\(\int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx\) [534]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {250}{81} \sqrt {1-2 x}+\frac {\sqrt {1-2 x}}{243 (2+3 x)^3}-\frac {158 \sqrt {1-2 x}}{1701 (2+3 x)^2}+\frac {3727 \sqrt {1-2 x}}{3969 (2+3 x)}-\frac {92996 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{3969 \sqrt {21}} \] Output:

250/81*(1-2*x)^(1/2)+1/243*(1-2*x)^(1/2)/(2+3*x)^3-158/1701*(1-2*x)^(1/2)/ 
(2+3*x)^2+3727*(1-2*x)^(1/2)/(7938+11907*x)-92996/83349*21^(1/2)*arctanh(1 
/7*21^(1/2)*(1-2*x)^(1/2))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (112187+484618 x+695043 x^2+330750 x^3\right )}{(2+3 x)^3}-92996 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{83349} \] Input:

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

Output:

((21*Sqrt[1 - 2*x]*(112187 + 484618*x + 695043*x^2 + 330750*x^3))/(2 + 3*x 
)^3 - 92996*Sqrt[21]*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/83349
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 166, 27, 163, 73, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {1}{9} \int \frac {(12-35 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^3}dx-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 166

\(\displaystyle \frac {1}{9} \left (\frac {1}{42} \int \frac {4 (136-745 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \int \frac {(136-745 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)^2}dx-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 163

\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \left (\frac {23249}{21} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx+\frac {\sqrt {1-2 x} (26075 x+18016)}{21 (3 x+2)}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \left (\frac {\sqrt {1-2 x} (26075 x+18016)}{21 (3 x+2)}-\frac {23249}{21} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{9} \left (\frac {2}{21} \left (\frac {\sqrt {1-2 x} (26075 x+18016)}{21 (3 x+2)}-\frac {46498 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{21 \sqrt {21}}\right )-\frac {53 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)^2}\right )-\frac {\sqrt {1-2 x} (5 x+3)^3}{9 (3 x+2)^3}\)

Input:

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

Output:

-1/9*(Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2 + 3*x)^3 + ((-53*Sqrt[1 - 2*x]*(3 + 5* 
x)^2)/(21*(2 + 3*x)^2) + (2*((Sqrt[1 - 2*x]*(18016 + 26075*x))/(21*(2 + 3* 
x)) - (46498*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(21*Sqrt[21])))/21)/9
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 108

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) 
, x] - Simp[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 163

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n 
+ 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* 
(m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + 
d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f 
*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* 
d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* 
d*(b*c - a*d)*(m + 1)*(m + n + 3))   Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 
1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]
 

rule 166

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> 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] - Simp[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] /; FreeQ[{a, b, c, d, 
e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.55

method	result	size
risch	\(-\frac {661500 x^{4}+1059336 x^{3}+274193 x^{2}-260244 x -112187}{3969 \left (2+3 x \right )^{3} \sqrt {1-2 x}}-\frac {92996 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{83349}\)	\(56\)
pseudoelliptic	\(\frac {-92996 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{3} \sqrt {21}+21 \sqrt {1-2 x}\, \left (330750 x^{3}+695043 x^{2}+484618 x +112187\right )}{83349 \left (2+3 x \right )^{3}}\)	\(60\)
derivativedivides	\(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{83349}\)	\(66\)
default	\(\frac {250 \sqrt {1-2 x}}{81}+\frac {-\frac {7454 \left (1-2 x \right )^{\frac {5}{2}}}{441}+\frac {44092 \left (1-2 x \right )^{\frac {3}{2}}}{567}-\frac {7246 \sqrt {1-2 x}}{81}}{\left (-4-6 x \right )^{3}}-\frac {92996 \sqrt {21}\, \operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right )}{83349}\)	\(66\)
trager	\(\frac {\left (330750 x^{3}+695043 x^{2}+484618 x +112187\right ) \sqrt {1-2 x}}{3969 \left (2+3 x \right )^{3}}-\frac {46498 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x +21 \sqrt {1-2 x}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )}{2+3 x}\right )}{83349}\)	\(77\)

Input:

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

Output:

-1/3969*(661500*x^4+1059336*x^3+274193*x^2-260244*x-112187)/(2+3*x)^3/(1-2 
*x)^(1/2)-92996/83349*21^(1/2)*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.88 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498 \, \sqrt {21} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (\frac {3 \, x + \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (330750 \, x^{3} + 695043 \, x^{2} + 484618 \, x + 112187\right )} \sqrt {-2 \, x + 1}}{83349 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:

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

Output:

1/83349*(46498*sqrt(21)*(27*x^3 + 54*x^2 + 36*x + 8)*log((3*x + sqrt(21)*s 
qrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(330750*x^3 + 695043*x^2 + 484618*x + 1 
12187)*sqrt(-2*x + 1))/(27*x^3 + 54*x^2 + 36*x + 8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498}{83349} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {2 \, {\left (33543 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}\right )}}{3969 \, {\left (27 \, {\left (2 \, x - 1\right )}^{3} + 189 \, {\left (2 \, x - 1\right )}^{2} + 882 \, x - 98\right )}} \] Input:

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

Output:

46498/83349*sqrt(21)*log(-(sqrt(21) - 3*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt 
(-2*x + 1))) + 250/81*sqrt(-2*x + 1) + 2/3969*(33543*(-2*x + 1)^(5/2) - 15 
4322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(27*(2*x - 1)^3 + 189*(2*x 
- 1)^2 + 882*x - 98)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {46498}{83349} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) + \frac {250}{81} \, \sqrt {-2 \, x + 1} + \frac {33543 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 154322 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 177527 \, \sqrt {-2 \, x + 1}}{15876 \, {\left (3 \, x + 2\right )}^{3}} \] Input:

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

Output:

46498/83349*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) 
 + 3*sqrt(-2*x + 1))) + 250/81*sqrt(-2*x + 1) + 1/15876*(33543*(2*x - 1)^2 
*sqrt(-2*x + 1) - 154322*(-2*x + 1)^(3/2) + 177527*sqrt(-2*x + 1))/(3*x + 
2)^3
 

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {250\,\sqrt {1-2\,x}}{81}-\frac {92996\,\sqrt {21}\,\mathrm {atanh}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}}{7}\right )}{83349}+\frac {\frac {7246\,\sqrt {1-2\,x}}{2187}-\frac {44092\,{\left (1-2\,x\right )}^{3/2}}{15309}+\frac {7454\,{\left (1-2\,x\right )}^{5/2}}{11907}}{\frac {98\,x}{3}+7\,{\left (2\,x-1\right )}^2+{\left (2\,x-1\right )}^3-\frac {98}{27}} \] Input:

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

Output:

(250*(1 - 2*x)^(1/2))/81 - (92996*21^(1/2)*atanh((21^(1/2)*(1 - 2*x)^(1/2) 
)/7))/83349 + ((7246*(1 - 2*x)^(1/2))/2187 - (44092*(1 - 2*x)^(3/2))/15309 
 + (7454*(1 - 2*x)^(5/2))/11907)/((98*x)/3 + 7*(2*x - 1)^2 + (2*x - 1)^3 - 
 98/27)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.06 \[ \int \frac {\sqrt {1-2 x} (3+5 x)^3}{(2+3 x)^4} \, dx=\frac {6945750 \sqrt {-2 x +1}\, x^{3}[phone] \sqrt {-2 x +1}\, x^{2}[phone] \sqrt {-2 x +1}\, x +2355927 \sqrt {-2 x +1}+1255446 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{3}+2510892 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x^{2}+1673928 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right ) x +371984 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}-\sqrt {21}\right )-1255446 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{3}-2510892 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x^{2}-1673928 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right ) x -371984 \sqrt {21}\, \mathrm {log}\left (3 \sqrt {-2 x +1}+\sqrt {21}\right )}{2250423 x^{3}+4500846 x^{2}+3000564 x +666792} \] Input:

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

Output:

(6945750*sqrt( - 2*x + 1)*x**3 + 14595903*sqrt( - 2*x + 1)*x**2 + 10176978 
*sqrt( - 2*x + 1)*x + 2355927*sqrt( - 2*x + 1) + 1255446*sqrt(21)*log(3*sq 
rt( - 2*x + 1) - sqrt(21))*x**3 + 2510892*sqrt(21)*log(3*sqrt( - 2*x + 1) 
- sqrt(21))*x**2 + 1673928*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21))*x + 
 371984*sqrt(21)*log(3*sqrt( - 2*x + 1) - sqrt(21)) - 1255446*sqrt(21)*log 
(3*sqrt( - 2*x + 1) + sqrt(21))*x**3 - 2510892*sqrt(21)*log(3*sqrt( - 2*x 
+ 1) + sqrt(21))*x**2 - 1673928*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21) 
)*x - 371984*sqrt(21)*log(3*sqrt( - 2*x + 1) + sqrt(21)))/(83349*(27*x**3 
+ 54*x**2 + 36*x + 8))