sqrt{1-2 x} (3 x+2) (5 x+3)}+\frac{3}{14 \sqrt{1-2 x} (3 x+2)^2 (5 x+3)}-\frac{70065}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{24000}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

245865/(41503*Sqrt[1 - 2*x]) - 36175/(1078*Sqrt[1 - 2*x]*(3 + 5*x)) + 3/(14*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x
)) + 165/(49*Sqrt[1 - 2*x]*(2 + 3*x)*(3 + 5*x)) - (70065*Sqrt[3/7]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/343 + (24
000*Sqrt[5/11]*ArcTanh[Sqrt[5/11]*Sqrt[1 - 2*x]])/121

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)^2} \, dx &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{1}{14} \int \frac{40-105 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^2} \, dx\\ &=\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{165}{49 \sqrt{1-2 x} (2+3 x) (3+5 x)}+\frac{1}{98} \int \frac{2285-8250 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^2} \, dx\\ &=-\frac{36175}{1078 \sqrt{1-2 x} (3+5 x)}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{165}{49 \sqrt{1-2 x} (2+3 x) (3+5 x)}-\frac{\int \frac{39855-325575 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)} \, dx}{1078}\\ &=\frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (3+5 x)}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{165}{49 \sqrt{1-2 x} (2+3 x) (3+5 x)}+\frac{\int \frac{-\frac{6019215}{2}+\frac{3687975 x}{2}}{\sqrt{1-2 x} (2+3 x) (3+5 x)} \, dx}{41503}\\ &=\frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (3+5 x)}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{165}{49 \sqrt{1-2 x} (2+3 x) (3+5 x)}+\frac{210195}{686} \int \frac{1}{\sqrt{1-2 x} (2+3 x)} \, dx-\frac{60000}{121} \int \frac{1}{\sqrt{1-2 x} (3+5 x)} \, dx\\ &=\frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (3+5 x)}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{165}{49 \sqrt{1-2 x} (2+3 x) (3+5 x)}-\frac{210195}{686} \operatorname{Subst}\left (\int \frac{1}{\frac{7}{2}-\frac{3 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )+\frac{60000}{121} \operatorname{Subst}\left (\int \frac{1}{\frac{11}{2}-\frac{5 x^2}{2}} \, dx,x,\sqrt{1-2 x}\right )\\ &=\frac{245865}{41503 \sqrt{1-2 x}}-\frac{36175}{1078 \sqrt{1-2 x} (3+5 x)}+\frac{3}{14 \sqrt{1-2 x} (2+3 x)^2 (3+5 x)}+\frac{165}{49 \sqrt{1-2 x} (2+3 x) (3+5 x)}-\frac{70065}{343} \sqrt{\frac{3}{7}} \tanh ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )+\frac{24000}{121} \sqrt{\frac{5}{11}} \tanh ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )\\ \end{align*}

Mathematica [C]  time = 0.0431089, size = 78, normalized size = 0.53 \[ \frac{16955730 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{3}{7}-\frac{6 x}{7}\right )-16464000 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{5}{11} (2 x-1)\right )-\frac{77 \left (325575 x^2+423210 x+137209\right )}{(3 x+2)^2 (5 x+3)}}{83006 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-77*(137209 + 423210*x + 325575*x^2))/((2 + 3*x)^2*(3 + 5*x)) + 16955730*Hypergeometric2F1[-1/2, 1, 1/2, 3/7
- (6*x)/7] - 16464000*Hypergeometric2F1[-1/2, 1, 1/2, (-5*(-1 + 2*x))/11])/(83006*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 91, normalized size = 0.6 \begin{align*}{\frac{486}{343\, \left ( -6\,x-4 \right ) ^{2}} \left ({\frac{49}{2} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}}}-{\frac{1043}{18}\sqrt{1-2\,x}} \right ) }-{\frac{70065\,\sqrt{21}}{2401}{\it Artanh} \left ({\frac{\sqrt{21}}{7}\sqrt{1-2\,x}} \right ) }+{\frac{32}{41503}{\frac{1}{\sqrt{1-2\,x}}}}+{\frac{250}{121}\sqrt{1-2\,x} \left ( -2\,x-{\frac{6}{5}} \right ) ^{-1}}+{\frac{24000\,\sqrt{55}}{1331}{\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)^2,x)

[Out]

486/343*(49/2*(1-2*x)^(3/2)-1043/18*(1-2*x)^(1/2))/(-6*x-4)^2-70065/2401*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*2
1^(1/2)+32/41503/(1-2*x)^(1/2)+250/121*(1-2*x)^(1/2)/(-2*x-6/5)+24000/1331*arctanh(1/11*55^(1/2)*(1-2*x)^(1/2)
)*55^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.57314, size = 185, normalized size = 1.27 \begin{align*} -\frac{12000}{1331} \, \sqrt{55} \log \left (-\frac{\sqrt{55} - 5 \, \sqrt{-2 \, x + 1}}{\sqrt{55} + 5 \, \sqrt{-2 \, x + 1}}\right ) + \frac{70065}{4802} \, \sqrt{21} \log \left (-\frac{\sqrt{21} - 3 \, \sqrt{-2 \, x + 1}}{\sqrt{21} + 3 \, \sqrt{-2 \, x + 1}}\right ) - \frac{11063925 \,{\left (2 \, x - 1\right )}^{3} + 50903010 \,{\left (2 \, x - 1\right )}^{2} + 117027330 \, x - 58496417}{41503 \,{\left (45 \,{\left (-2 \, x + 1\right )}^{\frac{7}{2}} - 309 \,{\left (-2 \, x + 1\right )}^{\frac{5}{2}} + 707 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 539 \, \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)^2,x, algorithm="maxima")

[Out]

-12000/1331*sqrt(55)*log(-(sqrt(55) - 5*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70065/4802*sqrt(21)*l
og(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 1/41503*(11063925*(2*x - 1)^3 + 50903010*(2
*x - 1)^2 + 117027330*x - 58496417)/(45*(-2*x + 1)^(7/2) - 309*(-2*x + 1)^(5/2) + 707*(-2*x + 1)^(3/2) - 539*s
qrt(-2*x + 1))

________________________________________________________________________________________

Fricas [A]  time = 1.59076, size = 508, normalized size = 3.48 \begin{align*} \frac{57624000 \, \sqrt{11} \sqrt{5}{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (-\frac{\sqrt{11} \sqrt{5} \sqrt{-2 \, x + 1} - 5 \, x + 8}{5 \, x + 3}\right ) + 93256515 \, \sqrt{7} \sqrt{3}{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\right )} \log \left (\frac{\sqrt{7} \sqrt{3} \sqrt{-2 \, x + 1} + 3 \, x - 5}{3 \, x + 2}\right ) - 77 \,{\left (22127850 \, x^{3} + 17711235 \, x^{2} - 5050290 \, x - 4664333\right )} \sqrt{-2 \, x + 1}}{6391462 \,{\left (90 \, x^{4} + 129 \, x^{3} + 25 \, x^{2} - 32 \, x - 12\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)^2,x, algorithm="fricas")

[Out]

1/6391462*(57624000*sqrt(11)*sqrt(5)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log(-(sqrt(11)*sqrt(5)*sqrt(-2*x
+ 1) - 5*x + 8)/(5*x + 3)) + 93256515*sqrt(7)*sqrt(3)*(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)*log((sqrt(7)*sqr
t(3)*sqrt(-2*x + 1) + 3*x - 5)/(3*x + 2)) - 77*(22127850*x^3 + 17711235*x^2 - 5050290*x - 4664333)*sqrt(-2*x +
1))/(90*x^4 + 129*x^3 + 25*x^2 - 32*x - 12)

________________________________________________________________________________________

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)**2,x)

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [A]  time = 2.07398, size = 182, normalized size = 1.25 \begin{align*} -\frac{12000}{1331} \, \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{70065}{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{2 \,{\left (428910 \, x - 214279\right )}}{41503 \,{\left (5 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 11 \, \sqrt{-2 \, x + 1}\right )}} + \frac{27 \,{\left (63 \,{\left (-2 \, x + 1\right )}^{\frac{3}{2}} - 149 \, \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)^2,x, algorithm="giac")

[Out]

-12000/1331*sqrt(55)*log(1/2*abs(-2*sqrt(55) + 10*sqrt(-2*x + 1))/(sqrt(55) + 5*sqrt(-2*x + 1))) + 70065/4802*
sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 2/41503*(428910*x - 2142
79)/(5*(-2*x + 1)^(3/2) - 11*sqrt(-2*x + 1)) + 27/196*(63*(-2*x + 1)^(3/2) - 149*sqrt(-2*x + 1))/(3*x + 2)^2