3.2409 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}+\frac{17 (5 x+3)^{5/2} (1-2 x)^{5/2}}{28 (3 x+2)^5}+\frac{935 (5 x+3)^{5/2} (1-2 x)^{3/2}}{224 (3 x+2)^4}+\frac{10285 (5 x+3)^{5/2} \sqrt{1-2 x}}{448 (3 x+2)^3}-\frac{113135 (5 x+3)^{3/2} \sqrt{1-2 x}}{12544 (3 x+2)^2}-\frac{3733455 \sqrt{5 x+3} \sqrt{1-2 x}}{175616 (3 x+2)}-\frac{41068005 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(-3733455*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (113135*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12544*(2 +
3*x)^2) + ((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/(14*(2 + 3*x)^6) + (17*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(28*(2 +
3*x)^5) + (935*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(224*(2 + 3*x)^4) + (10285*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(448
*(2 + 3*x)^3) - (41068005*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

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Rubi [A]  time = 0.0695618, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {96, 94, 93, 204} \[ \frac{(5 x+3)^{5/2} (1-2 x)^{7/2}}{14 (3 x+2)^6}+\frac{17 (5 x+3)^{5/2} (1-2 x)^{5/2}}{28 (3 x+2)^5}+\frac{935 (5 x+3)^{5/2} (1-2 x)^{3/2}}{224 (3 x+2)^4}+\frac{10285 (5 x+3)^{5/2} \sqrt{1-2 x}}{448 (3 x+2)^3}-\frac{113135 (5 x+3)^{3/2} \sqrt{1-2 x}}{12544 (3 x+2)^2}-\frac{3733455 \sqrt{5 x+3} \sqrt{1-2 x}}{175616 (3 x+2)}-\frac{41068005 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3733455*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (113135*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12544*(2 +
3*x)^2) + ((1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/(14*(2 + 3*x)^6) + (17*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(28*(2 +
3*x)^5) + (935*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(224*(2 + 3*x)^4) + (10285*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(448
*(2 + 3*x)^3) - (41068005*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

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)^{3/2}}{(2+3 x)^7} \, dx &=\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{85}{28} \int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^6} \, dx\\ &=\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935}{56} \int \frac{(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^5} \, dx\\ &=\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac{30855}{448} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^4} \, dx\\ &=\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac{10285 \sqrt{1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac{113135}{896} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{113135 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac{10285 \sqrt{1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac{3733455 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac{3733455 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{113135 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac{10285 \sqrt{1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac{41068005 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{351232}\\ &=-\frac{3733455 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{113135 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac{10285 \sqrt{1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}+\frac{41068005 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{175616}\\ &=-\frac{3733455 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{113135 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}+\frac{(1-2 x)^{7/2} (3+5 x)^{5/2}}{14 (2+3 x)^6}+\frac{17 (1-2 x)^{5/2} (3+5 x)^{5/2}}{28 (2+3 x)^5}+\frac{935 (1-2 x)^{3/2} (3+5 x)^{5/2}}{224 (2+3 x)^4}+\frac{10285 \sqrt{1-2 x} (3+5 x)^{5/2}}{448 (2+3 x)^3}-\frac{41068005 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{175616 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.135888, size = 138, normalized size = 0.66 \[ \frac{1}{28} \left (\frac{935 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (100159 x^3+213240 x^2+145940 x+32400\right )}{(3 x+2)^4}-43923 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{43904}+\frac{2 (5 x+3)^{5/2} (1-2 x)^{7/2}}{(3 x+2)^6}+\frac{17 (5 x+3)^{5/2} (1-2 x)^{5/2}}{(3 x+2)^5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((2*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^6 + (17*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^5 + (935*((7
*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(32400 + 145940*x + 213240*x^2 + 100159*x^3))/(2 + 3*x)^4 - 43923*Sqrt[7]*ArcTan[
Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/43904)/28

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Maple [B]  time = 0.013, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{2458624\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 29938575645\,\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+37695279552\,{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)^(3/2)/(2+3*x)^7,x)

[Out]

1/2458624*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(29938575645*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*
x^6+119754302580*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+199590504300*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+12212429390*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+41253428440*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+55752986016*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arcta
n(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+37695279552*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))/
(-10*x^2-x+3)^(1/2)/(2+3*x)^6

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Maxima [A]  time = 2.69103, size = 369, normalized size = 1.77 \begin{align*} \frac{7709075}{921984} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{6 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{47 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{84 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{2805 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{1568 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{103785 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{4625445 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{62789925}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{41068005}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{55323015}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{18300755 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{3687936 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^7,x, algorithm="maxima")

[Out]

7709075/921984*(-10*x^2 - x + 3)^(3/2) + 1/6*(-10*x^2 - x + 3)^(5/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3
+ 2160*x^2 + 576*x + 64) + 47/84*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32
) + 2805/1568*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 103785/21952*(-10*x^2 - x + 3
)^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 4625445/614656*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 62789925/61
4656*sqrt(-10*x^2 - x + 3)*x + 41068005/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 55
323015/1229312*sqrt(-10*x^2 - x + 3) + 18300755/3687936*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.71892, size = 501, normalized size = 2.4 \begin{align*} -\frac{41068005 \, \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 (872316385 \, x^{5} + 2946673460 \, x^{4} + 3982356144 \, x^{3} + 2692519968 \, x^{2} + 910641904 \, x + 123208128\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2458624 \,{\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)^(3/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/2458624*(41068005*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*(872316385*x^5 + 2946673460*x^4 + 398235
6144*x^3 + 2692519968*x^2 + 910641904*x + 123208128)*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)

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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)**(3/2)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [B]  time = 5.2767, size = 676, normalized size = 3.23 \begin{align*} \frac{8213601}{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{805255 \,{\left (51 \, \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} + 80920 \, \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} - 59615360 \, \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} - 14778086400 \, \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} - 1776355840000 \, \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} - 87772876800000 \, \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 )}}{87808 \,{\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)^(3/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

8213601/4917248*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)))) - 805255/87808*(51*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 + 80920*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 -
59615360*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 - 14778086400*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 - 1776355840000*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 - 87772876800000*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^6