\(\int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx\) [2285]

Optimal result

Integrand size = 26, antiderivative size = 209 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}-\frac {64645339 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \]

[Out]

-64645339/8605184*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-1/18*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+
3*x)^6-107/3780*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^5+4619/211680*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^4+42461/
423360*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^3+1460201/2370816*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^2+152571047/3
3191424*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {99, 154, 156, 12, 95, 210} \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {64645339 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{1229312 \sqrt {7}}-\frac {\sqrt {1-2 x} (5 x+3)^{3/2}}{18 (3 x+2)^6}+\frac {152571047 \sqrt {1-2 x} \sqrt {5 x+3}}{33191424 (3 x+2)}+\frac {1460201 \sqrt {1-2 x} \sqrt {5 x+3}}{2370816 (3 x+2)^2}+\frac {42461 \sqrt {1-2 x} \sqrt {5 x+3}}{423360 (3 x+2)^3}+\frac {4619 \sqrt {1-2 x} \sqrt {5 x+3}}{211680 (3 x+2)^4}-\frac {107 \sqrt {1-2 x} \sqrt {5 x+3}}{3780 (3 x+2)^5} \]

[In]

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

[Out]

(-107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3780*(2 + 3*x)^5) + (4619*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(211680*(2 + 3*x)^4
) + (42461*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(423360*(2 + 3*x)^3) + (1460201*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2370816*
(2 + 3*x)^2) + (152571047*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(33191424*(2 + 3*x)) - (Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/
(18*(2 + 3*x)^6) - (64645339*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1229312*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 99

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/(b*(m + 1))), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 154

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*((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 - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]

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 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*} \text {integral}& = -\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {1}{18} \int \frac {\left (\frac {9}{2}-20 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^6} \, dx \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {-\frac {2087}{4}-1360 x}{\sqrt {1-2 x} (2+3 x)^5 \sqrt {3+5 x}} \, dx}{1890} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {112467}{8}-\frac {69285 x}{2}}{\sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x}} \, dx}{52920} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {27328875}{16}-\frac {4458405 x}{2}}{\sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}} \, dx}{1111320} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {\frac {3295705245}{32}-\frac {766605525 x}{8}}{\sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}} \, dx}{15558480} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {\int \frac {183269536065}{64 \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{108909360} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {64645339 \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx}{2458624} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}+\frac {64645339 \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )}{1229312} \\ & = -\frac {107 \sqrt {1-2 x} \sqrt {3+5 x}}{3780 (2+3 x)^5}+\frac {4619 \sqrt {1-2 x} \sqrt {3+5 x}}{211680 (2+3 x)^4}+\frac {42461 \sqrt {1-2 x} \sqrt {3+5 x}}{423360 (2+3 x)^3}+\frac {1460201 \sqrt {1-2 x} \sqrt {3+5 x}}{2370816 (2+3 x)^2}+\frac {152571047 \sqrt {1-2 x} \sqrt {3+5 x}}{33191424 (2+3 x)}-\frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{18 (2+3 x)^6}-\frac {64645339 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{1229312 \sqrt {7}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.44 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {1331 \left (\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (2906375616+21497808880 x+63585046048 x^2+94045700016 x^3+69576897780 x^4+20597091345 x^5\right )}{1331 (2+3 x)^6}-728535 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )\right )}{129077760} \]

[In]

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

[Out]

(1331*((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2906375616 + 21497808880*x + 63585046048*x^2 + 94045700016*x^3 + 695768
97780*x^4 + 20597091345*x^5))/(1331*(2 + 3*x)^6) - 728535*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])
]))/129077760

Maple [A] (verified)

Time = 3.62 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (20597091345 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{18439680 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {64645339 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{17210368 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(139\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (706896781965 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{258155520 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\)	\(346\)

[In]

int((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^7,x,method=_RETURNVERBOSE)

[Out]

-1/18439680*(-1+2*x)*(3+5*x)^(1/2)*(20597091345*x^5+69576897780*x^4+94045700016*x^3+63585046048*x^2+2149780888
0*x+2906375616)/(2+3*x)^6/(-(-1+2*x)*(3+5*x))^(1/2)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)[phone]/17210368*7^
(1/2)*arctan(9/14*(20/3+37/3*x)*7^(1/2)/(-90*(2/3+x)^2+67+111*x)^(1/2))*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/
(3+5*x)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=-\frac {969680085 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (20597091345 \, x^{5} + 69576897780 \, x^{4} + 94045700016 \, x^{3} + 63585046048 \, x^{2} + 21497808880 \, x + 2906375616\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{258155520 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[In]

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

[Out]

-1/258155520*(969680085*sqrt(7)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*arctan(1/14
*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(20597091345*x^5 + 69576897780*x^4 +
94045700016*x^3 + 63585046048*x^2 + 21497808880*x + 2906375616)*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)

Sympy [F]

\[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1 - 2 x} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{7}}\, dx \]

[In]

integrate((3+5*x)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**7,x)

[Out]

Integral(sqrt(1 - 2*x)*(5*x + 3)**(3/2)/(3*x + 2)**7, x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {64645339}{17210368} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) + \frac {2671295}{921984} \, \sqrt {-10 \, x^{2} - x + 3} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{42 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {29 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{980 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {1273 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{7840 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {45245 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{65856 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {1602777 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac {19767583 \, \sqrt {-10 \, x^{2} - x + 3}}{3687936 \, {\left (3 \, x + 2\right )}} \]

[In]

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

[Out]

64645339/17210368*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 2671295/921984*sqrt(-10*x^2 - x
+ 3) - 1/42*(-10*x^2 - x + 3)^(3/2)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64) + 29/98
0*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 1273/7840*(-10*x^2 - x + 3)^
(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 45245/65856*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x +
8) + 1602777/614656*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 19767583/3687936*sqrt(-10*x^2 - x + 3)/(3*x
+ 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (164) = 328\).

Time = 0.67 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\frac {64645339}{172103680} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {1331 \, \sqrt {10} {\left (145707 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 231188440 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} - 144245619840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 41365512115200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 5067855403520000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {250767109017600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {1003068436070400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{1843968 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]

[In]

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

[Out]

64645339/172103680*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) -
sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 1331/1843968*sqrt(10)*(145707*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^11 + 231188440
*((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
- 144245619840*((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 - 41365512115200*((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 - 5067855403520000*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sq
rt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 250767109017600000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/
sqrt(5*x + 3) + 1003068436070400000*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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^7} \, dx=\int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^7} \,d x \]

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(3/2))/(3*x + 2)^7,x)

[Out]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(3/2))/(3*x + 2)^7, x)