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]

(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])

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Rubi [A]  time = 0.0639432, antiderivative size = 180, 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 = {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 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 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{(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*}

Mathematica [A]  time = 0.0674968, 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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Maple [B]  time = 0.015, size = 305, normalized size = 1.7 \begin{align*}{\frac{1}{2151296\, \left ( 2+3\,x \right ) ^{4} \left ( 2\,x-1 \right ) } \left ( 27344790\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-14041020\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-4051080\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-36100610\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-10802880\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-30578016\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-2700720\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -9629592\,x\sqrt{-10\,{x}^{2}-x+3}-743904\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}\sqrt{3+5\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2151296*(27344790*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+59247045*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+36459720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2
))*x^3-14041020*x^4*(-10*x^2-x+3)^(1/2)-4051080*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2
-36100610*x^3*(-10*x^2-x+3)^(1/2)-10802880*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-305780
16*x^2*(-10*x^2-x+3)^(1/2)-2700720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-9629592*x*(-10*x
^2-x+3)^(1/2)-743904*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4/(2*x-1)/(-10*x^2-x+3)^(1/2)

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Maxima [B]  time = 2.20298, size = 400, normalized size = 2.22 \begin{align*} \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 )}} \end{align*}

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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Fricas [A]  time = 1.80318, size = 412, normalized size = 2.29 \begin{align*} -\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 )}} \end{align*}

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

[Out]

Timed out

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Giac [B]  time = 5.11416, size = 547, normalized size = 3.04 \begin{align*} \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 \,{\left (10277 \, \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 )}^{7} + 10598840 \, \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 )}^{5} + 3966648000 \, \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} + 122821440000 \, \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 )}}{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{align*}

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*(10277*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)))^7 + 10598840*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)))^5 + 3966648000*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 + 122821440000*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)^4