3.2357 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\)

Optimal. Leaf size=178 \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{871 \sqrt{1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac{77269 \sqrt{1-2 x} \sqrt{5 x+3}}{254016 (3 x+2)}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{7}} \]

[Out]

(-77269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(254016*(2 + 3*x)) - (871*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(6048*(2 + 3*x)^
2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3
) + (100*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (1922677*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(762048*Sqrt[7])

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Rubi [A]  time = 0.063871, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{181 \sqrt{1-2 x} (5 x+3)^{5/2}}{216 (3 x+2)^3}-\frac{(1-2 x)^{3/2} (5 x+3)^{5/2}}{12 (3 x+2)^4}-\frac{871 \sqrt{1-2 x} (5 x+3)^{3/2}}{6048 (3 x+2)^2}-\frac{77269 \sqrt{1-2 x} \sqrt{5 x+3}}{254016 (3 x+2)}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{762048 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-77269*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(254016*(2 + 3*x)) - (871*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(6048*(2 + 3*x)^
2) - ((1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(12*(2 + 3*x)^4) + (181*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(216*(2 + 3*x)^3
) + (100*Sqrt[10]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/243 - (1922677*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x]
)])/(762048*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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{1}{12} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{1}{108} \int \frac{\left (-\frac{1511}{4}-240 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{\int \frac{\left (-\frac{166869}{8}-16800 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{4536}\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}-\frac{\int \frac{-\frac{8194677}{16}-588000 x}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{95256}\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1922677 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{1524096}+\frac{500}{243} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{1922677 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{762048}+\frac{1}{243} \left (200 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )\\ &=-\frac{77269 \sqrt{1-2 x} \sqrt{3+5 x}}{254016 (2+3 x)}-\frac{871 \sqrt{1-2 x} (3+5 x)^{3/2}}{6048 (2+3 x)^2}-\frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{12 (2+3 x)^4}+\frac{181 \sqrt{1-2 x} (3+5 x)^{5/2}}{216 (2+3 x)^3}+\frac{100}{243} \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )-\frac{1922677 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{762048 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.188224, size = 131, normalized size = 0.74 \[ \frac{-21 \sqrt{5 x+3} \left (26580294 x^4+33080973 x^3+3682800 x^2-8266660 x-2583760\right )-2195200 \sqrt{10-20 x} (3 x+2)^4 \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )-1922677 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{5334336 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-21*Sqrt[3 + 5*x]*(-2583760 - 8266660*x + 3682800*x^2 + 33080973*x^3 + 26580294*x^4) - 2195200*Sqrt[10 - 20*x
]*(2 + 3*x)^4*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]] - 1922677*Sqrt[7 - 14*x]*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt
[7]*Sqrt[3 + 5*x])])/(5334336*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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Maple [B]  time = 0.012, size = 315, normalized size = 1.8 \begin{align*}{\frac{1}{10668672\, \left ( 2+3\,x \right ) ^{4}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 155736837\,\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}^{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]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ){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]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \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+210739200\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x+973793520\,{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]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^5,x)

[Out]

1/10668672*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(155736837*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x
^4+177811200*10^(1/2)*arcsin(20/11*x+1/11)*x^4+415298232*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(
1/2))*x^3+474163200*10^(1/2)*arcsin(20/11*x+1/11)*x^3+415298232*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^2+474163200*10^(1/2)*arcsin(20/11*x+1/11)*x^2+558186174*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+210739200*10^(1/2)*arcsin(20/11*x+1/11)*x+973793520*x^
2*(-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))[phone]*10^(1/2)*ar
csin(20/11*x+1/11)[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 = 2.00873, size = 266, normalized size = 1.49 \begin{align*} \frac{27065}{148176} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{28 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{169 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1176 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{5413 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{32928 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{528205}{296352} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{50}{243} \, \sqrt{10} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{1922677}{10668672} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{802877}{1778112} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3667 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{197568 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27065/148176*(-10*x^2 - x + 3)^(3/2) - 1/28*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) +
169/1176*(-10*x^2 - x + 3)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 5413/32928*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 1
2*x + 4) + 528205/296352*sqrt(-10*x^2 - x + 3)*x + 50/243*sqrt(10)*arcsin(20/11*x + 1/11) + 1922677/10668672*s
qrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 802877/1778112*sqrt(-10*x^2 - x + 3) + 3667/197568*
(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.60707, size = 564, normalized size = 3.17 \begin{align*} -\frac{1922677 \, \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 ) + 2195200 \, \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 (13290147 \, x^{3} + 23185560 \, x^{2} + 13434180 \, x + 2583760\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{10668672 \,{\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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^5,x, algorithm="fricas")

[Out]

-1/10668672*(1922677*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)) + 2195200*sqrt(10)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*arctan(1/20
*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 42*(13290147*x^3 + 23185560*x^2 + 134341
80*x + 2583760)*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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**5,x)

[Out]

Timed out

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Giac [B]  time = 3.36465, size = 602, normalized size = 3.38 \begin{align*} \frac{1922677}{106686720} \, \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{50}{243} \, \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 (77269 \, \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} + 81002040 \, \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} + 31057924800 \, \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} - 8580356288000 \, \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 )}}{127008 \,{\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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^5,x, algorithm="giac")

[Out]

1922677/106686720*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - s
qrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 50/243*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)))) - 11/1270
08*(77269*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 + 81002040*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 + 31057924800*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 - 8580356288000*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