3.2626 \(\int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=159 \[ -\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{81}{28 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

[Out]

-2725/(3234*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 89945/(249018*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (46307675*Sqrt[1 - 2
*x])/(5478396*Sqrt[3 + 5*x]) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 81/(28*(1 - 2*x)^(3/2)*(2 +
3*x)*Sqrt[3 + 5*x]) + (79515*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Rubi [A]  time = 0.0602733, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{5 x+3}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{81}{28 (1-2 x)^{3/2} (3 x+2) \sqrt{5 x+3}}-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{5 x+3}}+\frac{3}{14 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2725/(3234*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) - 89945/(249018*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (46307675*Sqrt[1 - 2
*x])/(5478396*Sqrt[3 + 5*x]) + 3/(14*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 81/(28*(1 - 2*x)^(3/2)*(2 +
3*x)*Sqrt[3 + 5*x]) + (79515*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{1}{14} \int \frac{\frac{29}{2}-120 x}{(1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}+\frac{1}{98} \int \frac{-\frac{2065}{4}-8505 x}{(1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}} \, dx\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{-\frac{514885}{8}+286125 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx}{11319}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}+\frac{2 \int \frac{\frac{42164605}{16}-\frac{9444225 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{871563}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{4 \int \frac{2222523765}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{9587193}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{79515 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2744}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}-\frac{79515 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{1372}\\ &=-\frac{2725}{3234 (1-2 x)^{3/2} \sqrt{3+5 x}}-\frac{89945}{249018 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{46307675 \sqrt{1-2 x}}{5478396 \sqrt{3+5 x}}+\frac{3}{14 (1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}}+\frac{81}{28 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}}+\frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{1372 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0971551, size = 84, normalized size = 0.53 \[ \frac{79515 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{1372 \sqrt{7}}-\frac{1667076300 x^4+520073880 x^3-1053213025 x^2-169466391 x+178740084}{5478396 (1-2 x)^{3/2} (3 x+2)^2 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(178740084 - 169466391*x - 1053213025*x^2 + 520073880*x^3 + 1667076300*x^4)/(5478396*(1 - 2*x)^(3/2)*(2 + 3*x
)^2*Sqrt[3 + 5*x]) + (79515*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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Maple [B]  time = 0.016, size = 305, normalized size = 1.9 \begin{align*} -{\frac{1}{76697544\, \left ( 2+3\,x \right ) ^{2} \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 57150611100\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}+53340570360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}-25082768205\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}+23339068200\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-28257802155\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}+7281034320\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}+2540027160\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-14744982350\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+3810040740\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -2372529474\,x\sqrt{-10\,{x}^{2}-x+3}+2502361176\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/76697544*(1-2*x)^(1/2)*(57150611100*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+53340570
360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4-25082768205*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+23339068200*x^4*(-10*x^2-x+3)^(1/2)-28257802155*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+7281034320*x^3*(-10*x^2-x+3)^(1/2)+2540027160*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x-14744982350*x^2*(-10*x^2-x+3)^(1/2)+3810040740*7^(1/2)*arctan(1/14*(37*x+20)*7^(
1/2)/(-10*x^2-x+3)^(1/2))-2372529474*x*(-10*x^2-x+3)^(1/2)+2502361176*(-10*x^2-x+3)^(1/2))/(2+3*x)^2/(2*x-1)^2
/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{3}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.84949, size = 431, normalized size = 2.71 \begin{align*} \frac{317503395 \, \sqrt{7}{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (1667076300 \, x^{4} + 520073880 \, x^{3} - 1053213025 \, x^{2} - 169466391 \, x + 178740084\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{76697544 \,{\left (180 \, x^{5} + 168 \, x^{4} - 79 \, x^{3} - 89 \, x^{2} + 8 \, x + 12\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/76697544*(317503395*sqrt(7)*(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)*arctan(1/14*sqrt(7)*(37*x + 20)
*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1667076300*x^4 + 520073880*x^3 - 1053213025*x^2 - 169466
391*x + 178740084)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(180*x^5 + 168*x^4 - 79*x^3 - 89*x^2 + 8*x + 12)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 2.77947, size = 479, normalized size = 3.01 \begin{align*} -\frac{15903}{38416} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{625}{2662} \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )} - \frac{32 \,{\left (944 \, \sqrt{5}{\left (5 \, x + 3\right )} - 5577 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{239679825 \,{\left (2 \, x - 1\right )}^{2}} - \frac{891 \,{\left (337 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 75880 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{4802 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15903/38416*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 625/2662*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 32/239679825*(944*sqrt(5)*(5*x
 + 3) - 5577*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2 - 891/4802*(337*sqrt(10)*((sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 75880*sqrt(10)*((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sq
rt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 28
0)^2