3.2420 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=207 \[ \frac{2543 \sqrt{1-2 x} (5 x+3)^{5/2}}{1296 (3 x+2)^3}+\frac{37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{72 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}-\frac{32453 \sqrt{1-2 x} (5 x+3)^{3/2}}{36288 (3 x+2)^2}-\frac{3248687 \sqrt{1-2 x} \sqrt{5 x+3}}{1524096 (3 x+2)}-\frac{200}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{109715471 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4572288 \sqrt{7}} \]

[Out]

(-3248687*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1524096*(2 + 3*x)) - (32453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36288*(2 +
3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(15*(2 + 3*x)^5) + (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(72*(2 +
3*x)^4) + (2543*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1296*(2 + 3*x)^3) - (200*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/729 - (109715471*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4572288*Sqrt[7])

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Rubi [A]  time = 0.0810377, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {97, 149, 157, 54, 216, 93, 204} \[ \frac{2543 \sqrt{1-2 x} (5 x+3)^{5/2}}{1296 (3 x+2)^3}+\frac{37 (1-2 x)^{3/2} (5 x+3)^{5/2}}{72 (3 x+2)^4}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{15 (3 x+2)^5}-\frac{32453 \sqrt{1-2 x} (5 x+3)^{3/2}}{36288 (3 x+2)^2}-\frac{3248687 \sqrt{1-2 x} \sqrt{5 x+3}}{1524096 (3 x+2)}-\frac{200}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{109715471 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{4572288 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3248687*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1524096*(2 + 3*x)) - (32453*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(36288*(2 +
3*x)^2) - ((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(15*(2 + 3*x)^5) + (37*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(72*(2 +
3*x)^4) + (2543*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1296*(2 + 3*x)^3) - (200*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]])/729 - (109715471*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(4572288*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 157

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/((a_.) + (b_.)*(x_)), x_Symbol]
:> Dist[h/b, Int[(c + d*x)^n*(e + f*x)^p, x], x] + Dist[(b*g - a*h)/b, Int[((c + d*x)^n*(e + f*x)^p)/(a + b*x
), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}-\frac{1}{180} \int \frac{\left (-\frac{5305}{4}-400 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac{2543 \sqrt{1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac{\int \frac{\left (\frac{149465}{8}-2400 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{1620}\\ &=-\frac{32453 \sqrt{1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac{2543 \sqrt{1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac{\int \frac{\left (\frac{14451435}{16}-168000 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{68040}\\ &=-\frac{3248687 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)}-\frac{32453 \sqrt{1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac{2543 \sqrt{1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac{\int \frac{\frac{423137355}{32}-5880000 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1428840}\\ &=-\frac{3248687 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)}-\frac{32453 \sqrt{1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac{2543 \sqrt{1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac{1000}{729} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx+\frac{109715471 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{9144576}\\ &=-\frac{3248687 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)}-\frac{32453 \sqrt{1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac{2543 \sqrt{1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}+\frac{109715471 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{4572288}-\frac{1}{729} \left (400 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{3248687 \sqrt{1-2 x} \sqrt{3+5 x}}{1524096 (2+3 x)}-\frac{32453 \sqrt{1-2 x} (3+5 x)^{3/2}}{36288 (2+3 x)^2}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{15 (2+3 x)^5}+\frac{37 (1-2 x)^{3/2} (3+5 x)^{5/2}}{72 (2+3 x)^4}+\frac{2543 \sqrt{1-2 x} (3+5 x)^{5/2}}{1296 (2+3 x)^3}-\frac{200}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{109715471 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{4572288 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.223057, size = 136, normalized size = 0.66 \[ \frac{-21 \sqrt{5 x+3} \left (980826030 x^5+3128525325 x^4+2484445206 x^3-58943604 x^2-682484168 x-180761312\right )[phone] \sqrt{10-20 x} (3 x+2)^5 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-548577355 \sqrt{7-14 x} (3 x+2)^5 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{160030080 \sqrt{1-2 x} (3 x+2)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*Sqrt[3 + 5*x]*(-180761312 - 682484168*x - 58943604*x^2 + 2484445206*x^3 + 3128525325*x^4 + 980826030*x^5)
+ 43904000*Sqrt[10 - 20*x]*(2 + 3*x)^5*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] - 548577355*Sqrt[7 - 14*x]*(2 + 3*x)^
5*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(160030080*Sqrt[1 - 2*x]*(2 + 3*x)^5)

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Maple [B]  time = 0.012, size = 377, normalized size = 1.8 \begin{align*} -{\frac{1}{320060160\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 10668672000\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{5}-133304297265\,\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{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}-444347657550\,\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{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{3}-592463543400\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}-20597346630\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-394975695600\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}-75997705140\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-131658565200\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x-90172201896\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -17554475360\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -43848285264\,x\sqrt{-10\,{x}^{2}-x+3}-7591975104\,\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((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^6,x)

[Out]

-1/320060160*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(10668672000*10^(1/2)*arcsin(20/11*x+1/11)*x^5-133304297265*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+35562240000*10^(1/2)*arcsin(20/11*x+1/11)*x^4-4443476575
50*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+47416320000*10^(1/2)*arcsin(20/11*x+1/11)*x^
3-592463543400*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3-20597346630*x^4*(-10*x^2-x+3)^(1
/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x^2-394975695600*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-
x+3)^(1/2))*x^2-75997705140*x^3*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-131658565200*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x-90172201896*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^
(1/2)*arcsin(20/11*x+1/11)-17554475360*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))-43848285264*
x*(-10*x^2-x+3)^(1/2)-7591975104*(-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 = 1.72456, size = 360, normalized size = 1.74 \begin{align*} \frac{44881}{691488} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{333 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{1960 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{6347 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{27440 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{44881 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{768320 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{3156205}{1382976} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{52017151}{24893568} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{9235489 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{13829760 \,{\left (3 \, x + 2\right )}} + \frac{17832215}{1778112} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{100}{729} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{109715471}{64012032} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{49508071}{10668672} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

44881/691488*(-10*x^2 - x + 3)^(5/2) + 3/35*(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 +
240*x + 32) + 333/1960*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 6347/27440*(-10*x^2
- x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 44881/768320*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 315620
5/1382976*(-10*x^2 - x + 3)^(3/2)*x + 52017151/24893568*(-10*x^2 - x + 3)^(3/2) - 9235489/13829760*(-10*x^2 -
x + 3)^(5/2)/(3*x + 2) + 17832215/1778112*sqrt(-10*x^2 - x + 3)*x - 100/729*sqrt(10)*arcsin(20/11*x + 1/11) +
109715471/64012032*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 49508071/10668672*sqrt(-10*x^2
- x + 3)

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Fricas [A]  time = 1.89538, size = 655, normalized size = 3.16 \begin{align*} -\frac{548577355 \, \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 ) - 43904000 \, \sqrt{10}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 42 \,{\left (490413015 \, x^{4} + 1809469170 \, x^{3} + 2146957188 \, x^{2} + 1044006792 \, x + 180761312\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{320060160 \,{\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((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/320060160*(548577355*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)) - 43904000*sqrt(10)*(243*x^5 + 810*x^4 + 1080*x^3 + 720
*x^2 + 240*x + 32)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(490413
015*x^4 + 1809469170*x^3 + 2146957188*x^2 + 1044006792*x + 180761312)*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((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**6,x)

[Out]

Timed out

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Giac [B]  time = 4.93981, size = 684, normalized size = 3.3 \begin{align*} \frac{109715471}{640120320} \, \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{100}{729} \, \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{4 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{11 \,{\left (3248687 \, \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} + 4238260880 \, \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} + 2165236899840 \, \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} - 364930179712000 \, \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} - 12258004702720000 \, \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 )}}{762048 \,{\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((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^6,x, algorithm="giac")

[Out]

109715471/640120320*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)))) - 100/729*sqrt(10)*(pi + 2*arctan(-1/4*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)))) - 11/7
62048*(3248687*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^9 + 4238260880*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 + 2165236899840*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/s
qrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 - 364930179712000*sqrt(10)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 1225800470
2720000*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))))/(((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)^5