3.2421 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{7/2}}{32 (3 x+2)^4}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{7/2}}{12 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{5/2}}{1344 (3 x+2)^3}-\frac{73205 \sqrt{1-2 x} (5 x+3)^{3/2}}{37632 (3 x+2)^2}-\frac{805255 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{8857805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(-805255*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (73205*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(37632*(2 + 3
*x)^2) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1344*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(6*(2 + 3
*x)^6) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(12*(2 + 3*x)^5) + (121*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(32*(2 +
3*x)^4) - (8857805*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0666988, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {94, 93, 204} \[ \frac{121 \sqrt{1-2 x} (5 x+3)^{7/2}}{32 (3 x+2)^4}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{7/2}}{12 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6}-\frac{1331 \sqrt{1-2 x} (5 x+3)^{5/2}}{1344 (3 x+2)^3}-\frac{73205 \sqrt{1-2 x} (5 x+3)^{3/2}}{37632 (3 x+2)^2}-\frac{805255 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{8857805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]

[Out]

(-805255*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (73205*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(37632*(2 + 3
*x)^2) - (1331*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(1344*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(6*(2 + 3
*x)^6) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(12*(2 + 3*x)^5) + (121*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(32*(2 +
3*x)^4) - (8857805*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

Rule 94

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[(n*(d*e - c*f))/((m + 1)*(b*e - a*
f)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] &&  !(SumSimplerQ[p, 1] &&  !SumSimplerQ[m, 1])

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)^7} \, dx &=\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{55}{12} \int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^6} \, dx\\ &=\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121}{8} \int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121 \sqrt{1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac{1331}{64} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{1331 \sqrt{1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121 \sqrt{1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac{73205 \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx}{2688}\\ &=-\frac{73205 \sqrt{1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121 \sqrt{1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac{805255 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac{805255 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{73205 \sqrt{1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121 \sqrt{1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac{8857805 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{351232}\\ &=-\frac{805255 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{73205 \sqrt{1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121 \sqrt{1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}+\frac{8857805 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{175616}\\ &=-\frac{805255 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{73205 \sqrt{1-2 x} (3+5 x)^{3/2}}{37632 (2+3 x)^2}-\frac{1331 \sqrt{1-2 x} (3+5 x)^{5/2}}{1344 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{6 (2+3 x)^6}+\frac{11 (1-2 x)^{3/2} (3+5 x)^{7/2}}{12 (2+3 x)^5}+\frac{121 \sqrt{1-2 x} (3+5 x)^{7/2}}{32 (2+3 x)^4}-\frac{8857805 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{175616 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.13171, size = 138, normalized size = 0.66 \[ \frac{121 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (814395 x^3+1285720 x^2+654436 x+105552\right )}{(3 x+2)^4}-219615 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{3687936}+\frac{11 (1-2 x)^{3/2} (5 x+3)^{7/2}}{12 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{6 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^7,x]

[Out]

((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(6*(2 + 3*x)^6) + (11*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(12*(2 + 3*x)^5) + (1
21*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(105552 + 654436*x + 1285720*x^2 + 814395*x^3))/(2 + 3*x)^4 - 219615*Sqrt[7
]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/3687936

________________________________________________________________________________________

Maple [B]  time = 0.011, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{7375872\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 19372019535\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{6}[phone]\,\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]\,{x}^{5}\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}^{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+24404657984\,{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((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^7,x)

[Out]

1/7375872*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(19372019535*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^6+77488078140*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+129146796900*7^(1/2)*arctan(1/1
4*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+7960010170*x^5*(-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^3+26676039880*x^4*(-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^2+36035545056*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+24404657984*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]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-1
0*x^2-x+3)^(1/2)/(2+3*x)^6

________________________________________________________________________________________

Maxima [A]  time = 3.3927, size = 408, normalized size = 1.95 \begin{align*} \frac{3304795}{19361664} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{14 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{37 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{196 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{4387 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{10976 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{81733 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{153664 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{660959 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}}}{4302592 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{59208325}{12907776} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{113659535}{25815552} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{109715471 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{77446656 \,{\left (3 \, x + 2\right )}} + \frac{13542925}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{8857805}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{11932415}{1229312} \, \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)^7,x, algorithm="maxima")

[Out]

3304795/19361664*(-10*x^2 - x + 3)^(5/2) + 1/14*(-10*x^2 - x + 3)^(7/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*
x^3 + 2160*x^2 + 576*x + 64) + 37/196*(-10*x^2 - x + 3)^(7/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x
+ 32) + 4387/10976*(-10*x^2 - x + 3)^(7/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 81733/153664*(-10*x^2 -
x + 3)^(7/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 660959/4302592*(-10*x^2 - x + 3)^(7/2)/(9*x^2 + 12*x + 4) - 592083
25/12907776*(-10*x^2 - x + 3)^(3/2)*x + 113659535/25815552*(-10*x^2 - x + 3)^(3/2) - 109715471/77446656*(-10*x
^2 - x + 3)^(5/2)/(3*x + 2) + 13542925/614656*sqrt(-10*x^2 - x + 3)*x + 8857805/2458624*sqrt(7)*arcsin(37/11*x
/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 11932415/1229312*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 2.15614, size = 500, normalized size = 2.39 \begin{align*} -\frac{26573415 \, \sqrt{7}{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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 (568572155 \, x^{5} + 1905431420 \, x^{4} + 2573967504 \, x^{3} + 1743189856 \, x^{2} + 589734736 \, x + 79536960\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{7375872 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\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)^7,x, algorithm="fricas")

[Out]

-1/7375872*(26573415*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*sq
rt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(568572155*x^5 + 1905431420*x^4 + 257396
7504*x^3 + 1743189856*x^2 + 589734736*x + 79536960)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(729*x^6 + 2916*x^5 + 4860*x
^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)

________________________________________________________________________________________

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)**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 4.28826, size = 676, normalized size = 3.23 \begin{align*} \frac{1771561}{4917248} \, \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{8857805 \,{\left (3 \, \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 )}^{11} + 4760 \, \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} + 3104640 \, \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} - 869299200 \, \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} - 104491520000 \, \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} - 5163110400000 \, \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 )}}{263424 \,{\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 )}^{6}} \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)^7,x, algorithm="giac")

[Out]

1771561/4917248*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)))) - 8857805/263424*(3*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)))^11 + 4760*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 +
3104640*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 - 869299200*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 - 104491520000*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 - 5163110400000*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)^6