3.24.87 \(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^4 (3+5 x)^{5/2}} \, dx\) [2387]

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

[Out]

-4246733/392*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-204595/168*(1-2*x)^(1/2)/(3+5*x)^(3/2)+7/
9*(1-2*x)^(1/2)/(2+3*x)^3/(3+5*x)^(3/2)+301/36*(1-2*x)^(1/2)/(2+3*x)^2/(3+5*x)^(3/2)+24469/168*(1-2*x)^(1/2)/(
2+3*x)/(3+5*x)^(3/2)+618645/56*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 166, normalized size of antiderivative = 1.00, 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 = {100, 156, 157, 12, 95, 210} \begin {gather*} -\frac {4246733 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{56 \sqrt {7}}+\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}} \end {gather*}

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 12

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

Rule 95

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 100

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 156

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] && ILtQ[m, -1]

Rule 157

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 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} \text {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*}

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Mathematica [A]
time = 0.26, size = 84, normalized size = 0.51 \begin {gather*} \frac {\sqrt {1-2 x} \left (43006496+267610802 x+623901861 x^2+645909120 x^3+250551225 x^4\right )}{168 (2+3 x)^3 (3+5 x)^{3/2}}-\frac {4246733 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{56 \sqrt {7}} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(127)=254\).
time = 0.11, size = 298, normalized size = 1.80

method	result	size
default	\(\frac {\left (8599634325 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right ) \sqrt {1-2 x}}{2352 \left (2+3 x \right )^{3} \sqrt {-10 x^{2}-x +3}\, \left (3+5 x \right )^{\frac {3}{2}}}\)	\(298\)

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,method=_RETURNVERBOSE)

[Out]

1/2352*(8599634325*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(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 = 0.52, size = 240, normalized size = 1.45 \begin {gather*} \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 {gather*}

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 = 0.86, size = 131, normalized size = 0.79 \begin {gather*} -\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 {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (127) = 254\).
time = 0.74, size = 434, normalized size = 2.61 \begin {gather*} \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 )} - \frac {5}{48} \, \sqrt {10} {\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 )}^{3} - \frac {3216 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {12864 \, \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 {gather*}

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]

4246733/7840*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 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 - 3216*(sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))/sqrt(5*x + 3) + 12864*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(-10*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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^4\,{\left (5\,x+3\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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