3.2387 \(\int \frac{(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=166 \[ \frac{618645 \sqrt{1-2 x}}{56 \sqrt{5 x+3}}-\frac{204595 \sqrt{1-2 x}}{168 (5 x+3)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (3 x+2) (5 x+3)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{4246733 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

[Out]

(-204595*Sqrt[1 - 2*x])/(168*(3 + 5*x)^(3/2)) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (301*Sqrt[
1 - 2*x])/(36*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (24469*Sqrt[1 - 2*x])/(168*(2 + 3*x)*(3 + 5*x)^(3/2)) + (618645*S
qrt[1 - 2*x])/(56*Sqrt[3 + 5*x]) - (4246733*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Rubi [A]  time = 0.0591056, antiderivative size = 166, 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} \[ \frac{618645 \sqrt{1-2 x}}{56 \sqrt{5 x+3}}-\frac{204595 \sqrt{1-2 x}}{168 (5 x+3)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (3 x+2) (5 x+3)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (3 x+2)^2 (5 x+3)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{4246733 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^(5/2)),x]

[Out]

(-204595*Sqrt[1 - 2*x])/(168*(3 + 5*x)^(3/2)) + (7*Sqrt[1 - 2*x])/(9*(2 + 3*x)^3*(3 + 5*x)^(3/2)) + (301*Sqrt[
1 - 2*x])/(36*(2 + 3*x)^2*(3 + 5*x)^(3/2)) + (24469*Sqrt[1 - 2*x])/(168*(2 + 3*x)*(3 + 5*x)^(3/2)) + (618645*S
qrt[1 - 2*x])/(56*Sqrt[3 + 5*x]) - (4246733*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[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)^4 (3+5 x)^{5/2}} \, dx &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{1}{9} \int \frac{\frac{345}{2}-268 x}{\sqrt{1-2 x} (2+3 x)^3 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{1}{126} \int \frac{\frac{87003}{4}-31605 x}{\sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2}} \, dx\\ &=\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac{1}{882} \int \frac{\frac{16024491}{8}-2569245 x}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2}} \, dx\\ &=-\frac{204595 \sqrt{1-2 x}}{168 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}-\frac{\int \frac{\frac{1808711289}{16}-\frac{425353005 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{14553}\\ &=-\frac{204595 \sqrt{1-2 x}}{168 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac{618645 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}+\frac{2 \int \frac{97118536977}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{160083}\\ &=-\frac{204595 \sqrt{1-2 x}}{168 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac{618645 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}+\frac{4246733}{112} \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx\\ &=-\frac{204595 \sqrt{1-2 x}}{168 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac{618645 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}+\frac{4246733}{56} \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )\\ &=-\frac{204595 \sqrt{1-2 x}}{168 (3+5 x)^{3/2}}+\frac{7 \sqrt{1-2 x}}{9 (2+3 x)^3 (3+5 x)^{3/2}}+\frac{301 \sqrt{1-2 x}}{36 (2+3 x)^2 (3+5 x)^{3/2}}+\frac{24469 \sqrt{1-2 x}}{168 (2+3 x) (3+5 x)^{3/2}}+\frac{618645 \sqrt{1-2 x}}{56 \sqrt{3+5 x}}-\frac{4246733 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{56 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0728577, size = 84, normalized size = 0.51 \[ \frac{\sqrt{1-2 x} \left (250551225 x^4+645909120 x^3+623901861 x^2+267610802 x+43006496\right )}{168 (3 x+2)^3 (5 x+3)^{3/2}}-\frac{4246733 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{56 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(3/2)/((2 + 3*x)^4*(3 + 5*x)^(5/2)),x]

[Out]

(Sqrt[1 - 2*x]*(43006496 + 267610802*x + 623901861*x^2 + 645909120*x^3 + 250551225*x^4))/(168*(2 + 3*x)^3*(3 +
5*x)^(3/2)) - (4246733*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(56*Sqrt[7])

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Maple [B]  time = 0.014, size = 298, normalized size = 1.8 \begin{align*}{\frac{1}{2352\, \left ( 2+3\,x \right ) ^{3}} \left ( 8599634325\,\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+8734626054\,{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}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x)

[Out]

1/2352*(8599634325*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+27518829840*7^(1/2)*arctan(1
/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+35201169837*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)
^(1/2))*x^3+3507717150*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+9042727680*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+8734626054*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)^3/(-10*x^2-x+3)^(1/2)
/(3+5*x)^(3/2)

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Maxima [A]  time = 2.56429, size = 324, normalized size = 1.95 \begin{align*} \frac{4246733}{784} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) - \frac{618645 \, x}{28 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1937773}{168 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{199895 \, x}{36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} + \frac{343}{81 \,{\left (27 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} + 54 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 36 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 8 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{4655}{108 \,{\left (9 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} + 12 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 4 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} + \frac{165739}{216 \,{\left (3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + 2 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}\right )}} - \frac{1943461}{648 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

4246733/784*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 618645/28*x/sqrt(-10*x^2 - x + 3) + 19
37773/168/sqrt(-10*x^2 - x + 3) + 199895/36*x/(-10*x^2 - x + 3)^(3/2) + 343/81/(27*(-10*x^2 - x + 3)^(3/2)*x^3
+ 54*(-10*x^2 - x + 3)^(3/2)*x^2 + 36*(-10*x^2 - x + 3)^(3/2)*x + 8*(-10*x^2 - x + 3)^(3/2)) + 4655/108/(9*(-
10*x^2 - x + 3)^(3/2)*x^2 + 12*(-10*x^2 - x + 3)^(3/2)*x + 4*(-10*x^2 - x + 3)^(3/2)) + 165739/216/(3*(-10*x^2
- x + 3)^(3/2)*x + 2*(-10*x^2 - x + 3)^(3/2)) - 1943461/648/(-10*x^2 - x + 3)^(3/2)

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Fricas [A]  time = 1.56328, size = 440, normalized size = 2.65 \begin{align*} -\frac{12740199 \, \sqrt{7}{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\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 (250551225 \, x^{4} + 645909120 \, x^{3} + 623901861 \, x^{2} + 267610802 \, x + 43006496\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{2352 \,{\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

-1/2352*(12740199*sqrt(7)*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*arctan(1/14*sqrt(7)*(37*x +
20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(250551225*x^4 + 645909120*x^3 + 623901861*x^2 + 26761
0802*x + 43006496)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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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)**4/(3+5*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 2.71504, size = 591, normalized size = 3.56 \begin{align*} -\frac{5}{48} \, \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} + \frac{4246733}{7840} \, \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 )} + 335 \, \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{99 \,{\left (21713 \, \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} + 10391360 \, \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} + 1283172800 \, \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 )}}{28 \,{\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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

-5/48*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 + 4246733/7840*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) + 335*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))) + 99/28*(21713*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 + 10391360*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22)))^3 + 1283172800*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)^3