3.2359 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\)

Optimal. Leaf size=209 \[ \frac{297 \sqrt{1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac{9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac{1089 \sqrt{1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac{11979 \sqrt{1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac{395307 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{4348377 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

[Out]

(-395307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (11979*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12544*(2 + 3
*x)^2) - (1089*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2240*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 +
3*x)^6) + (9*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (297*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(160*(2 +
3*x)^4) - (4348377*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(175616*Sqrt[7])

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Rubi [A]  time = 0.0708929, 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} \[ \frac{297 \sqrt{1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac{9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac{1089 \sqrt{1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac{11979 \sqrt{1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac{395307 \sqrt{1-2 x} \sqrt{5 x+3}}{175616 (3 x+2)}-\frac{4348377 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{175616 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-395307*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(175616*(2 + 3*x)) - (11979*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(12544*(2 + 3
*x)^2) - (1089*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(2240*(2 + 3*x)^3) + ((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 +
3*x)^6) + (9*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (297*Sqrt[1 - 2*x]*(3 + 5*x)^(7/2))/(160*(2 +
3*x)^4) - (4348377*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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx &=\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac{9}{4} \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}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297}{40} \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}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297 \sqrt{1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac{3267}{320} \int \frac{(3+5 x)^{5/2}}{\sqrt{1-2 x} (2+3 x)^4} \, dx\\ &=-\frac{1089 \sqrt{1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297 \sqrt{1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac{11979}{896} \int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^3} \, dx\\ &=-\frac{11979 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac{1089 \sqrt{1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297 \sqrt{1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac{395307 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^2} \, dx}{25088}\\ &=-\frac{395307 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{11979 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac{1089 \sqrt{1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297 \sqrt{1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac{4348377 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{351232}\\ &=-\frac{395307 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{11979 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac{1089 \sqrt{1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297 \sqrt{1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}+\frac{4348377 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{175616}\\ &=-\frac{395307 \sqrt{1-2 x} \sqrt{3+5 x}}{175616 (2+3 x)}-\frac{11979 \sqrt{1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac{1089 \sqrt{1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac{(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac{9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac{297 \sqrt{1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}-\frac{4348377 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{175616 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.128388, size = 138, normalized size = 0.66 \[ \frac{99 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (814395 x^3+1285720 x^2+654436 x+105552\right )}{(3 x+2)^4}-219615 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{6146560}+\frac{9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac{(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((1 - 2*x)^(5/2)*(3 + 5*x)^(7/2))/(14*(2 + 3*x)^6) + (9*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2))/(20*(2 + 3*x)^5) + (9
9*((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])]))/6146560

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Maple [B]  time = 0.013, size = 346, normalized size = 1.7 \begin{align*}{\frac{1}{12293120\, \left ( 2+3\,x \right ) ^{6}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 15849834165\,\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+19993885632\,{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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7,x)

[Out]

1/12293120*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(15849834165*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))
*x^6+63399336660*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+105665561100*7^(1/2)*arctan(1/
14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+6448875230*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+21774762520*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+29513642144*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+19993885632*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 = 2.41115, size = 369, normalized size = 1.77 \begin{align*} \frac{272085}{307328} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} - \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{42 \,{\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac{23 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{420 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{297 \,{\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{10989 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{21952 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{489753 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}}}{614656 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{6648345}{614656} \, \sqrt{-10 \, x^{2} - x + 3} x + \frac{4348377}{2458624} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{5857731}{1229312} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{645909 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{1229312 \,{\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)^7,x, algorithm="maxima")

[Out]

272085/307328*(-10*x^2 - x + 3)^(3/2) - 1/42*(-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) + 23/420*(-10*x^2 - x + 3)^(5/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 3
2) + 297/1568*(-10*x^2 - x + 3)^(5/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 10989/21952*(-10*x^2 - x + 3)
^(5/2)/(27*x^3 + 54*x^2 + 36*x + 8) + 489753/614656*(-10*x^2 - x + 3)^(5/2)/(9*x^2 + 12*x + 4) + 6648345/61465
6*sqrt(-10*x^2 - x + 3)*x + 4348377/2458624*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 585773
1/1229312*sqrt(-10*x^2 - x + 3) + 645909/1229312*(-10*x^2 - x + 3)^(3/2)/(3*x + 2)

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Fricas [A]  time = 1.64535, size = 501, normalized size = 2.4 \begin{align*} -\frac{21741885 \, \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 (460633945 \, x^{5} + 1555340180 \, x^{4} + 2108117296 \, x^{3} + 1428134688 \, x^{2} + 482263920 \, x + 64829376\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{12293120 \,{\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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="fricas")

[Out]

-1/12293120*(21741885*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14*s
qrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(460633945*x^5 + 1555340180*x^4 + 21081
17296*x^3 + 1428134688*x^2 + 482263920*x + 64829376)*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)**(3/2)*(3+5*x)**(5/2)/(2+3*x)**7,x)

[Out]

Timed out

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Giac [B]  time = 4.5488, size = 676, normalized size = 3.23 \begin{align*} \frac{4348377}{24586240} \, \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{161051 \,{\left (27 \, \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} + 42840 \, \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} + 27941760 \, \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} - 6539187200 \, \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} - 940423680000 \, \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} - 46467993600000 \, \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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^7,x, algorithm="giac")

[Out]

4348377/24586240*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 161051/87808*(27*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 + 42840*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
+ 27941760*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 - 6539187200*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 - 940423680000*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 - 46467993600000*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