3.2284 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}+\frac{1852307 \sqrt{1-2 x} \sqrt{5 x+3}}{1185408 (3 x+2)}+\frac{17981 \sqrt{1-2 x} \sqrt{5 x+3}}{84672 (3 x+2)^2}+\frac{641 \sqrt{1-2 x} \sqrt{5 x+3}}{15120 (3 x+2)^3}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{2520 (3 x+2)^4}-\frac{783959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2520*(2 + 3*x)^4) + (641*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15120*(2 + 3*x)^3)
+ (17981*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 + 3*x)^2) + (1852307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1185408*(2
+ 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(15*(2 + 3*x)^5) - (783959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(43904*Sqrt[7])

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Rubi [A]  time = 0.0637865, 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 = {97, 149, 151, 12, 93, 204} \[ -\frac{\sqrt{1-2 x} (5 x+3)^{3/2}}{15 (3 x+2)^5}+\frac{1852307 \sqrt{1-2 x} \sqrt{5 x+3}}{1185408 (3 x+2)}+\frac{17981 \sqrt{1-2 x} \sqrt{5 x+3}}{84672 (3 x+2)^2}+\frac{641 \sqrt{1-2 x} \sqrt{5 x+3}}{15120 (3 x+2)^3}-\frac{107 \sqrt{1-2 x} \sqrt{5 x+3}}{2520 (3 x+2)^4}-\frac{783959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2520*(2 + 3*x)^4) + (641*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15120*(2 + 3*x)^3)
+ (17981*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(84672*(2 + 3*x)^2) + (1852307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1185408*(2
+ 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(15*(2 + 3*x)^5) - (783959*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*
x])])/(43904*Sqrt[7])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx &=-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{\left (\frac{9}{2}-20 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^5} \, dx\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{\int \frac{-\frac{1691}{4}-1195 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{1260}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}+\frac{641 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{\int \frac{\frac{90125}{8}-22435 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{26460}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}+\frac{641 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{17981 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{\int \frac{\frac{13219115}{16}-\frac{3146675 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{370440}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}+\frac{641 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{17981 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{1852307 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{\int \frac{740841255}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2593080}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}+\frac{641 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{17981 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{1852307 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{783959 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{87808}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}+\frac{641 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{17981 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{1852307 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}+\frac{783959 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{43904}\\ &=-\frac{107 \sqrt{1-2 x} \sqrt{3+5 x}}{2520 (2+3 x)^4}+\frac{641 \sqrt{1-2 x} \sqrt{3+5 x}}{15120 (2+3 x)^3}+\frac{17981 \sqrt{1-2 x} \sqrt{3+5 x}}{84672 (2+3 x)^2}+\frac{1852307 \sqrt{1-2 x} \sqrt{3+5 x}}{1185408 (2+3 x)}-\frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{15 (2+3 x)^5}-\frac{783959 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{43904 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.109475, size = 133, normalized size = 0.74 \[ \frac{589 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (4223 x^2+4478 x+1152\right )}{(3 x+2)^3}-3993 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{921984}+\frac{81 (1-2 x)^{3/2} (5 x+3)^{5/2}}{280 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{5/2}}{35 (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^6,x]

[Out]

(3*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(35*(2 + 3*x)^5) + (81*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(280*(2 + 3*x)^4)
+ (589*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1152 + 4478*x + 4223*x^2))/(2 + 3*x)^3 - 3993*Sqrt[7]*ArcTan[Sqrt[1 -
2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/921984

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Maple [B]  time = 0.013, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{9219840\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2857530555\,\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}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3221121848\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\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)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^6,x)

[Out]

1/9219840*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2857530555*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x
^5+9525101850*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+12700135800*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1166953410*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(3
7*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3164739900*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37
*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3221121848*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x+2
0)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^
(1/2)/(2+3*x)^5

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Maxima [A]  time = 2.01841, size = 267, normalized size = 1.48 \begin{align*} \frac{783959}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{32395}{32928} \, \sqrt{-10 \, x^{2} - x + 3} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{13 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{280 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{545 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{2352 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{19437 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{239723 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="maxima")

[Out]

783959/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 32395/32928*sqrt(-10*x^2 - x + 3) -
1/35*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 13/280*(-10*x^2 - x + 3)^
(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 545/2352*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x + 8)
+ 19437/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 239723/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.87353, size = 437, normalized size = 2.43 \begin{align*} -\frac{11759385 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (83353815 \, x^{4} + 226052850 \, x^{3} + 230080132 \, x^{2} + 103856008 \, x + 17507808\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{9219840 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/9219840*(11759385*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x +
20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(83353815*x^4 + 226052850*x^3 + 230080132*x^2 + 10385
6008*x + 17507808)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**6,x)

[Out]

Timed out

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Giac [B]  time = 3.74261, size = 594, normalized size = 3.3 \begin{align*} \frac{783959}{6146560} \, \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{1331 \,{\left (1767 \, \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 )}^{9} + 2308880 \, \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} - 925245440 \, \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} - 177804928000 \, \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} - 10860971520000 \, \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 )}}{65856 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

783959/6146560*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)))) - 1331/65856*(1767*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)))^9 + 2308880*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 -
925245440*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 - 177804928000*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 - 10860971520000*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sq
rt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^5