3.2379 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{5 x+3}}+\frac{1403963 \sqrt{1-2 x}}{3136 (3 x+2) \sqrt{5 x+3}}+\frac{8063 \sqrt{1-2 x}}{224 (3 x+2)^2 \sqrt{5 x+3}}+\frac{33 \sqrt{1-2 x}}{8 (3 x+2)^3 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{145708761 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

[Out]

(-63678595*Sqrt[1 - 2*x])/(9408*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (33*Sqrt[1
- 2*x])/(8*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (8063*Sqrt[1 - 2*x])/(224*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (1403963*Sqrt[
1 - 2*x])/(3136*(2 + 3*x)*Sqrt[3 + 5*x]) + (145708761*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqr
t[7])

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Rubi [A]  time = 0.0646826, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {98, 151, 152, 12, 93, 204} \[ -\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{5 x+3}}+\frac{1403963 \sqrt{1-2 x}}{3136 (3 x+2) \sqrt{5 x+3}}+\frac{8063 \sqrt{1-2 x}}{224 (3 x+2)^2 \sqrt{5 x+3}}+\frac{33 \sqrt{1-2 x}}{8 (3 x+2)^3 \sqrt{5 x+3}}+\frac{7 \sqrt{1-2 x}}{12 (3 x+2)^4 \sqrt{5 x+3}}+\frac{145708761 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{3136 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-63678595*Sqrt[1 - 2*x])/(9408*Sqrt[3 + 5*x]) + (7*Sqrt[1 - 2*x])/(12*(2 + 3*x)^4*Sqrt[3 + 5*x]) + (33*Sqrt[1
- 2*x])/(8*(2 + 3*x)^3*Sqrt[3 + 5*x]) + (8063*Sqrt[1 - 2*x])/(224*(2 + 3*x)^2*Sqrt[3 + 5*x]) + (1403963*Sqrt[
1 - 2*x])/(3136*(2 + 3*x)*Sqrt[3 + 5*x]) + (145708761*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(3136*Sqr
t[7])

Rule 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

Rule 151

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

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 + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[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*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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}}{(2+3 x)^5 (3+5 x)^{3/2}} \, dx &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{1}{12} \int \frac{\frac{341}{2}-264 x}{\sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{1}{252} \int \frac{\frac{86163}{4}-31185 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{8063 \sqrt{1-2 x}}{224 (2+3 x)^2 \sqrt{3+5 x}}+\frac{\int \frac{\frac{15937383}{8}-2539845 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{3/2}} \, dx}{3528}\\ &=\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{8063 \sqrt{1-2 x}}{224 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1403963 \sqrt{1-2 x}}{3136 (2+3 x) \sqrt{3+5 x}}+\frac{\int \frac{\frac{1880555061}{16}-\frac{442248345 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{24696}\\ &=-\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{8063 \sqrt{1-2 x}}{224 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1403963 \sqrt{1-2 x}}{3136 (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{100976171373}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{135828}\\ &=-\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{8063 \sqrt{1-2 x}}{224 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1403963 \sqrt{1-2 x}}{3136 (2+3 x) \sqrt{3+5 x}}-\frac{145708761 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6272}\\ &=-\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{8063 \sqrt{1-2 x}}{224 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1403963 \sqrt{1-2 x}}{3136 (2+3 x) \sqrt{3+5 x}}-\frac{145708761 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{3136}\\ &=-\frac{63678595 \sqrt{1-2 x}}{9408 \sqrt{3+5 x}}+\frac{7 \sqrt{1-2 x}}{12 (2+3 x)^4 \sqrt{3+5 x}}+\frac{33 \sqrt{1-2 x}}{8 (2+3 x)^3 \sqrt{3+5 x}}+\frac{8063 \sqrt{1-2 x}}{224 (2+3 x)^2 \sqrt{3+5 x}}+\frac{1403963 \sqrt{1-2 x}}{3136 (2+3 x) \sqrt{3+5 x}}+\frac{145708761 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{3136 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0696399, size = 84, normalized size = 0.49 \[ \frac{145708761 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )-\frac{7 \sqrt{1-2 x} \left (1719322065 x^4+4546951839 x^3+4508028900 x^2+1985778980 x+327908240\right )}{(3 x+2)^4 \sqrt{5 x+3}}}{21952} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*Sqrt[1 - 2*x]*(327908240 + 1985778980*x + 4508028900*x^2 + 4546951839*x^3 + 1719322065*x^4))/((2 + 3*x)^4
*Sqrt[3 + 5*x]) + 145708761*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/21952

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Maple [B]  time = 0.015, size = 298, normalized size = 1.7 \begin{align*} -{\frac{1}{43904\, \left ( 2+3\,x \right ) ^{4}} \left ( 59012048205\,\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]\,\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+63112404600\,{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 ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)/(2+3*x)^5/(3+5*x)^(3/2),x)

[Out]

-1/43904*(59012048205*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+192772690803*7^(1/2)*arct
an(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+251784739008*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^3+24070508910*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+63657325746*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+63112404600*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))*(1-2*x)^(1/2)/(2+3*x)^4/(-10*x
^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [B]  time = 1.8602, size = 400, normalized size = 2.31 \begin{align*} -\frac{145708761}{43904} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{63678595 \, x}{4704 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{66486521}{9408 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{49}{36 \,{\left (81 \, \sqrt{-10 \, x^{2} - x + 3} x^{4} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 216 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 96 \, \sqrt{-10 \, x^{2} - x + 3} x + 16 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{665}{72 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{7799}{96 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{457237}{448 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-145708761/43904*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 63678595/4704*x/sqrt(-10*x^2 - x
+ 3) - 66486521/9408/sqrt(-10*x^2 - x + 3) + 49/36/(81*sqrt(-10*x^2 - x + 3)*x^4 + 216*sqrt(-10*x^2 - x + 3)*x
^3 + 216*sqrt(-10*x^2 - x + 3)*x^2 + 96*sqrt(-10*x^2 - x + 3)*x + 16*sqrt(-10*x^2 - x + 3)) + 665/72/(27*sqrt(
-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) +
7799/96/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 457237/448/(3*s
qrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

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Fricas [A]  time = 1.5397, size = 448, normalized size = 2.59 \begin{align*} \frac{145708761 \, \sqrt{7}{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\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 (1719322065 \, x^{4} + 4546951839 \, x^{3} + 4508028900 \, x^{2} + 1985778980 \, x + 327908240\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{43904 \,{\left (405 \, x^{5} + 1323 \, x^{4} + 1728 \, x^{3} + 1128 \, x^{2} + 368 \, x + 48\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/43904*(145708761*sqrt(7)*(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)*arctan(1/14*sqrt(7)*(37*x +
20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1719322065*x^4 + 4546951839*x^3 + 4508028900*x^2 + 1
985778980*x + 327908240)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(405*x^5 + 1323*x^4 + 1728*x^3 + 1128*x^2 + 368*x + 48)

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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)/(2+3*x)**5/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [B]  time = 3.14843, size = 591, normalized size = 3.42 \begin{align*} -\frac{145708761}{439040} \, \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{275}{2} \, \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 )} - \frac{11 \,{\left (13252949 \, \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} + 8830442040 \, \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} + 2086818820800 \, \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} + 170309125952000 \, \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 )}}{1568 \,{\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)/(2+3*x)^5/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-145708761/439040*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)))) - 275/2*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))) - 11/1568*(13252949*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 +
8830442040*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 + 2086818820800*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 + 170309125952000*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