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

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

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Rubi [A]  time = 0.06, antiderivative size = 159, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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 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 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*}

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Mathematica [A]  time = 0.15, size = 84, normalized size = 0.53 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/5478396*(178740084 - 169466391*x - 1053213025*x^2 + 520073880*x^3 + 1667076300*x^4)/((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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IntegrateAlgebraic [A]  time = 0.24, size = 138, normalized size = 0.87 \begin {gather*} \frac {\left (-\frac {25725000 (1-2 x)^4}{(5 x+3)^4}-\frac {531421605 (1-2 x)^3}{(5 x+3)^3}-\frac {2224548725 (1-2 x)^2}{(5 x+3)^2}+\frac {457856 (1-2 x)}{5 x+3}+12544\right ) (5 x+3)^{3/2}}{5478396 (1-2 x)^{3/2} \left (\frac {1-2 x}{5 x+3}+7\right )^2}+\frac {79515 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1372 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((3 + 5*x)^(3/2)*(12544 - (25725000*(1 - 2*x)^4)/(3 + 5*x)^4 - (531421605*(1 - 2*x)^3)/(3 + 5*x)^3 - (22245487
25*(1 - 2*x)^2)/(3 + 5*x)^2 + (457856*(1 - 2*x))/(3 + 5*x)))/(5478396*(1 - 2*x)^(3/2)*(7 + (1 - 2*x)/(3 + 5*x)
)^2) + (79515*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1372*Sqrt[7])

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fricas [A]  time = 2.20, size = 131, normalized size = 0.82 \begin {gather*} \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 {gather*}

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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giac [B]  time = 3.21, size = 355, normalized size = 2.23 \begin {gather*} -\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 {gather*}

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

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maple [B]  time = 0.02, size = 305, normalized size = 1.92 \begin {gather*} -\frac {\sqrt {-2 x +1}\, \left (57150611100 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}-25082768205 \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}-28257802155 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-14744982350 \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2372529474 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {-10 x^{2}-x +3}\right )}{76697544 \left (3 x +2\right )^{2} \left (2 x -1\right )^{2} \sqrt {-10 x^{2}-x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/76697544*(-2*x+1)^(1/2)*(57150611100*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))+5334057
0360*7^(1/2)*x^4*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-25082768205*7^(1/2)*x^3*arctan(1/14*(37*x+
20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^4-28257802155*7^(1/2)*x^2*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*(-10*x^2-x+3)^(1/2)*x^3+2540027160*7^(1/2)*x*arctan(1/14*(37*x+20
)*7^(1/2)/(-10*x^2-x+3)^(1/2))-14744982350*(-10*x^2-x+3)^(1/2)*x^2+3810040740*7^(1/2)*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))-2372529474*(-10*x^2-x+3)^(1/2)*x+2502361176*(-10*x^2-x+3)^(1/2))/(3*x+2)^2/(2*x-1)^
2/(-10*x^2-x+3)^(1/2)/(5*x+3)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^3\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (1 - 2 x\right )^{\frac {5}{2}} \left (3 x + 2\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \end {gather*}

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]

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

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