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

Optimal. Leaf size=180 \[ \frac {16985 \sqrt {1-2 x} \sqrt {5 x+3}}{153664 (3 x+2)}-\frac {745 \sqrt {1-2 x} \sqrt {5 x+3}}{10976 (3 x+2)^2}-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)^3}-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^4}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {279015 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]

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Rubi [A]  time = 0.07, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \begin {gather*} \frac {16985 \sqrt {1-2 x} \sqrt {5 x+3}}{153664 (3 x+2)}-\frac {745 \sqrt {1-2 x} \sqrt {5 x+3}}{10976 (3 x+2)^2}-\frac {89 \sqrt {1-2 x} \sqrt {5 x+3}}{392 (3 x+2)^3}-\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)^4}+\frac {11 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {279015 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(11*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (131*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)^4) - (89*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(392*(2 + 3*x)^3) - (745*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10976*(2 + 3*x)^2) + (16985
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(153664*(2 + 3*x)) - (279015*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(153
664*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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] + Dist[1/(b*(b*e - a*
f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 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 {(3+5 x)^{3/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{7} \int \frac {-338-\frac {1145 x}{2}}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {1}{196} \int \frac {-\frac {4617}{2}-3930 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {\int \frac {-\frac {44625}{4}-18690 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{4116}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {745 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}-\frac {\int \frac {-\frac {327495}{8}-\frac {78225 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{57624}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {745 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {16985 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}-\frac {\int -\frac {5859315}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{403368}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {745 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {16985 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}+\frac {279015 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{307328}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {745 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {16985 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}+\frac {279015 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{153664}\\ &=\frac {11 \sqrt {3+5 x}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)^4}-\frac {89 \sqrt {1-2 x} \sqrt {3+5 x}}{392 (2+3 x)^3}-\frac {745 \sqrt {1-2 x} \sqrt {3+5 x}}{10976 (2+3 x)^2}+\frac {16985 \sqrt {1-2 x} \sqrt {3+5 x}}{153664 (2+3 x)}-\frac {279015 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{153664 \sqrt {7}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 95, normalized size = 0.53 \begin {gather*} \frac {7 \sqrt {5 x+3} \left (-917190 x^4-1188045 x^3+60048 x^2+538276 x+163152\right )-279015 \sqrt {7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1075648 \sqrt {1-2 x} (3 x+2)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[3 + 5*x]*(163152 + 538276*x + 60048*x^2 - 1188045*x^3 - 917190*x^4) - 279015*Sqrt[7 - 14*x]*(2 + 3*x)^
4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1075648*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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IntegrateAlgebraic [A]  time = 4.37, size = 235, normalized size = 1.31 \begin {gather*} \frac {5 \sqrt {11-2 (5 x+3)} \left (183438 \sqrt {5} (5 x+3)^{9/2}-1013211 \sqrt {5} (5 x+3)^{7/2}-1086993 \sqrt {5} (5 x+3)^{5/2}+610451 \sqrt {5} (5 x+3)^{3/2}+55803 \sqrt {5} \sqrt {5 x+3}\right )}{153664 (2 (5 x+3)-11) (3 (5 x+3)+1)^4}-\frac {279015 \tan ^{-1}\left (\frac {\sqrt {\frac {2}{34+\sqrt {1155}}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{153664 \sqrt {7}}-\frac {279015 \tan ^{-1}\left (\frac {\sqrt {68+2 \sqrt {1155}} \sqrt {5 x+3}}{\sqrt {11}-\sqrt {11-2 (5 x+3)}}\right )}{153664 \sqrt {7}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(5*Sqrt[11 - 2*(3 + 5*x)]*(55803*Sqrt[5]*Sqrt[3 + 5*x] + 610451*Sqrt[5]*(3 + 5*x)^(3/2) - 1086993*Sqrt[5]*(3 +
5*x)^(5/2) - 1013211*Sqrt[5]*(3 + 5*x)^(7/2) + 183438*Sqrt[5]*(3 + 5*x)^(9/2)))/(153664*(-11 + 2*(3 + 5*x))*(
1 + 3*(3 + 5*x))^4) - (279015*ArcTan[(Sqrt[2/(34 + Sqrt[1155])]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 + 5*
x)])])/(153664*Sqrt[7]) - (279015*ArcTan[(Sqrt[68 + 2*Sqrt[1155]]*Sqrt[3 + 5*x])/(Sqrt[11] - Sqrt[11 - 2*(3 +
5*x)])])/(153664*Sqrt[7])

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fricas [A]  time = 1.28, size = 131, normalized size = 0.73 \begin {gather*} -\frac {279015 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\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 (917190 \, x^{4} + 1188045 \, x^{3} - 60048 \, x^{2} - 538276 \, x - 163152\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2151296 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2151296*(279015*sqrt(7)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*arctan(1/14*sqrt(7)*(37*x + 20)*
sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(917190*x^4 + 1188045*x^3 - 60048*x^2 - 538276*x - 163152)
*sqrt(5*x + 3)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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giac [B]  time = 3.70, size = 394, normalized size = 2.19 \begin {gather*} \frac {55803}{4302592} \, \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 {176 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{84035 \, {\left (2 \, x - 1\right )}} - \frac {11 \, \sqrt {10} {\left (178579 \, {\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 )}^{7} + 183436680 \, {\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 )}^{5} - 17824632000 \, {\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} - \frac {2829942080000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {11319768320000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\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 )}^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

55803/4302592*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(
22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 176/84035*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)
/(2*x - 1) - 11/537824*sqrt(10)*(178579*((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)))^7 + 183436680*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq
rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 17824632000*((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 - 2829942080000*(sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))/sqrt(5*x + 3) + 11319768320000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^4

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maple [B]  time = 0.02, size = 305, normalized size = 1.69 \begin {gather*} \frac {\left (45200430 \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}[phone] \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}-6696360 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-840672 \sqrt {-10 x^{2}-x +3}\, x^{2}-17856960 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-7535864 \sqrt {-10 x^{2}-x +3}\, x -4464240 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-2284128 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{2151296 \left (3 x +2\right )^{4} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2151296*(45200430*7^(1/2)*x^5*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*7^(1/2)*x^4*arctan
(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*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^4-6696360*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^3-17856960*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-840672
*(-10*x^2-x+3)^(1/2)*x^2-4464240*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-7535864*(-10*x^2-x
+3)^(1/2)*x-2284128*(-10*x^2-x+3)^(1/2))*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(3*x+2)^4/(2*x-1)/(-10*x^2-x+3)^(1/2)

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maxima [B]  time = 1.41, size = 296, normalized size = 1.64 \begin {gather*} \frac {279015}{2151296} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {84925 \, x}{230496 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {131015}{460992 \, \sqrt {-10 \, x^{2} - x + 3}} - \frac {1}{252 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {169}{3528 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {649}{4704 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {2475}{21952 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

279015/2151296*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 84925/230496*x/sqrt(-10*x^2 - x + 3
) + 131015/460992/sqrt(-10*x^2 - x + 3) - 1/252/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x^3
+ 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 169/3528/(27*sqrt(-
10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) - 6
49/4704/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 2475/21952/(3*s
qrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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