3.2548 \(\int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx\)

Optimal. Leaf size=180 \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {167155 \sqrt {1-2 x} \sqrt {5 x+3}}{1382976 (3 x+2)}-\frac {38365 \sqrt {1-2 x} \sqrt {5 x+3}}{98784 (3 x+2)^2}-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3528 (3 x+2)^3}+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^4}-\frac {168795 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]

[Out]

-168795/1075648*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+11/7*(3+5*x)^(3/2)/(2+3*x)^4/(1-2*x)^(
1/2)+131/588*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4-3653/3528*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3-38365/98784
*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2-167155/1382976*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]  time = 0.06, antiderivative size = 180, 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 = {98, 149, 151, 12, 93, 204} \[ \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^4}-\frac {167155 \sqrt {1-2 x} \sqrt {5 x+3}}{1382976 (3 x+2)}-\frac {38365 \sqrt {1-2 x} \sqrt {5 x+3}}{98784 (3 x+2)^2}-\frac {3653 \sqrt {1-2 x} \sqrt {5 x+3}}{3528 (3 x+2)^3}+\frac {131 \sqrt {1-2 x} \sqrt {5 x+3}}{588 (3 x+2)^4}-\frac {168795 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{153664 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(131*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(588*(2 + 3*x)^4) - (3653*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3528*(2 + 3*x)^3) -
(38365*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(98784*(2 + 3*x)^2) - (167155*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1382976*(2 + 3
*x)) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^4) - (168795*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x
])])/(153664*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 149

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*(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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegerQ[m]

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)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^5} \, dx &=\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{7} \int \frac {\left (-228-\frac {815 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^5} \, dx\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {1}{588} \int \frac {-\frac {62303}{2}-53120 x}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {\int \frac {-\frac {592375}{4}-255710 x}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{12348}\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}-\frac {38365 \sqrt {1-2 x} \sqrt {3+5 x}}{98784 (2+3 x)^2}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {\int \frac {-\frac {3190705}{8}-\frac {1342775 x}{2}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{172872}\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}-\frac {38365 \sqrt {1-2 x} \sqrt {3+5 x}}{98784 (2+3 x)^2}-\frac {167155 \sqrt {1-2 x} \sqrt {3+5 x}}{1382976 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {\int -\frac {10634085}{16 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{1210104}\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}-\frac {38365 \sqrt {1-2 x} \sqrt {3+5 x}}{98784 (2+3 x)^2}-\frac {167155 \sqrt {1-2 x} \sqrt {3+5 x}}{1382976 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {168795 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{307328}\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}-\frac {38365 \sqrt {1-2 x} \sqrt {3+5 x}}{98784 (2+3 x)^2}-\frac {167155 \sqrt {1-2 x} \sqrt {3+5 x}}{1382976 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}+\frac {168795 \operatorname {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{153664}\\ &=\frac {131 \sqrt {1-2 x} \sqrt {3+5 x}}{588 (2+3 x)^4}-\frac {3653 \sqrt {1-2 x} \sqrt {3+5 x}}{3528 (2+3 x)^3}-\frac {38365 \sqrt {1-2 x} \sqrt {3+5 x}}{98784 (2+3 x)^2}-\frac {167155 \sqrt {1-2 x} \sqrt {3+5 x}}{1382976 (2+3 x)}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^4}-\frac {168795 \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 = 0.07, size = 95, normalized size = 0.53 \[ \frac {7 \sqrt {5 x+3} \left (1002930 x^4+2578615 x^3+2184144 x^2+687828 x+53136\right )-168795 \sqrt {7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1075648 \sqrt {1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(7*Sqrt[3 + 5*x]*(53136 + 687828*x + 2184144*x^2 + 2578615*x^3 + 1002930*x^4) - 168795*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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fricas [A]  time = 1.32, size = 131, normalized size = 0.73 \[ -\frac {168795 \, \sqrt {7} {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) + 14 \, {\left (1002930 \, x^{4} + 2578615 \, x^{3} + 2184144 \, x^{2} + 687828 \, x + 53136\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2151296 \, {\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2151296*(168795*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*(1002930*x^4 + 2578615*x^3 + 2184144*x^2 + 687828*x + 5313
6)*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 = 4.35, size = 394, normalized size = 2.19 \[ \frac {33759}{4302592} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {968 \, \sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{84035 \, {\left (2 \, x - 1\right )}} - \frac {121 \, \sqrt {10} {\left (10277 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} + 10598840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} + 3966648000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} + \frac {122821440000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} - \frac {491285760000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{537824 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

33759/4302592*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)))) - 968/84035*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5)
/(2*x - 1) - 121/537824*sqrt(10)*(10277*((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 + 10598840*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 3966648000*((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 + 122821440000*(sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3) - 491285760000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((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)))^2 + 280)^4

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maple [B]  time = 0.02, size = 305, normalized size = 1.69 \[ \frac {\left (27344790 \sqrt {7}\, x^{5} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] \sqrt {7}\, x^{4} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-14041020 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {7}\, x^{3} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-36100610 \sqrt {-10 x^{2}-x +3}\, x^{3}-4051080 \sqrt {7}\, x^{2} \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-30578016 \sqrt {-10 x^{2}-x +3}\, x^{2}-10802880 \sqrt {7}\, x \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-9629592 \sqrt {-10 x^{2}-x +3}\, x -2700720 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )-743904 \sqrt {-10 x^{2}-x +3}\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{2151296 \left (3 x +2\right )^{4} \left (2 x -1\right ) \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2151296*(27344790*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))-14041020*(-10*x^2-x+3)^(1/2)*x^4-4051080*7^(1/2)*x^2*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
-36100610*(-10*x^2-x+3)^(1/2)*x^3-10802880*7^(1/2)*x*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-305780
16*(-10*x^2-x+3)^(1/2)*x^2-2700720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-9629592*(-10*x^2
-x+3)^(1/2)*x-743904*(-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 = 1.10, size = 296, normalized size = 1.64 \[ \frac {168795}{2151296} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {835775 \, x}{2074464 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {843155}{4148928 \, \sqrt {-10 \, x^{2} - x + 3}} + \frac {1}{756 \, {\left (81 \, \sqrt {-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt {-10 \, x^{2} - x + 3} x + 16 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {787}{31752 \, {\left (27 \, \sqrt {-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt {-10 \, x^{2} - x + 3} x + 8 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} + \frac {20681}{127008 \, {\left (9 \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt {-10 \, x^{2} - x + 3} x + 4 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} - \frac {69575}{197568 \, {\left (3 \, \sqrt {-10 \, x^{2} - x + 3} x + 2 \, \sqrt {-10 \, x^{2} - x + 3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

168795/2151296*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 835775/2074464*x/sqrt(-10*x^2 - x +
3) + 843155/4148928/sqrt(-10*x^2 - x + 3) + 1/756/(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)) - 787/31752/(27*sq
rt(-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))
+ 20681/127008/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) - 69575/1
97568/(3*sqrt(-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 \[ \int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^5} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((5*x + 3)^(5/2)/((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 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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