3.2419 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=200 \[ \frac{1165 \sqrt{1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{3485 \sqrt{1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac{249575 \sqrt{1-2 x} \sqrt{5 x+3}}{108864}+\frac{1850}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{3304795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{326592 \sqrt{7}} \]

[Out]

(249575*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/108864 - (3485*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(4032*(2 + 3*x)) - ((1 - 2*
x)^(5/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3) + (1165*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2592*(2 + 3*x)^2) + (1850*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/729 + (330
4795*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(326592*Sqrt[7])

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Rubi [A]  time = 0.08079, antiderivative size = 200, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {97, 149, 154, 157, 54, 216, 93, 204} \[ \frac{1165 \sqrt{1-2 x} (5 x+3)^{5/2}}{2592 (3 x+2)^2}+\frac{185 (1-2 x)^{3/2} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{5/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{3485 \sqrt{1-2 x} (5 x+3)^{3/2}}{4032 (3 x+2)}+\frac{249575 \sqrt{1-2 x} \sqrt{5 x+3}}{108864}+\frac{1850}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )+\frac{3304795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{326592 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(249575*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/108864 - (3485*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(4032*(2 + 3*x)) - ((1 - 2*
x)^(5/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + (185*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3) + (1165*S
qrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2592*(2 + 3*x)^2) + (1850*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/729 + (330
4795*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(326592*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 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

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)^5} \, dx &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{1}{108} \int \frac{\left (-\frac{3655}{4}-1225 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^3} \, dx\\ &=-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1165 \sqrt{1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac{1}{648} \int \frac{\left (\frac{20765}{8}-\frac{3975 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^2} \, dx\\ &=-\frac{3485 \sqrt{1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1165 \sqrt{1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac{\int \frac{\left (\frac{1308195}{16}-\frac{748725 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)} \, dx}{13608}\\ &=\frac{249575 \sqrt{1-2 x} \sqrt{3+5 x}}{108864}-\frac{3485 \sqrt{1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1165 \sqrt{1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac{\int \frac{-\frac{13271205}{8}-3108000 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{81648}\\ &=\frac{249575 \sqrt{1-2 x} \sqrt{3+5 x}}{108864}-\frac{3485 \sqrt{1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1165 \sqrt{1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac{3304795 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{653184}+\frac{9250}{729} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=\frac{249575 \sqrt{1-2 x} \sqrt{3+5 x}}{108864}-\frac{3485 \sqrt{1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1165 \sqrt{1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}-\frac{3304795 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{326592}+\frac{1}{729} \left (3700 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=\frac{249575 \sqrt{1-2 x} \sqrt{3+5 x}}{108864}-\frac{3485 \sqrt{1-2 x} (3+5 x)^{3/2}}{4032 (2+3 x)}-\frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{185 (1-2 x)^{3/2} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1165 \sqrt{1-2 x} (3+5 x)^{5/2}}{2592 (2+3 x)^2}+\frac{1850}{729} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )+\frac{3304795 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{326592 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.206844, size = 136, normalized size = 0.68 \[ \frac{-21 \sqrt{5 x+3} \left (7257600 x^5+55321830 x^4+61652541 x^3+6433776 x^2-15812708 x-5093072\right )-5801600 \sqrt{10-20 x} (3 x+2)^4 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )+3304795 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2286144 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*Sqrt[3 + 5*x]*(-5093072 - 15812708*x + 6433776*x^2 + 61652541*x^3 + 55321830*x^4 + 7257600*x^5) - 5801600
*Sqrt[10 - 20*x]*(2 + 3*x)^4*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] + 3304795*Sqrt[7 - 14*x]*(2 + 3*x)^4*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2286144*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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Maple [B]  time = 0.012, size = 332, normalized size = 1.7 \begin{align*}{\frac{1}{4572288\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 469929600\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{4}-267688395\,\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}-713835720\,\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{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){x}^{2}-713835720\,\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{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-317260320\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+1913684976\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) -52876720\,\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((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^5,x)

[Out]

1/4572288*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(469929600*10^(1/2)*arcsin(20/11*x+1/11)*x^4-267688395*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+1253145600*10^(1/2)*arcsin(20/11*x+1/11)*x^3-713835720*7^(1/2)*a
rctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+152409600*x^4*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*ar
csin(20/11*x+1/11)*x^2-713835720*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+1237963230*x^3
*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)*x-317260320*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)
/(-10*x^2-x+3)^(1/2))*x+1913684976*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)-52876720*7^(
1/2)*arctan(1/14*(37*x+20)*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)^4

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Maxima [A]  time = 2.31267, size = 305, normalized size = 1.52 \begin{align*} \frac{5755}{49392} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{392 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{1151 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{10976 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{182225}{98784} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{1488395}{1778112} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{44881 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{197568 \,{\left (3 \, x + 2\right )}} - \frac{28675}{127008} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{925}{729} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) - \frac{3304795}{4572288} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{1643795}{762048} \, \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)^5,x, algorithm="maxima")

[Out]

5755/49392*(-10*x^2 - x + 3)^(5/2) + 3/28*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 3
7/392*(-10*x^2 - x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 1151/10976*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x
+ 4) + 182225/98784*(-10*x^2 - x + 3)^(3/2)*x - 1488395/1778112*(-10*x^2 - x + 3)^(3/2) + 44881/197568*(-10*x^
2 - x + 3)^(5/2)/(3*x + 2) - 28675/127008*sqrt(-10*x^2 - x + 3)*x + 925/729*sqrt(10)*arcsin(20/11*x + 1/11) -
3304795/4572288*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1643795/762048*sqrt(-10*x^2 - x +
3)

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Fricas [A]  time = 1.90476, size = 581, normalized size = 2.9 \begin{align*} \frac{3304795 \, \sqrt{7}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, 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 ) - 5801600 \, \sqrt{10}{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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 (3628800 \, x^{4} + 29475315 \, x^{3} + 45563928 \, x^{2} + 25998852 \, x + 5093072\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4572288 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\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)^5,x, algorithm="fricas")

[Out]

1/4572288*(3304795*sqrt(7)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x +
3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 5801600*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20*s
qrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) + 42*(3628800*x^4 + 29475315*x^3 + 45563928*
x^2 + 25998852*x + 5093072)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(81*x^4 + 216*x^3 + 216*x^2 + 96*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((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [B]  time = 5.17586, size = 628, normalized size = 3.14 \begin{align*} -\frac{660959}{9144576} \, \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{925}{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{20}{243} \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{55 \,{\left (8191 \, \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} + 7386792 \, \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} + 2164545600 \, \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} + 2731201984000 \, \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 )}}{54432 \,{\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((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^5,x, algorithm="giac")

[Out]

-660959/9144576*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 925/729*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)))) + 20/243*s
qrt(5)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 55/54432*(8191*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 + 7386792*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 + 2164545600*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 + 2731
201984000*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)^4