\(\int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx\) [2359]

Optimal result

Integrand size = 26, antiderivative size = 209 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {395307 \sqrt {1-2 x} \sqrt {3+5 x}}{175616 (2+3 x)}-\frac {11979 \sqrt {1-2 x} (3+5 x)^{3/2}}{12544 (2+3 x)^2}-\frac {1089 \sqrt {1-2 x} (3+5 x)^{5/2}}{2240 (2+3 x)^3}+\frac {(1-2 x)^{5/2} (3+5 x)^{7/2}}{14 (2+3 x)^6}+\frac {9 (1-2 x)^{3/2} (3+5 x)^{7/2}}{20 (2+3 x)^5}+\frac {297 \sqrt {1-2 x} (3+5 x)^{7/2}}{160 (2+3 x)^4}-\frac {4348377 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{175616 \sqrt {7}} \]

[Out]

1/14*(1-2*x)^(5/2)*(3+5*x)^(7/2)/(2+3*x)^6+9/20*(1-2*x)^(3/2)*(3+5*x)^(7/2)/(2+3*x)^5-4348377/1229312*arctan(1
/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)-11979/12544*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-1089/2240*(3
+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^3+297/160*(3+5*x)^(7/2)*(1-2*x)^(1/2)/(2+3*x)^4-395307/175616*(1-2*x)^(1/2)*
(3+5*x)^(1/2)/(2+3*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {4348377 \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{175616 \sqrt {7}}+\frac {297 \sqrt {1-2 x} (5 x+3)^{7/2}}{160 (3 x+2)^4}+\frac {9 (1-2 x)^{3/2} (5 x+3)^{7/2}}{20 (3 x+2)^5}+\frac {(1-2 x)^{5/2} (5 x+3)^{7/2}}{14 (3 x+2)^6}-\frac {1089 \sqrt {1-2 x} (5 x+3)^{5/2}}{2240 (3 x+2)^3}-\frac {11979 \sqrt {1-2 x} (5 x+3)^{3/2}}{12544 (3 x+2)^2}-\frac {395307 \sqrt {1-2 x} \sqrt {5 x+3}}{175616 (3 x+2)} \]

[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 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 96

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[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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 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 {(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 \text {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] (verified)

Time = 0.36 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.43 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} \left (64829376+482263920 x+1428134688 x^2+2108117296 x^3+1555340180 x^4+460633945 x^5\right )}{(2+3 x)^6}-21741885 \sqrt {7} \arctan \left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{6146560} \]

[In]

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

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(64829376 + 482263920*x + 1428134688*x^2 + 2108117296*x^3 + 1555340180*x^4 + 4
60633945*x^5))/(2 + 3*x)^6 - 21741885*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/6146560

Maple [A] (verified)

Time = 1.15 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.67

method	result	size
risch	\(-\frac {\left (-1+2 x \right ) \sqrt {3+5 x}\, \left (460633945 x^{5}[phone] x^{4}[phone] x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{878080 \left (2+3 x \right )^{6} \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right )}\, \sqrt {1-2 x}}+\frac {4348377 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2458624 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(139\)
default	\(\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (15849834165 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{6}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{5}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{4}[phone] x^{5} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{3}[phone] x^{4} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}[phone] x^{3} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x [phone] x^{2} \sqrt {-10 x^{2}-x +3}[phone] \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )[phone] x \sqrt {-10 x^{2}-x +3}[phone] \sqrt {-10 x^{2}-x +3}\right )}{12293120 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{6}}\)	\(346\)

[In]

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

[Out]

-1/878080*(-1+2*x)*(3+5*x)^(1/2)*(460633945*x^5+1555340180*x^4+2108117296*x^3+1428134688*x^2+482263920*x+64829
376)/(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)+4348377/2458624*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 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=-\frac {21741885 \, \sqrt {7} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (460633945 \, x^{5} + 1555340180 \, x^{4} + 2108117296 \, x^{3} + 1428134688 \, x^{2} + 482263920 \, x + 64829376\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{12293120 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.31 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {272085}{307328} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {{\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{42 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} + \frac {23 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{420 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac {297 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{1568 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac {10989 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{21952 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac {489753 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {5}{2}}}{614656 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {6648345}{614656} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {4348377}{2458624} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {5857731}{1229312} \, \sqrt {-10 \, x^{2} - x + 3} + \frac {645909 \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{1229312 \, {\left (3 \, x + 2\right )}} \]

[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)

Giac [B] (verification not implemented)

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

Time = 0.77 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.32 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\frac {4348377}{24586240} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {161051 \, \sqrt {10} {\left (27 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{11} + 42840 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{9} + 27941760 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{7} - 6539187200 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{5} - 940423680000 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {46467993600000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {185871974400000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{87808 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{6}} \]

[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*sqrt(10)*(27*((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(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 + 2794176
0*((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(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(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*s
qrt(-10*x + 5) - sqrt(22)))^3 - 46467993600000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 1858719744
00000*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. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^7} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^7} \,d x \]

[In]

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

[Out]

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